Conference Publications


Dynamical Systems and Differential Equations, AIMS Proceedings 2015, Proceedings of the 10th AIMS International Conference (Madrid, Spain)

Editors: Manuel de León, Wei Feng, Zhaosheng Feng, Julian Lopez Gomez, Xin Lu, J.M. Martell, Javier Parcet, Daniel Peralta-Salas and Weihua Ruan

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Cover page and Preface
Shouchuan Hu and Xin Lu
2015(special): i-i doi: 10.3934/ +[Abstract](2505) +[PDF](50.4KB)
The Tenth AIMS International Conference on Dynamical Systems, Di eren- tial Equations and Applications took place in the magni cent Madrid, Spain, July 7 - 11, 2014. The present volume is the Proceedings, consisting of some carefully selected submissions after a rigorous refereeing process.

For more information please click the “Full Text” above.
Thermopower of a graphene monolayer with inhomogeneous spin-orbit interaction
M. I. Alomar and David Sánchez
2015(special): 1-9 doi: 10.3934/proc.2015.0001 +[Abstract](2856) +[PDF](429.6KB)
We consider a single layer of graphene with a Rashba spin-orbit coupling localized in the central region. Generally, a spin-orbit interaction induces a spin splitting and modifies the band structure of graphene, opening a gap between the two sublattices. We investigate the transport properties within the scattering approach and calculate the linear electric and thermoelectric conductances. We observe a weak dependence of the electric conductance with both the length of the spin-orbit region and the Rashba strength. Strikingly, the thermoelectric conductance is much more sensitive to variations of these two parameters. Our results are relevant in view of recent developments that emphasize thermoelectric effects in graphene.
Existence and multiplicity of stationary solutions for a Cahn--Hilliard-type equation in $\mathbb{R}^N$
Pablo Álvarez-Caudevilla
2015(special): 10-18 doi: 10.3934/proc.2015.0010 +[Abstract](3090) +[PDF](456.5KB)
Solutions of the stationary semilinear Cahn--Hilliard-type equation $$ -\Delta^2 u - u -\Delta(|u|^{p-1}u)=0 \quad \mbox{in}     \mathbb{R}^N, \quad \mbox{with} \quad p>1, $$ which are exponentially decaying at infinity, are studied. Using the Mounting Pass Theorem allows us the determination of two different solutions. On the other hand, the application of Lusternik--Schnirel'man (L--S) Category Theory shows the existence of, at least, a countable family of solutions.
Noncommutative bi-symplectic $\mathbb{N}Q$-algebras of weight 1
Luis Álvarez–cónsul and David Fernández
2015(special): 19-28 doi: 10.3934/proc.2015.0019 +[Abstract](3559) +[PDF](290.9KB)
It is well known that symplectic $\mathbb{N}Q$-manifolds of weight 1 are in 1-1 correspondence with Poisson manifolds. In this article, we prove a version of this correspondence in the framework of noncommutative algebraic geometry based on double derivations, as introduced by W. Crawley-Boevey, P. Etingof and V. Ginzburg. More precisely, we define noncommutative bi-symplectic $\mathbb{N}Q$-algebras and prove that bi-symplectic $\mathbb{N}Q$-algebras of weight 1 are in 1-1 correspondence with double Poisson algebras, as previously defined by M. Van den Bergh.
A nonlinear generalization of the Camassa-Holm equation with peakon solutions
Stephen C. Anco, Elena Recio, María L. Gandarias and María S. Bruzón
2015(special): 29-37 doi: 10.3934/proc.2015.0029 +[Abstract](3660) +[PDF](350.0KB)
A nonlinearly generalized Camassa-Holm equation, depending an arbitrary nonlinearity power $p \neq 0$, is considered. This equation reduces to the Camassa-Holm equation when $p=1$ and shares one of the Hamiltonian structures of the Camassa-Holm equation. Two main results are obtained. A classification of point symmetries is presented and a peakon solution is derived, for all powers $p \neq 0$.
Hartman-type conditions for multivalued Dirichlet problem in abstract spaces
Jan Andres, Luisa Malaguti and Martina Pavlačková
2015(special): 38-55 doi: 10.3934/proc.2015.0038 +[Abstract](3037) +[PDF](461.5KB)
The classical Hartman's Theorem in [18] for the solvability of the vector Dirichlet problem will be generalized and extended in several directions. We will consider its multivalued versions for Marchaud and upper-Carathéodory right-hand sides with only certain amount of compactness in Banach spaces. Advanced topological methods are combined with a bound sets technique. Besides the existence, the localization of solutions can be obtained in this way.
Subexponential growth rates in functional differential equations
John A. D. Appleby and Denis D. Patterson
2015(special): 56-65 doi: 10.3934/proc.2015.0056 +[Abstract](3274) +[PDF](310.1KB)
This paper determines the rate of growth to infinity of a scalar autonomous nonlinear functional differential equation with finite delay, where the right hand side is a positive continuous linear functional of $f(x)$. We assume $f$ grows sublinearly, and is such that solutions should exhibit growth faster than polynomial, but slower than exponential. Under some technical conditions on $f$, it is shown that the solution of the functional differential equation is asymptotic to that of an auxiliary autonomous ordinary differential equation with righthand side proportional to $f$ (with the constant of proportionality equal to the mass of the finite measure associated with the linear functional), provided $f$ grows more slowly than $l(x)=x/\log x$. This linear--logarithmic growth rate is also shown to be critical: if $f$ grows more rapidly than $l$, the ODE dominates the FDE; if $f$ is asymptotic to a constant multiple of $l$, the FDE and ODE grow at the same rate, modulo a constant non--unit factor.
Stabilization of a hyperbolic/elliptic system modelling the viscoelastic-gravitational deformation in a multilayered Earth
Alicia Arjona and Jesús Ildefonso Díaz
2015(special): 66-74 doi: 10.3934/proc.2015.0066 +[Abstract](2638) +[PDF](351.5KB)
In the last 30 years several mathematical studies have been devoted to the viscoelastic-gravitational coupling in stationary and transient regimes either for static case or for hyperbolic case. However, to the best of our knowledge there is a lack of mathematical study of the stabilization as $t$ goes to infinity of a viscoelastic-gravitational models crustal deformations of multilayered Earth. Here we prove that, under some additional conditions on the data, the difference of the viscoelastic and elastic solutions converges to zero, as $t$ goes to infinity, in a suitable functional space. The proof of that uses a reformulation of the hyperbolic/elliptic system in terms of a nonlocal hyperbolic system.
Vectorized and parallel particle filter SMC parameter estimation for stiff ODEs
Andrea Arnold, Daniela Calvetti and Erkki Somersalo
2015(special): 75-84 doi: 10.3934/proc.2015.0075 +[Abstract](3617) +[PDF](383.5KB)
Particle filter (PF) sequential Monte Carlo (SMC) methods are very attractive for estimating parameters of time-dependent systems where the data is either not all available at once, or the range of time constants is wide enough to create problems in the numerical time propagation of the states. The need to evolve (and hence integrate) a large number of particles makes PF-based methods computationally challenging, and parallelization is often advocated to speed up computing time. While careful parallelization may indeed improve performance, vectorization of the algorithm on a single processor may result in even larger speedups for certain problems. In this paper we demonstrate how the PF-SMC class of algorithms proposed in [2] can be implemented in both parallel and vectorized computing environments, illustrating the performance with computed examples in MATLAB. In particular, two stiff test problems with different features show that both the size and structure of the problem affect which version of the algorithm is more efficient.
Canard-type solutions in epidemiological models
Jacek Banasiak and Eddy Kimba Phongi
2015(special): 85-93 doi: 10.3934/proc.2015.0085 +[Abstract](2849) +[PDF](436.9KB)
The paper concerns an epidemiological model with an age structure and with two time scales: the slow one refers to the demographical processes and the fast describes the dynamics of a fast disease such as flu or common cold. The model in the singular limit corresponding to the infinite disease-related rates has two intersecting quasi-stationary steady states. We investigate the asymptotic behaviour of solutions passing close to the intersection manifold and show that in the models with increasing total populations there is a delay in switching between the quasi-stationary states which resembles the so-called canard solutions.
Infinitely many solutions for a perturbed Schrödinger equation
Rossella Bartolo, Anna Maria Candela and Addolorata Salvatore
2015(special): 94-102 doi: 10.3934/proc.2015.0094 +[Abstract](3109) +[PDF](341.6KB)
We find multiple solutions for a nonlinear perturbed Schrödinger equation by means of the so--called Bolle's method.
Nonlocal problems in Hilbert spaces
Irene Benedetti, Luisa Malaguti and Valentina Taddei
2015(special): 103-111 doi: 10.3934/proc.2015.0103 +[Abstract](3073) +[PDF](318.4KB)
An existence result for differential inclusions in a separable Hilbert space is furnished. A wide family of nonlocal boundary value problems is treated, including periodic, anti-periodic, mean value and multipoint conditions. The study is based on an approximation solvability method. Advanced topological methods are used as well as a Scorza Dragoni-type result for multivalued maps. The conclusions are original also in the single-valued setting. An application to a nonlocal dispersal model is given.
The role of aerodynamic forces in a mathematical model for suspension bridges
Elvise Berchio and Filippo Gazzola
2015(special): 112-121 doi: 10.3934/proc.2015.0112 +[Abstract](2568) +[PDF](653.4KB)
In a fish-bone model for suspension bridges previously studied by us in [3] we introduce linear aerodynamic forces. We numerically analyze the role of these forces and we theoretically show that they do not influence the onset of torsional oscillations. This suggests a new explanation for the origin of instability in suspension bridges: it is a combined interaction between structural nonlinearity and aerodynamics and it follows a precise pattern.
Optimal control in a free boundary fluid-elasticity interaction
Lorena Bociu, Lucas Castle, Kristina Martin and Daniel Toundykov
2015(special): 122-131 doi: 10.3934/proc.2015.0122 +[Abstract](3333) +[PDF](359.5KB)
We establish existence of an optimal control for the problem of minimizing flow turbulence in the case of a nonlinear fluid-structure interaction model in the framework of the known local well-posedness theory. If the initial configuration is regular, in an appropriate sense, then a class of sufficiently smooth control inputs contains an element that minimizes, within the control class, the vorticity of the fluid flow around a moving and deforming elastic solid.
Analysis of the archetypal functional equation in the non-critical case
Leonid V. Bogachev, Gregory Derfel and Stanislav A. Molchanov
2015(special): 132-141 doi: 10.3934/proc.2015.0132 +[Abstract](2810) +[PDF](420.9KB)
We study the archetypal functional equation of the form $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(da,db)$ ($x\in\mathbb{R}$), where $\mu$ is a probability measure on $\mathbb{R}^2$; equivalently, $y(x)=\mathbb{E}\{y(\alpha(x-\beta))\}$, where $\mathbb{E}$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value $K:=\iint_{\mathbb{R}^2}\ln|a|\,\mu(da,db) =\mathbb{E}\{\ln|\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K<0$, whereas if $K>0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $\mathbb{P}(\alpha<0)>0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x ⇝ \alpha(x-\beta)$.
Some regularity results for a singular elliptic problem
Brahim Bougherara, Jacques Giacomoni and Jesus Hernández
2015(special): 142-150 doi: 10.3934/proc.2015.0142 +[Abstract](3153) +[PDF](362.0KB)
In the present paper we investigate the following singular elliptic problem with $p$-Laplacian operator: \begin{equation*} (P)\qquad \left \{ \begin{array}{l} -\Delta_p u = \frac{K(x)}{ u^{\alpha}}\quad \text{ in } \Omega \\ u = 0\ \text{ on } \partial\Omega,\ u>0 \text{ on } \Omega, \end{array} \right . \end{equation*} where $\Omega$ is a regular bounded domain of $\mathbb R^{N}$, $\alpha\in\mathbb R$, $K\in L^\infty_{\rm loc}(\Omega)$ a non-negative function. We discuss below the existence, the regularity and the uniqueness of a weak solution $u$ to the problem (P).
Classical and nonclassical symmetries and exact solutions for a generalized Benjamin equation
M. S. Bruzón, M. L. Gandarias and J. C. Camacho
2015(special): 151-158 doi: 10.3934/proc.2015.0151 +[Abstract](2617) +[PDF](621.6KB)
We apply the Lie-group formalism to deduce symmetries of a generalized Benjamin equation. We make an analysis of the symmetry reductions of the equation. In order to obtain travelling wave solutions we apply an indirect F-function method. We obtained in an unified way simultaneously many periodic wave solutions expressed by various single and combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions. We compare these solutions with the solutions derived by other authors by using different methods and we observe that we have obtained new solutions for this equation.
Stochastic control of individual's health investments
Christine Burggraf, Wilfried Grecksch and Thomas Glauben
2015(special): 159-168 doi: 10.3934/proc.2015.0159 +[Abstract](2926) +[PDF](460.3KB)
Grossman's health investment model has been one of the most important developments in health economics. However, the model's derived demand function for medical care predicts the demand for medical care to increase if the individual's health status increases. Yet, empirical studies indicate the opposite relationship. Therefore, this study improves the informative value of the health investment model by introducing a reworked Grossman model, which assumes a more realistic Cobb-Douglas health investment function with decreasing returns to scale. Because we introduced uncertainty surrounding individual's health status the resulting dynamic utility maximization problem is tackled by optimal stochastic control theory.
Matter-wave solitons with a minimal number of particles in a time-modulated quasi-periodic potential
Gennadiy Burlak and Salomon García-Paredes
2015(special): 169-175 doi: 10.3934/proc.2015.0169 +[Abstract](2786) +[PDF](524.0KB)
The two-dimensional (2D) matter-wave soliton families supported by an external potential are systematically studied, in a vicinity of the junction between stable and unstable branches of the families. In this case the norm of the solution attains a minimum, facilitating the creation of such excitation. We study the dynamics and stability boundaries for fundamental solitons in a 2D self-attracting Bose-Einstein condensate (BEC), trapped in an quasiperiodic optical lattice (OL), with the amplitude subject to periodic time modulation.
Similarity reductions of a nonlinear model for vibrations of beams
Jose Carlos Camacho and Maria de los Santos Bruzon
2015(special): 176-184 doi: 10.3934/proc.2015.0176 +[Abstract](2717) +[PDF](401.0KB)
In this paper we make a full analysis of the symmetry reductions of this equation by using the classical Lie method of infinitesimals. We consider travelling wave reductions depending on the constants. We present some reductions and explicit solutions.
Construction of highly stable implicit-explicit general linear methods
Angelamaria Cardone, Zdzisław Jackiewicz, Adrian Sandu and Hong Zhang
2015(special): 185-194 doi: 10.3934/proc.2015.0185 +[Abstract](3675) +[PDF](1157.0KB)
This paper deals with the numerical solution of systems of differential equations with a stiff part and a non-stiff one, typically arising from the semi-discretization of certain partial differential equations models. It is illustrated the construction and analysis of highly stable and high-stage order implicit-explicit (IMEX) methods based on diagonally implicit multistage integration methods (DIMSIMs), a subclass of general linear methods (GLMs). Some examples of methods with optimal stability properties are given. Finally numerical experiments confirm the theoretical expectations.
Stochastic modeling of the firing activity of coupled neurons periodically driven
Maria Francesca Carfora and Enrica Pirozzi
2015(special): 195-203 doi: 10.3934/proc.2015.0195 +[Abstract](3350) +[PDF](619.0KB)
A stochastic model for describing the firing activity of a couple of interacting neurons subject to time-dependent stimuli is proposed. Two stochastic differential equations suitably coupled and including periodic terms to represent stimuli imposed to one or both neurons are considered to describe the problem. We investigate the first passage time densities through specified firing thresholds for the involved time non-homogeneous Gauss-Markov processes. We provide simulation results and numerical approximations of the firing densities. Asymptotic behaviors of the first passage times are also given.
On the virial theorem for nonholonomic Lagrangian systems
José F. Cariñena, Irina Gheorghiu, Eduardo Martínez and Patrícia Santos
2015(special): 204-212 doi: 10.3934/proc.2015.0204 +[Abstract](3282) +[PDF](330.8KB)
A generalization of the virial theorem to nonholonomic Lagrangian systems is given. We will first establish the theorem in terms of Lagrange multipliers and later on in terms of the nonholonomic bracket.
Jacobi fields for second-order differential equations on Lie algebroids
José F. Cariñena, Irina Gheorghiu and Eduardo Martínez
2015(special): 213-222 doi: 10.3934/proc.2015.0213 +[Abstract](3246) +[PDF](329.2KB)
We generalize the concept of Jacobi field for general second-order differential equations on a manifold and on a Lie algebroid. The Jacobi equation is expressed in terms of the dynamical covariant derivative and the generalized Jacobi endomorphism associated to the given differential equation.
Complete recuperation after the blow up time for semilinear problems
Alfonso C. Casal, Jesús Ildefonso Díaz and José Manuel Vegas
2015(special): 223-229 doi: 10.3934/proc.2015.0223 +[Abstract](2535) +[PDF](348.6KB)
We consider explosive solutions $y^{0}(t)$, $t\in \lbrack 0,T_{y^{0}}),$ of some ordinary differential equations \begin{equation*} P(T_{y^{0}}): \begin{array}{lc} \frac{dy}{dt}(t)=f(y(t)),y(0)=y_{0}, & \end{array} \end{equation*} where $f:$ $\mathbb{R}^{d}\rightarrow \mathbb{R}^{d}$ is a locally Lipschitz superlinear function and $d\geq 1$. In this work we analyze the following question of controlability: given $\epsilon >0$, a continuous deformation $y(t)$ de $y^{0}(t)$, built as a solution of the perturbed control problem obtained by replacing $f(y(t))$ by $f(y(t))+u(t),$ for a suitable control $u$, such that $y(t)=y^{0}(t)$ for any $t\in \lbrack 0,T_{y^{0}}-\epsilon ]$ and such that $y(t)$ also blows up in $t=T_{y_{0}}$ but in such a way that $y(t)$ could be extended beyond $T_{y_{0}}$ as a function $y\in L_{loc}^{1}(0,+\infty :\mathbb{R}^{d})$?
Branches of positive solutions of subcritical elliptic equations in convex domains
Alfonso Castro and Rosa Pardo
2015(special): 230-238 doi: 10.3934/proc.2015.0230 +[Abstract](2944) +[PDF](337.5KB)
We provide sufficient conditions for the existence of $L^{\infty}$ a priori estimates for positive solutions to a class of subcritical elliptic equations in bounded $C^2$ convex domains. These sufficient conditions widen the range of nonlinearities for which a priori bounds are known. Using these a priori bounds we prove the existence of positive solutions for a class of problems depending on a parameter
Bridges between subriemannian geometry and algebraic geometry: Now and then
Alex L Castro, Wyatt Howard and Corey Shanbrom
2015(special): 239-247 doi: 10.3934/proc.2015.0239 +[Abstract](3199) +[PDF](330.4KB)
We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the field of algebraic geometry emerge here organically in an attempt to elucidate the geometric structures underlying a large class of nonholonomic distributions known as Goursat constraints. Among our new results is a regularization theorem for curves stated and proved using tools exclusively from nonholonomic geometry, and a computation of topological invariants that answer a question on the global topology of our classifying space. Last but not least we present for the first time some experimental results connecting the discrete invariants of nonholonomic plane fields such as the RVT code and the Milnor number of complex plane algebraic curves.
Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays
Parin Chaipunya and Poom Kumam
2015(special): 248-257 doi: 10.3934/proc.2015.0248 +[Abstract](3860) +[PDF](373.4KB)
We investigate the existence of fixed points for a very general class of cyclic implicit contractive set-valued operators. We also point out that this class contains an important case of ordered contractions. As an application, we show the solvability of delayed fractional integral inclusion problems.
Stability of interacting traveling waves in reaction-convection-diffusion systems
Grigori Chapiro, Lucas Furtado, Dan Marchesin and Stephen Schecter
2015(special): 258-266 doi: 10.3934/proc.2015.0258 +[Abstract](3668) +[PDF](331.3KB)
The stability of isolated combustion traveling waves has been exhaustively studied in the literature of reaction-diffusion systems. The analysis has been done mainly by neglecting other waves that are usually present in the solution and that can influence the stability of the combustion wave. In this paper, a numerical example on the influence of such interaction on wave stability are presented.
    The paper is illustrated through a simple model for the injection of air into a porous medium that contains a solid fuel. The model considered here reproduces a variety of observed phenomena and yet is simple enough to allow rigorous investigation. We refer on earlier work containing proofs of existence of traveling waves corresponding to combustion waves by phase plane analysis were presented; wave sequences that can occur as solutions of Riemann problems were identified.
Interaction of oscillatory packets of water waves
Martina Chirilus-Bruckner and Guido Schneider
2015(special): 267-275 doi: 10.3934/proc.2015.0267 +[Abstract](2848) +[PDF](599.6KB)
For surface gravity water waves we give a detailed analysis of the interaction of two NLS described wave packets with different carrier waves. We separate the internal dynamics of each wave packet from the dynamics caused by the interaction and prove the validity of a formula for the envelope shift caused by the interaction of the wave packets.
Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition
Eun Bee Choi and Yun-Ho Kim
2015(special): 276-286 doi: 10.3934/proc.2015.0276 +[Abstract](3091) +[PDF](374.6KB)
We study the following elliptic equations with variable exponents \begin{equation*} \begin{cases} -\text{div}(\varphi(x,\nabla u))+{|u|}^{p(x)-2}u= f(x,u) \quad &\text{in } \Omega \\ \varphi(x,\nabla u) \frac{\partial u}{\partial n}= g(x,u) & \text{on }\partial\Omega. \end{cases} \tag{P} \end{equation*} Under suitable conditions on $\phi$, $f$, and $g$, by employing the mountain pass theorem, the problem (P) has at least one nontrivial weak solution without assuming the Ambrosetti and Rabinowitz type condition.
On the properties of solutions set for measure driven differential inclusions
Mieczysław Cichoń and Bianca Satco
2015(special): 287-296 doi: 10.3934/proc.2015.0287 +[Abstract](3061) +[PDF](328.8KB)
The aim of the paper is to present properties of solutions set for differential inclusions driven by a positive finite Borel measure. We provide for the most natural type of solution results concerning the continuity of the solution set with respect to the data similar to some already known results, available for different types of solutions. As consequence, the solution set is shown to be compact as a subset of the space of regulated functions. The results allow one (by taking the measure $\mu$ of a particular form) to obtain information on the solution set for continuous or discrete problems, as well as impulsive or retarded set-valued problems.
Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape
Chiara Corsato, Colette De Coster and Pierpaolo Omari
2015(special): 297-303 doi: 10.3934/proc.2015.0297 +[Abstract](3429) +[PDF](301.4KB)
We prove existence and uniqueness of classical solutions of the anisotropic prescribed mean curvature problem \begin{equation*} {\rm -div}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}}, \ \text{ in } B, \quad u=0, \ \text{ on } \partial B, \end{equation*} where $a,b>0$ are given parameters and $B$ is a ball in ${\mathbb R}^N$. The solution we find is positive, radially symmetric, radially decreasing and concave. This equation has been proposed as a model of the corneal shape in the recent papers [13,14,15,18,17], where however a linearized version of the equation has been investigated.
An equation unifying both Camassa-Holm and Novikov equations
Priscila Leal da Silva and Igor Leite Freire
2015(special): 304-311 doi: 10.3934/proc.2015.0304 +[Abstract](3041) +[PDF](364.1KB)
In this paper we derive a new equation unifying the Camassa-Holm and Novikov equations invariant under the scaling transformation $(x,t,u)\mapsto(x,\lambda^{-b}t,\lambda u)$ and admitting a certain multiplier.
NLWE with a special scale invariant damping in odd space dimension
Marcello D'Abbicco and Sandra Lucente
2015(special): 312-319 doi: 10.3934/proc.2015.0312 +[Abstract](3122) +[PDF](328.5KB)
Let $p_0(k)$ be the critical Strauss exponent for the nonlinear wave equation $u_{t t}-\Delta u=|u|^p$ in $\mathbb{R}_t\times \mathbb{R}_x^k$. In this note we prove global existence for small data radial solutions to $v_{t t}-\Delta v+2(1+t)^{-1}v_t=|v|^p$ in $\mathbb{R}_t\times \mathbb{R}_x^n$, provided that $p>p_0(n+2)$ and $n\geq5$ is odd. This result is a counterpart of the non-existence result for $p\in(1,p_0(n+2)]$ in [2]. In particular we show that the scale invariant damping term $2(1+t)^{-1}u_t$ shifts by 2 the critical exponent of NLWE.
A note on a weakly coupled system of structurally damped waves
Marcello D'Abbicco
2015(special): 320-329 doi: 10.3934/proc.2015.0320 +[Abstract](2867) +[PDF](374.5KB)
In this note, we find the critical exponent for a system of weakly coupled structurally damped waves.
A symmetric nearly preserving general linear method for Hamiltonian problems
Raffaele D’Ambrosio, Giuseppe De Martino and Beatrice Paternoster
2015(special): 330-339 doi: 10.3934/proc.2015.0330 +[Abstract](2555) +[PDF](660.0KB)
This paper is concerned with the numerical solution of Hamiltonian problems, by means of nearly conservative multivalue numerical methods. In particular, the method we propose is symmetric, G-symplectic, diagonally implicit and generates bounded parasitic components over suitable time intervals. Numerical experiments on a selection of separable Hamiltonian problems are reported, also based on real data provided by Nasa Horizons System.
Bifurcation without parameters in circuits with memristors: A DAE approach
Ignacio García de la Vega and Ricardo Riaza
2015(special): 340-348 doi: 10.3934/proc.2015.0340 +[Abstract](4088) +[PDF](273.3KB)
Bifurcations without parameters describe qualitative changes in the local dynamics of nonlinear ODEs when normal hyperbolicity of a manifold of equilibria fails. Non-isolated equilibrium points are systematically exhibited by nonlinear circuits with memristors; a memristor is a nonlinear device recently introduced in circuit theory and which is expected to play a key role in electronics in the near future. In this communication we provide a graph-theoretic analysis of the transcritical bifurcation without parameters in memristive circuits, owing to the presence of a locally active memristor. The results are crucially based on the use of differential-algebraic circuit models.
Anisotropically diffused and damped Navier-Stokes equations
Hermenegildo Borges de Oliveira
2015(special): 349-358 doi: 10.3934/proc.2015.0349 +[Abstract](4120) +[PDF](351.1KB)
The incompressible Navier-Stokes equations with anisotropic diffusion and anisotropic damping is considered in this work. For the associated initial-boundary value problem, we prove the existence of weak solutions and we establish an energy inequality satisfied by these solutions. We prove also under what conditions the solutions of this problem extinct in a finite time.
Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation
Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova and Atanas Stefanov
2015(special): 359-368 doi: 10.3934/proc.2015.0359 +[Abstract](3458) +[PDF](362.9KB)
In the present work, we introduce a new $\mathcal{PT}$-symmetric variant of the Klein-Gordon field theoretic problem. We identify the standing wave solutions of the proposed class of equations and analyze their stability. In particular, we obtain an explicit frequency condition, somewhat reminiscent of the classical Vakhitov-Kolokolov criterion, which sharply separates the regimes of spectral stability and instability. Our numerical computations corroborate the relevant theoretical result.
Parabolic Monge-Ampere equations giving rise to a free boundary: The worn stone model
Gregorio Díaz and Jesús Ildefonso Díaz
2015(special): 369-378 doi: 10.3934/proc.2015.0369 +[Abstract](2672) +[PDF](338.5KB)
This paper deals with several qualitative properties of solutions of some parabolic equations associated to the Monge--Ampère operator arising in suitable formulations of the Gauss curvature flow and the worn stone problems.
Steiner symmetrization for concave semilinear elliptic and parabolic equations and the obstacle problem
J.I. Díaz and D. Gómez-Castro
2015(special): 379-386 doi: 10.3934/proc.2015.0379 +[Abstract](2791) +[PDF](293.1KB)
We extend some previous results in the literature on the Steiner rearrangement of linear second order elliptic equations to the semilinear concave parabolic problems and the obstacle problem.
Global existence and low Mach number limit to the 3D compressible magnetohydrodynamic equations in a bounded domain
Jishan Fan, Fucai Li and Gen Nakamura
2015(special): 387-394 doi: 10.3934/proc.2015.0387 +[Abstract](3203) +[PDF](290.6KB)
In this paper we establish the global existence of strong solutions to the three-dimensional compressible magnetohydrodynamic equations in a bounded domain with small initial data. Moreover, we study the low Mach number limit to the corresponding problem.
A regularity criterion for 3D density-dependent MHD system with zero viscosity
Jishan Fan and Tohru Ozawa
2015(special): 395-399 doi: 10.3934/proc.2015.0395 +[Abstract](3197) +[PDF](288.3KB)
This paper proves a regularity criterion $\nabla u,\nabla b\in L^\infty(0,T;L^\infty)$ for 3D density-dependent MHD system with zero viscosity and positive initial density.
Existence and uniqueness of positive solutions for singular biharmonic elliptic systems
Luiz F. O. Faria
2015(special): 400-408 doi: 10.3934/proc.2015.0400 +[Abstract](3216) +[PDF](350.6KB)
In this paper we prove existence and uniqueness of positive solutions of nonlinear singular biharmonic elliptic system in smooth bounded domains, with coupling of the equations, under Navier boundary condition. The solution is constructed through an approximating process based on a priori estimates, regularity up to the boundary and Hardy-Sobolev inequality.
On explicit lower bounds and blow-up times in a model of chemotaxis
Maria Antonietta Farina, Monica Marras and Giuseppe Viglialoro
2015(special): 409-417 doi: 10.3934/proc.2015.0409 +[Abstract](3157) +[PDF](1263.4KB)
This paper is concerned with a parabolic Keller-Segel system in $\mathbb{R}^n$, with $n=2$ or $3$, under Neumann boundary conditions. First, important theoretical and general results dealing with lower bounds for blow-up time estimates are summarized and analyzed. Next, a resolution method is proposed and used to both compute the real blow-up times of such unbounded solutions and analyze and discuss some of their properties.
Singular limit of Allen--Cahn equation with constraint and its Lagrange multiplier
Mohammad Hassan Farshbaf-Shaker, Takeshi Fukao and Noriaki Yamazaki
2015(special): 418-427 doi: 10.3934/proc.2015.0418 +[Abstract](3186) +[PDF](332.6KB)
We consider the Allen--Cahn equation with a constraint. Our constraint is provided by the subdifferential of the indicator function on a closed interval, which is the multivalued function. In this paper we give the characterization of the Lagrange multiplier for our equation. Moreover, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier for our problem.
A general approach to identification problems and applications to partial differential equations
Angelo Favini
2015(special): 428-435 doi: 10.3934/proc.2015.0428 +[Abstract](2583) +[PDF](316.7KB)
An abstract method to deal with identification problems related to evolution equations with multivalued linear operators (or linear relations) is described. Some applications to partial differential equations are presented.
Existence of positive solutions of a superlinear boundary value problem with indefinite weight
Guglielmo Feltrin
2015(special): 436-445 doi: 10.3934/proc.2015.0436 +[Abstract](3734) +[PDF](338.9KB)
We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u''+a(x)g(u)=0$. The weight $a(x)$ is allowed to change sign. We assume that the function $g\colon\mathopen{[}0,+\infty\mathclose{[}\to\mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth conditions, including the superlinear case $g(s)=s^{p}$, with $p>1$. In particular we suppose that $g(s)/s$ is large near infinity, but we do not require that $g(s)$ is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.
High order periodic impulsive problems
João Fialho and Feliz Minhós
2015(special): 446-454 doi: 10.3934/proc.2015.0446 +[Abstract](2976) +[PDF](351.7KB)
The theory of impulsive problem is experiencing a rapid development in the last few years. Mainly because they have been used to describe some phenomena, arising from different disciplines like physics or biology, subject to instantaneous change at some time instants called moments. Second order periodic impulsive problems were studied to some extent, however very few papers were dedicated to the study of third and higher order impulsive problems.
    The high order impulsive problem considered is composed by the fully nonlinear equation \begin{equation*} u^{\left( n\right) }\left( x\right) =f\left( x,u\left( x\right) ,u^{\prime }\left( x\right) ,...,u^{\left( n-1\right) }\left( x\right) \right) \end{equation*} for a. e. $x\in I:=\left[ 0,1\right] ~\backslash ~\left\{ x_{1},...,x_{m}\right\} $ where $f:\left[ 0,1\right] \times \mathbb{R} ^{n}\rightarrow \mathbb{R}$ is $L^{1}$-Carathéodory function, along with the periodic boundary conditions \begin{equation*} u^{\left( i\right) }\left( 0\right) =u^{\left( i\right) }\left( 1\right) ,         i=0,...,n-1, \end{equation*} and the impulsive conditions \begin{equation*} \begin{array}{c} u^{\left( i\right) }\left( x_{j}^{+}\right) =g_{j}^{i}\left( u\left( x_{j}\right) \right) ,        i=0,...,n-1, \end{array} \end{equation*} where $g_{j}^{i},$ for $j=1,...,m,$are given real valued functions satisfying some adequate conditions, and $x_{j}\in \left( 0,1\right) ,$ such that $0 = x_0 < x_1 <...< x_m < x_{m+1}=1.$
     The arguments applied make use of the lower and upper solution method combined with an iterative technique, which is not necessarily monotone, together with classical results such as Lebesgue Dominated Convergence Theorem, Ascoli-Arzela Theorem and fixed point theory.
Well-posedness for a class of nonlinear degenerate parabolic equations
Giuseppe Floridia
2015(special): 455-463 doi: 10.3934/proc.2015.0455 +[Abstract](3184) +[PDF](381.3KB)
In this paper we obtain well-posedness for a class of semilinear weakly degenerate reaction-diffusion systems with Robin boundary conditions. This result is obtained through a Gagliardo-Nirenberg interpolation inequality and some embedding results for weighted Sobolev spaces.
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$
Kentarou Fujie, Chihiro Nishiyama and Tomomi Yokota
2015(special): 464-472 doi: 10.3934/proc.2015.0464 +[Abstract](3536) +[PDF](347.5KB)
This paper is concerned with global existence and boundedness of classical solutions to the quasilinear fully parabolic Keller-Segel system $u_t = \nabla \cdot(D(u)\nabla u) -\nabla \cdot (v^{-1}S(u)\nabla v)$, $v_t= \Delta v-v+u$. In [7,4], global existence and boundedness were established in the system without $v^{-1}$. In this paper the signal-dependent sensitivity $v^{-1}$ is taken into account via the Weber-Fechner law. A uniform-in-time estimate for $v$ obtained in [2] defeats the singularity of $v^{-1}$.
Remark on a semirelativistic equation in the energy space
Kazumasa Fujiwara, Shuji Machihara and Tohru Ozawa
2015(special): 473-478 doi: 10.3934/proc.2015.0473 +[Abstract](3143) +[PDF](318.8KB)
Well-posedness of the Cauchy problem for a semirelativistic equation with cubic nonlinearity is shown in the energy space $H^{1/2}$. Solutions are constructed as a limit of approximation solutions, where the argument on the convergence depends on the completeness of $L^2$ and is independent of compactness. The Yudovitch type argument plays an important role for the convergence arguments.
Estimates for solutions of nonautonomous semilinear ill-posed problems
Matthew A. Fury
2015(special): 479-488 doi: 10.3934/proc.2015.0479 +[Abstract](2925) +[PDF](354.1KB)
The nonautonomous, semilinear problem $\frac{du}{dt}=A(t)u(t)+h(t,u(t))$, $0 \leq s \leq t < T$, $u(s)=\chi$ in Hilbert space with a Lipschitz condition on $h$, is generally ill-posed under prescribed conditions on the operators $A(t)$. Hence, regularization techniques are sought out in order to estimate known solutions of the problem. We study two quasi-reversibility methods of approximation which have successfully established regularization in the linear case, and provide an estimate on a solution $u(t)$ of the problem under these approximations in the nonlinear case. The results apply to partial differential equations of arbitrary even order including the nonlinear backward heat equation with a time-dependent diffusion coefficient.
Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets
Evgeny Galakhov and Olga Salieva
2015(special): 489-494 doi: 10.3934/proc.2015.0489 +[Abstract](3012) +[PDF](275.2KB)
Nonexistence results for nontrivial solutions for some classes of nonlinear partial differential inequalities with gradient terms and coefficients possessing singularities on unbounded sets are obtained.
3D reconstruction for partial data electrical impedance tomography using a sparsity prior
Henrik Garde and Kim Knudsen
2015(special): 495-504 doi: 10.3934/proc.2015.0495 +[Abstract](4246) +[PDF](2129.5KB)
In electrical impedance tomography the electrical conductivity inside a physical body is computed from electro-static boundary measurements. The focus of this paper is to extend recent results for the 2D problem to 3D: prior information about the sparsity and spatial distribution of the conductivity is used to improve reconstructions for the partial data problem with Cauchy data measured only on a subset of the boundary. A sparsity prior is enforced using the $\ell_1$ norm in the penalty term of a Tikhonov functional, and spatial prior information is incorporated by applying a spatially distributed regularization parameter. The optimization problem is solved numerically using a generalized conditional gradient method with soft thresholding. Numerical examples show the effectiveness of the suggested method even for the partial data problem with measurements affected by noise.
Manakov solitons and effects of external potential wells
V. S. Gerdjikov, A. V. Kyuldjiev and M. D. Todorov
2015(special): 505-514 doi: 10.3934/proc.2015.0505 +[Abstract](2379) +[PDF](525.9KB)
The effects of the external potential wells on the Manakov soliton interactions using the perturbed complex Toda chain (PCTC) model are analyzed. The superposition of a large number of wells/humps influences stronger the motion of the soliton envelopes and can cause a transition from asymptotically free and mixed asymptotic regime to a bound state regime and vice versa. Such external potentials are easier to implement in experiments and can be used to control the soliton motion in a given direction and to achieve a predicted motion of the optical pulse. A general feature of the conducted numerical experiments is that the long-time evolution of both CTC and PCTC match very well with the Manakov model numerics, often much longer than expected even for 9-soliton train configurations. This means that PCTC is reliable dynamical model for predicting the evolution of the multisoliton solutions of Manakov model in adiabatic approximation.
Numerical optimal control of a coupled ODE-PDE model of a truck with a fluid basin
Matthias Gerdts and Sven-Joachim Kimmerle
2015(special): 515-524 doi: 10.3934/proc.2015.0515 +[Abstract](3946) +[PDF](3738.0KB)
We consider a numerical study of an optimal control problem for a truck with a fluid basin, which leads to an optimal control problem with a coupled system of partial differential equations (PDEs) and ordinary differential equations (ODEs). The motion of the fluid in the basin is modeled by the nonlinear hyperbolic Saint-Venant (shallow water) equations while the vehicle dynamics are described by the equations of motion of a mechanical multi-body system. These equations are fully coupled through boundary conditions and force terms. We pursue a first-discretize-then-optimize approach using a Lax-Friedrich scheme. To this end a reduced optimization problem is obtained by a direct shooting approach and solved by a sequential quadratic programming method. For the computation of gradients we employ an efficient adjoint scheme. Numerical case studies for optimal braking maneuvers of the truck and the basin filled with a fluid are presented.
A posteriori error analysis of a stabilized mixed FEM for convection-diffusion problems
M. González, J. Jansson and S. Korotov
2015(special): 525-532 doi: 10.3934/proc.2015.0525 +[Abstract](2983) +[PDF](490.5KB)
We present an augmented dual-mixed variational formulation for a linear convection-diffusion equation with homogeneous Dirichlet boundary conditions. The approach is based on the addition of suitable least squares type terms. We prove that for appropriate values of the stabilization parameters, that depend on the diffusion coefficient and the magnitude of the convective velocity, the new variational formulation and the corresponding Galerkin scheme are well-posed, and a Céa estimate holds. In particular, we derive the rate of convergence when the flux and the concentration are approximated, respectively, by Raviart-Thomas and continuous piecewise polynomials. In addition, we introduce a simple a posteriori error estimator which is reliable and locally efficient. Finally, we provide numerical experiments that illustrate the behavior of the method.
Existence of homoclinic solutions for second order difference equations with $p$-laplacian
John R. Graef, Lingju Kong and Min Wang
2015(special): 533-539 doi: 10.3934/proc.2015.0533 +[Abstract](2765) +[PDF](342.3KB)
Using the variational method and critical point theory, the authors study the existence of infinitely many homoclinic solutions to the difference equation \begin{equation*} -\Delta \big(a(k)\phi_p(\Delta u(k-1))\big)+b(k)\phi_p(u(k))=\lambda f(k,u(k))),\quad k\in\mathbb{Z}, \end{equation*} where $p>1$ is a real number, $\phi_p(t)=|t|^{p-2}t$ for $t\in\mathbb{R}$, $\lambda>0$ is a parameter, $a, b:\mathbb{Z}\to (0,\infty)$, and $f: \mathbb{Z}\times\mathbb{R}\to\mathbb{R}$ is continuous in the second variable. Related results in the literature are extended.
Real cocycles of point-distal minimal flows
Gernot Greschonig
2015(special): 540-548 doi: 10.3934/proc.2015.0540 +[Abstract](2830) +[PDF](316.2KB)
We generalise the structure theorem for topologically recurrent real skew product extensions of distal minimal compact metric flows in [9] to a class of point distal minimal compact metric flows. While the general case of a point distal flow according to the Veech structure theorem seems hopeless, we prove a result for cocycles of minimal point distal flows without strong Li-Yorke pairs which can be obtained by an almost 1-1 extension of a distal flow with connected fibres. Moreover, a stronger condition on recurrence is necessary. We shall assume that every non-distal point in the point distal compact metric flow is proximal to a point which lifts to recurrent points in the skew product. However, we shall prove that the usual notion of topological recurrence is sufficient for locally connected almost 1-1 extensions of an isometry. This setting includes a well-known example of a point-distal flow by Mary Rees.
Optimal control for an epidemic in populations of varying size
Ellina Grigorieva, Evgenii Khailov and Andrei Korobeinikov
2015(special): 549-561 doi: 10.3934/proc.2015.0549 +[Abstract](4331) +[PDF](326.2KB)
For a Susceptible-Infected-Recovered (SIR) control model with varying population size, the optimal control problem of minimization of the infected individuals at a terminal time is stated and solved. Three distinctive control policies are considered, namely the vaccination of the susceptible individuals, treatment of the infected individuals and an indirect policy aimed at reduction of the transmission. Such values of the model parameters and control constraints are used, for which the optimal controls are bang-bang. We estimated the maximal possible number of switchings of these controls, which task is related to the estimation of the number of zeros of the corresponding switching functions. Different approaches of estimating the number of zeros of the switching functions are applied. The found estimates enable us to reduce the optimal control problem to a considerably simpler problem of the finite-dimensional constrained minimization.
The Nehari solutions and asymmetric minimizers
Armands Gritsans and Felix Sadyrbaev
2015(special): 562-568 doi: 10.3934/proc.2015.0562 +[Abstract](2992) +[PDF](354.9KB)
We consider the boundary value problem $x'' = -q(t,h) x^3,$ $x(-1)=x(1)=0$ which exhibits bifurcation of the Nehari solutions. The Nehari solution of the problem is a solution which minimizes certain functional. We show that for $h$ small there is exactly one Nehari solution. Then under the increase of $h$ there appear two Nehari solutions which supply the functional smaller value than the remaining symmetrical solution does. So the bifurcation of the Nehari solutions is observed and the previously studied in the literature phenomenon of asymmetrical Nehari solutions is confirmed.
Modeling HIV: Determining the factors affecting the racial disparity in the prevalence of infected women
K.F. Gurski, K.A. Hoffman and E.K. Thomas
2015(special): 569-578 doi: 10.3934/proc.2015.0569 +[Abstract](2730) +[PDF](353.9KB)
We present a mathematical model of the transmission of HIV through sexual contact in a population stratified by sexual behavior and by race/ethnicity. We consider two theories for the disproportionate prevalence of HIV in the population of African American women and Hispanic/Latino women compared with U.S. women of other races/ethnicities. First, we consider that minority women are being adversely affected by incurable STDs due to the non-disclosure of risky homosexual activities of their male sex partners. Second, we consider the effect of sexual network factors, such as the racially homophilic networks through the use of a partnership mixing matrix. Both analytic and numerical results indicate that the effect of the down low population on the disproportionate spread of HIV in women is small compared to the effect of homophilic racial mixing.
On reachability analysis for nonlinear control systems with state constraints
Mikhail Gusev
2015(special): 579-587 doi: 10.3934/proc.2015.0579 +[Abstract](3073) +[PDF](422.3KB)
The paper is devoted to the problem of approximating reachable sets of a nonlinear control system with state constraints given as a solution set for a nonlinear inequality. A procedure to remove state constraints is proposed; this procedure consists in replacing a primary system by an auxiliary system without state constraints. The equations of the auxiliary system depend on a small parameter. It is shown that a reachable set of the primary system may be approximated in the Hausdorff metric by reachable sets of the auxiliary system when the small parameter tends to zero. The estimates of the rate of convergence are given.
Noncontrollability for the Colemann-Gurtin model in several dimensions
Andrei Halanay and Luciano Pandolfi
2015(special): 588-595 doi: 10.3934/proc.2015.0588 +[Abstract](2325) +[PDF](296.5KB)
It is proved that, for every smooth kernel $M(t)$, the corresponding Colemann-Gurtin model in several spatial dimensions cannot be controlled to zero.
Existence of positive solutions for a system of nonlinear second-order integral boundary value problems
Johnny Henderson and Rodica Luca
2015(special): 596-604 doi: 10.3934/proc.2015.0596 +[Abstract](3157) +[PDF](349.7KB)
We study the existence and multiplicity of positive solutions of a system of nonlinear second-order ordinary differential equations subject to Riemann-Stieltjes integral boundary conditions.
Jacobi--Lie systems: Fundamentals and low-dimensional classification
F.J. Herranz, J. de Lucas and C. Sardón
2015(special): 605-614 doi: 10.3934/proc.2015.0605 +[Abstract](2848) +[PDF](408.1KB)
A Lie system is a system of differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields, a Vessiot--Guldberg Lie algebra. We define and analyze Lie systems possessing a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a Jacobi manifold, the hereafter called Jacobi--Lie systems. We classify Jacobi--Lie systems on $\mathbb{R}$ and $\mathbb{R}^2$. Our results shall be illustrated through examples of physical and mathematical interest.
Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition
Christina A. Hollon and Jeffrey T. Neugebauer
2015(special): 615-620 doi: 10.3934/proc.2015.0615 +[Abstract](4403) +[PDF](302.7KB)
In this paper, we apply Krasnosel'skii's cone expansion and compression fixed point theorem to show the existence of at least one positive solution to the nonlinear fractional boundary value problem $D^\alpha_{0^+} u + a(t)f(u)=0$, $0 < t < 1$, $1 < \alpha \le 2$, satisfying boundary conditions $u(0)=D^\beta_{0^+} u(1)=0$, $0\le\beta\le1$.
Optimal control and stability analysis of an epidemic model with education campaign and treatment
Sanjukta Hota, Folashade Agusto, Hem Raj Joshi and Suzanne Lenhart
2015(special): 621-634 doi: 10.3934/proc.2015.0621 +[Abstract](4590) +[PDF](574.5KB)
In this paper we investigated a SIR epidemic model in which education campaign and treatment are both important for the disease management. Optimal control theory was used on the system of differential equations to achieve the goal of minimizing the infected population and slow down the epidemic outbreak. Stability analysis of the disease free equilibrium of the system was completed. Numerical results with education campaign levels and treatment rates as controls are illustrated.
An iterative approach to $L^\infty$-boundedness in quasilinear Keller-Segel systems
Sachiko Ishida
2015(special): 635-643 doi: 10.3934/proc.2015.0635 +[Abstract](3409) +[PDF](381.1KB)
This paper mainly considers the uniform bound on solutions of non-degenerate Keller-Segel systems on the whole space. In the case that the domain is bounded, Tao-Winkler (2012) proved existence of globally bounded solutions of non-degenerate systems. More precisely, they gave the result on boundedness in quasilinear parabolic equations by using the $L^p$-bounds on the solution for some large $p>1$. In Ishida-Yokota (2012), they dealt with the same system as this paper on the whole space, however, their $L^\infty$-estimate possibly grows up even if the solutions have the uniform bounds in $L^p(\mathbb{R}^N)$ for all $p\in[1,\infty)$. The present work asserts the uniform in time $L^\infty$-bound on solutions. Moreover, this paper covers the degenerate Keller-Segel systems and constructs the uniformly bounded weak solutions.
Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field
Tetsuya Ishiwata and Kota Kumazaki
2015(special): 644-651 doi: 10.3934/proc.2015.0644 +[Abstract](3438) +[PDF](330.9KB)
In this short paper we propose a finite difference scheme for the Landau-Lifshitz equation and an iteration procedure to solve the scheme. The key concept is ``structure-preserving''. We show that the proposed method inherits important mathematical structures from the original problem and also analysis the iteration.
On global dynamics in a multi-dimensional discrete map
Anatoli F. Ivanov
2015(special): 652-659 doi: 10.3934/proc.2015.0652 +[Abstract](2959) +[PDF](298.9KB)
We derive preliminary results on global dynamics of the multi-dimensional discrete map $$ F:\; (x_1,x_2,\dots,x_{k-1},x_k)\mapsto (x_1+af(x_k),x_1,x_2,\dots,x_{k-1}) $$ where the continuous real-valued function $f$ is one-sided bounded and satisfying the negative feedback condition, $x\cdot f(x)<0, x\ne0$, $a$ is a positive parameter. We show the existence of a compact global attractor for map $F$, and derive a condition for the global attractivity of the zero fixed point.
Characterizing chaos in a type of fractional Duffing's equation
S. Jiménez and Pedro J. Zufiria
2015(special): 660-669 doi: 10.3934/proc.2015.0660 +[Abstract](2922) +[PDF](450.2KB)
We characterize the chaos in a fractional Duffing's equation computing the Lyapunov exponents and the dimension of the strange attractor in the effective phase space of the system. We develop a specific analytical method to estimate all Lyapunov exponents and check the results with the fiduciary orbit technique and a time series estimation method.
Linear model of traffic flow in an isolated network
Ángela Jiménez-Casas and Aníbal Rodríguez-Bernal
2015(special): 670-677 doi: 10.3934/proc.2015.0670 +[Abstract](2947) +[PDF](284.9KB)
We obtain a mathematical linear model which describes automatic operation of the traffic of material objects in a network. Existence and global solutions is obtained for such model. A related model which used outdated information is shown to collapse in finite time.
Stability of neutral delay differential equations modeling wave propagation in cracked media
Stéphane Junca and Bruno Lombard
2015(special): 678-685 doi: 10.3934/proc.2015.0678 +[Abstract](3242) +[PDF](370.7KB)
Propagation of elastic waves is studied in a 1D medium containing $N$ cracks modeled by nonlinear jump conditions. The case $N=1$ is fully understood. When $N>1$, the evolution equations are written as a system of nonlinear neutral delay differential equations, leading to a well-posed Cauchy problem. In the case $N=2$, some mathematical results about the existence, uniqueness and attractivity of periodic solutions have been obtained in 2012 by the authors, under the assumption of small sources. The difficulty of analysis follows from the fact that the spectrum of the linear operator is asymptotically closed to the imaginary axis. Here we propose a new result of stability in the homogeneous case, based on an energy method. One deduces the asymptotic stability of the zero steady-state. Extension to $N=3$ cracks is also considered, leading to new results in particular configurations.
Enhanced choice of the parameters in an iteratively regularized Newton-Landweber iteration in Banach space
Barbara Kaltenbacher and Ivan Tomba
2015(special): 686-695 doi: 10.3934/proc.2015.0686 +[Abstract](2735) +[PDF](539.1KB)
This paper is a close follow-up of [9] and [11], where Newton-Landweber iterations have been shown to converge either (unconditionally) without rates or (under an additional regularity assumption) with rates. The choice of the parameters in the method were different in each of these two cases. We now found a unified and more general strategy for choosing these parameters that enables both convergence and convergence rates. Moreover, as opposed to the previous one, this choice yields strong convergence as the noise level tends to zero, also in the case of no additional regularity. Additionally, the resulting method appears to be more efficient than the one from [9], as our numerical tests show.
Non-holonomic constraints and their impact on discretizations of Klein-Gordon lattice dynamical models
Panayotis G. Kevrekidis, Vakhtang Putkaradze and Zoi Rapti
2015(special): 696-704 doi: 10.3934/proc.2015.0696 +[Abstract](3129) +[PDF](382.4KB)
We explore a new type of discretizations of lattice dynamical models of the Klein-Gordon type relevant to the existence and long-term mobility of nonlinear waves. The discretization is based on non-holonomic constraints and is shown to retrieve the ``proper'' continuum limit of the model. Such discretizations are useful in exactly preserving a discrete analogue of the momentum. It is also shown that for generic initial data, the momentum and energy conservation laws cannot be achieved concurrently. Finally, direct numerical simulations illustrate that our models yield considerably higher mobility of strongly nonlinear solutions than the well-known ``standard'' discretizations, even in the case of highly discrete systems when the coupling between the adjacent nodes is weak. Thus, our approach is better suited for cases where an accurate description of mobility for nonlinear traveling waves is important.
Reduction of a kinetic model of active export of importins
Sarbaz H. A. Khoshnaw
2015(special): 705-722 doi: 10.3934/proc.2015.0705 +[Abstract](3179) +[PDF](2942.9KB)
We study a kinetic model of active export of importins. The kinetic model is written as a system of ordinary differential equations. We developed some model reduction techniques to simplify the system. The techniques are: removal of very slow reactions, quasi-steady state approximation and simplification of kinetic equations based on stoichiometric conservation laws. Local sensitivity analysis is used for the identification of critical model parameters. After model reduction, the numbers of reactions and species are reduced from $28$ and $29$ to $20$ and $20$, respectively. The reduced model and original model are compared in numerical simulations using SBedit tools for Matlab, and the methods of further model reduction are discussed. Interestingly, we investigate an iterative algorithm based on the Duhamel iterates to calculate the analytical approximate solutions of the complex non--linear chemical kinetics. This technique plays as an explicit formula that can be studied in detail for wide regions of concentrations for optimization and parameter identification purposes. It seems that the third iterative solution of the suggested method is significantly close to the actual solution of the kinetic models in most cases.
On control synthesis for uncertain dynamical discrete-time systems through polyhedral techniques
Elena K. Kostousova
2015(special): 723-732 doi: 10.3934/proc.2015.0723 +[Abstract](2924) +[PDF](398.6KB)
Problems of feedback terminal target control for linear discrete-time systems without and with uncertainties are considered. We continue the development of methods of control synthesis using polyhedral (parallelotope-valued) solvability tubes. The cases without uncertainties, with additive parallelotope-bounded uncertainties, and also with interval uncertainties in coefficients of the system are considered. Also the same systems under state constraints are considered. Nonlinear recurrent relations are presented for polyhedral solvability tubes for each of the mentioned cases. Two types of control strategies, which can be calculated on the base of the mentioned tubes, are proposed. Controls of the second type can be calculated by explicit formulas. Results of computer simulations are presented.
Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to a tumour invasion model
Akisato Kubo, Hiroki Hoshino and Katsutaka Kimura
2015(special): 733-744 doi: 10.3934/proc.2015.0733 +[Abstract](3245) +[PDF](380.3KB)
We study the global existence in time and asymptotic behaviour of solutions of nonlinear evolution equations with strong dissipation and proliferation terms arising in mathematical models of biology and medicine including tumour invasion models.
Strategic games in a competitive market: Feedback from the users' environment
Natalia Kudryashova
2015(special): 745-753 doi: 10.3934/proc.2015.0745 +[Abstract](2784) +[PDF](459.7KB)
We propose a model for the dynamics of a competitive market, in which strategic decision making of market players may be affected by the users' switching behaviour. A novel concept, habitual attraction, is introduced to reflect the resistance of users against changing their routinely used provider. The model can be adopted for a particular market and allows for dynamic learning. We demonstrate applications of the model to the internet search market in the context of competition law and regulation, where a user centred description is essential, for instance, in the anti-trust proceedings.
Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces
Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota and Kentarou Yoshii
2015(special): 754-763 doi: 10.3934/proc.2015.0754 +[Abstract](3279) +[PDF](374.0KB)
This paper is concerned with the Cauchy problem for the complex Ginzburg-Landau type equation $u_t = (\delta _{1}+i\delta _{2})\Delta u -i\mu |u| ^{2\sigma}u$ in $(0,\infty)\times\mathbb{R}^d$, where $\delta_{1}>0$, $\delta_{2}, \mu \in \mathbb{R}$ and $d\in\mathbb{N}$. Existence and uniqueness of spatially periodic solutions to the problem are established in a space which corresponds to the Sobolev space on the $d$-dimensional torus when $0<\sigma<\infty$ ($d=1, 2$) and $0<\sigma<1/(d-2)$ ($d \ge 3$). The result improves the case $p=2$ of the result in the space $W^{1,p}$ given by Gao-Wang [2,Theorem 1] in which it is assumed that $d < p$ and $\sigma < p/d$.
Decomposition of discrete linear-quadratic optimal control problems for switching systems
Galina Kurina and Sahlar Meherrem
2015(special): 764-774 doi: 10.3934/proc.2015.0764 +[Abstract](3370) +[PDF](306.0KB)
A discrete linear-quadratic optimal control problem for two controlled systems acting sequentially is considered. Matching conditions for trajectories at the switching point are absent, however the minimized functional depends on values of a state trajectory at the left and right sides from the switching point. State trajectories have fixed left and right points. We derive control optimality conditions in the maximum principle form. The unique solvability of the considered problem is established. The algorithm for solving the problem is given, which is based on solving sequentially some initial value problems. The formula for the minimal value of the performance index is also obtained. The transformation reducing the discrete problem with switching point to a problem without one is presented only in a special case.
An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem
Jeffrey W. Lyons
2015(special): 775-782 doi: 10.3934/proc.2015.0775 +[Abstract](3222) +[PDF](300.2KB)
In this article, we show the existence of an antisymmetric solution to the second order boundary value problem $x''+f(x(t))=0,\; t\in(0,n)$ satisfying antiperiodic boundary conditions $x(0)+x(n)=0,\; x'(0)+x'(n)=0$ using an Avery et. al. fixed point theorem which itself is an extension of the traditional Leggett-Williams fixed point theorem. The antisymmetric solution satisfies $x(t)=-x(n-t)$ for $t\in[0,n]$ and is nonnegative, nonincreasing, and concave for $t\in[0,n/2]$. To conclude, we present an example.
Optimal control of system governed by the Gao beam equation
Jitka Machalová and Horymír Netuka
2015(special): 783-792 doi: 10.3934/proc.2015.0783 +[Abstract](3626) +[PDF](365.1KB)
In this contribution several optimal control problems are mathematically formulated and analyzed for a nonlinear beam which was introduced in 1996 by David Y. Gao. The beam model is given by a static nonlinear fourth-order differential equation with some boundary conditions. The beam is here subjected to a vertical load and possibly to an axial tension load as well. A cost functional is constructed in such a way that the lower its value is, the better model we obtain. Both existence and uniqueness are studied for the solution to the proposed control problems along with optimality conditions. Due to the fact that analytical solution is not available for the nonlinear Gao beam, a finite element approximation is provided for the proposed problems. Numerical results are compared with Euler-Bernoulli beam as well as the authors' previous considerations.
Potential estimates and applications to elliptic equations
Farman Mamedov, Sara Monsurrò and Maria Transirico
2015(special): 793-800 doi: 10.3934/proc.2015.0793 +[Abstract](3175) +[PDF](324.4KB)
In this paper we prove a potential type estimate for the solutions of some classes of Dirichlet problems associated to certain non divergence structure elliptic equations with smooth datum. As a consequence of our potential bound, we can get an a priori estimate for the solutions of the same kind of Dirichlet problem, but with less regular datum.
Strong solutions to a class of boundary value problems on a mixed Riemannian--Lorentzian metric
Antonella Marini and Thomas H. Otway
2015(special): 801-808 doi: 10.3934/proc.2015.0801 +[Abstract](2400) +[PDF](341.8KB)
A first-order elliptic-hyperbolic system in extended projective space is shown to possess strong solutions to a natural class of Guderley--Morawetz--Keldysh problems on a typical domain.
Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system
Monica Marras, Stella Vernier Piro and Giuseppe Viglialoro
2015(special): 809-816 doi: 10.3934/proc.2015.0809 +[Abstract](3476) +[PDF](330.6KB)
This paper deals with a parabolic-parabolic Keller-Segel system, modeling chemotaxis, with time dependent coefficients. We consider non-negative solutions of the system which blow up in finite time $t^*$ and an explicit lower bound for $t^*$ is derived under sufficient conditions on the coefficients and the spatial domain.
Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations
Jochen Merker and Aleš Matas
2015(special): 817-825 doi: 10.3934/proc.2015.0817 +[Abstract](2839) +[PDF](400.6KB)
The aim of this short note is to study self-similiar radially symmetric solutions of the scalar doubly nonlinear reaction-diffusion equation \begin{equation*} \frac{\partial u^{m-1}}{\partial t} - \Delta_p u = \lambda u^{q-1} \end{equation*} on $\mathbb{R}^n$, where the parameters $1 < m, p,q < \infty$ and $0 < \lambda < \infty$ are fixed. Particularly, for $m < p < q < q_c := p(1+\frac{m-1}{n})$ (where $q_c$ is Fujita's critical exponent of blow-up) we show that there exist self-similar and radially symmetric solutions $u$, which do not blow up in finite time, but instantly become sign-changing for $t>0$ inside some subdomain.
Numerical simulation of a SIS epidemic model based on a nonlinear Volterra integral equation
Eleonora Messina
2015(special): 826-834 doi: 10.3934/proc.2015.0826 +[Abstract](3236) +[PDF](376.6KB)
We consider a SIS epidemic model based on a Volterra integral equation and we compare the dynamical behavior of the analytical solution and its numerical approximation obtained by direct quadrature methods. We prove that, under suitable assumptions, the numerical scheme preserves the qualitative properties of the continuous equation and we show that, as the stepsize tends to zero, the numerical bifurcation points tend to the continuous ones.
Averaging in random systems of nonnegative matrices
Janusz Mierczyński
2015(special): 835-840 doi: 10.3934/proc.2015.0835 +[Abstract](2990) +[PDF](279.3KB)
It is proved that the top Lyapunov exponent of a random matrix system of the form $\{A D(\omega)\}$, where $A$ is a nonnegative matrix and $D(\omega)$ is a diagonal matrix with positive diagonal entries, is bounded from below by the top Lyapunov exponent of the averaged system. This is in contrast to what one should expect of systems describing biological metapopulations.
Solvability of higher-order BVPs in the half-line with unbounded nonlinearities
Feliz Minhós and Hugo Carrasco
2015(special): 841-850 doi: 10.3934/proc.2015.0841 +[Abstract](3002) +[PDF](312.5KB)
This work presents sufficient conditions for the existence of unbounded solutions of a Sturm-Liouville type boundary value problem on the half-line. One-sided Nagumo condition plays a special role because it allows an asymmetric unbounded behavior on the nonlinearity. The arguments are based on fixed point theory and lower and upper solutions method. An example is given to show the applicability of our results.
On higher order nonlinear impulsive boundary value problems
Feliz Minhós and Rui Carapinha
2015(special): 851-860 doi: 10.3934/proc.2015.0851 +[Abstract](3497) +[PDF](310.5KB)
This work studies some two point impulsive boundary value problems composed by a fully differential equation, which higher order contains an increasing homeomorphism, by two point boundary conditions and impulsive effects. We point out that the impulsive conditions are given via multivariate generalized functions, including impulses on the referred homeomorphism. The method used apply lower and upper solutions technique together with fixed point theory. Therefore we have not only the existence of solutions but also the localization and qualitative data on their behavior. Moreover a Nagumo condition will play a key role in the arguments.
Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization
Tatsuki Mori, Kousuke Kuto, Masaharu Nagayama, Tohru Tsujikawa and Shoji Yotsutani
2015(special): 861-877 doi: 10.3934/proc.2015.0861 +[Abstract](3605) +[PDF](690.1KB)
We are interested in wave-pinning in a reaction-diffusion model for cell polarization proposed by Y.Mori, A.Jilkine and L.Edelstein-Keshet. They showed interesting bifurcation diagrams and stability results for stationary solutions for a limiting equation by numerical computations. Kuto and Tsujikawa showed several mathematical bifurcation results of stationary solutions of this problem. We show exact expressions of all the solution by using the Jacobi elliptic functions and complete elliptic integrals. Moreover, we construct a bifurcation sheet which gives bifurcation diagram. Furthermore, we show numerical results of the stability of stationary solutions.
Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition
Minoru Murai, Kunimochi Sakamoto and Shoji Yotsutani
2015(special): 878-900 doi: 10.3934/proc.2015.0878 +[Abstract](3443) +[PDF](800.3KB)
This paper deals with a derivative nonlinear Schrödinger equation under periodic boundary conditions. Taking advantage of the symmetries of the equation, we search for the traveling wave solutions. The problem is reduced to second order nonlinear nonlocal differential equations. By solving the equations, explicit formulas for the traveling waves are obtained. These formulas allow us to visualize the global structure of the traveling waves with various speeds and profiles.
Remarks on a dispersive equation in de Sitter spacetime
Makoto Nakamura
2015(special): 901-905 doi: 10.3934/proc.2015.0901 +[Abstract](2671) +[PDF](289.1KB)
Some nonlinear Schrödinger equations are derived from the nonrelativistic limit of nonlinear Klein-Gordon equations in de Sitter spacetime. Time local solutions for the Cauchy problem are considered in Sobolev spaces for power type nonlinear terms. The roles of spatial expansion and contraction on the problem are studied.
Oscillation results for second order nonlinear neutral differential equations with delay
Saroj Panigrahi and Rakhee Basu
2015(special): 906-912 doi: 10.3934/proc.2015.0906 +[Abstract](3108) +[PDF](298.2KB)
In this paper, oscillatory and asymptotic behavior of solutions of a class of nonlinear second order neutral differential equations with positive and negative coefficients of the form \begin{eqnarray*} (r_{1}(t)(x(t)+p_{1}(t)x(\tau(t)))^{\prime})^{\prime}+r_{2}(t)(x(t)+p_{2}(t)x(\sigma(t)))^{\prime} \\+p(t)G(x(\alpha(t)))-q(t)H(x(\beta(t)))=0 \end{eqnarray*} and \begin{eqnarray*} (r_{1}(t)(x(t)+p_{1}(t)x(\tau(t)))^{\prime})^{\prime}+r_{2}(t)(x(t)+p_{2}(t)x(\sigma(t)))^{\prime} \\+p(t)G(x(\alpha(t)))-q(t)H(x(\beta(t)))=f(t) \end{eqnarray*} are studied for various ranges of $p_{1}(t), p_{2}(t)$.
An in-host model of HIV incorporating latent infection and viral mutation
Stephen Pankavich and Deborah Shutt
2015(special): 913-922 doi: 10.3934/proc.2015.0913 +[Abstract](2892) +[PDF](594.0KB)
We construct a seven-component model of the in-host dynamics of the Human Immunodeficiency Virus Type-1 (i.e, HIV) that accounts for latent infection and the propensity of viral mutation. A dynamical analysis is conducted and a theorem is presented which characterizes the long time behavior of the model. Finally, we study the effects of an antiretroviral drug and treatment implications.
A functional-analytic technique for the study of analytic solutions of PDEs
Eugenia N. Petropoulou and Panayiotis D. Siafarikas
2015(special): 923-935 doi: 10.3934/proc.2015.0923 +[Abstract](2264) +[PDF](363.7KB)
A functional-analytic method is used to study the existence and the uniqueness of bounded, analytic and entire complex solutions of partial differential equations. As a benchmark problem, this method is applied to the nonlinear Benjamin--Bona--Mahony equation and the associated to this, linear equation. The predicted solutions are in power series form and two concrete examples are given for specific initial conditions.
Optimal design of sensors for a damped wave equation
Yannick Privat and Emmanuel Trélat
2015(special): 936-944 doi: 10.3934/proc.2015.0936 +[Abstract](2892) +[PDF](368.7KB)
In this paper we model and solve the problem of shaping and placing in an optimal way sensors for a wave equation with constant damping in a bounded open connected subset $\Omega$ of $\mathbb{R}^n$. Sensors are modeled by subdomains of $\Omega$ of a given measure $L|\Omega|$, with $0 < L < 1$. We prove that, if $L$ is close enough to $1$, then the optimal design problem has a unique solution, which is characterized by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.
Approximation and model order reduction for second order systems with Levy-noise
Martin Redmann and Peter Benner
2015(special): 945-953 doi: 10.3934/proc.2015.0945 +[Abstract](3932) +[PDF](379.2KB)
We consider a controlled second order stochastic partial differential equation (SPDE) with Levy noise. To solve this system numerically, we apply a Galerkin scheme leading to a sequence of ordinary SDEs of large order. To reduce the high dimension we use balanced truncation.
Infinitely many multi-pulses near a bifocal cycle
Alexandre A. P. Rodrigues
2015(special): 954-964 doi: 10.3934/proc.2015.0954 +[Abstract](2391) +[PDF](360.8KB)
The entire dynamics in a neighbourhood of a reversible heteroclinic cycle involving a bifocus is far from being understood. In this article, using the well known theory of reversing symmetries, we prove that there are infinitely many pulses near a cycle involving two symmetric equilibria, a real saddle and a bifocus, giving rise to a complex network. We also conjecture that suspended blenders might appear in the neighbourhood of the network.
Exact lumping of feller semigroups: A $C^{\star}$-algebras approach
Lavinia Roncoroni
2015(special): 965-973 doi: 10.3934/proc.2015.0965 +[Abstract](2365) +[PDF](309.4KB)
In this note we analyze a particular exact lumping of Feller semigroups in the context of $C^{\star}$-algebras, in order to pass from a space of functions defined on a locally compact Hausdorff space ${X}$ to a space of functions defined on a closed subspace ${\mathscr{C}}\subset X$. We want our reduction to preserve the essential properties of the Feller semigroup.
A model of malignant gliomas throug symmetry reductions
María Rosa, María S. Bruzón and M. L. Gandarias
2015(special): 974-980 doi: 10.3934/proc.2015.0974 +[Abstract](2463) +[PDF](264.2KB)
A glioma is a kind of tumor that starts in the brain or spine. The most common site of gliomas is in the brain. Most of the mathematical models in use for malignant gliomas are based on a simple reaction-diffusion equation: the Fisher equation [3]. A nonlinear wave model describing the fundamental features of these tumors has been introduced in [5], by V.M. Pérez and collaborators. In this work, we study this model from the point of view of the theory of symmetry reductions in partial differential equations. We obtain the classical symmetries admitted by the system, then, we use the transformations groups to reduce the equations to ordinary differential equations. Some exact solutions are derived from the solutions of a simple non-linear ordinary differential equation.
Symmetries and solutions of a third order equation
Júlio Cesar Santos Sampaio and Igor Leite Freire
2015(special): 981-989 doi: 10.3934/proc.2015.0981 +[Abstract](2711) +[PDF](371.1KB)
In this paper we study a new third order evolution equation discovered a couple of years ago using a genetic programming. We show that the Lie symmetries of this equation corresponds to space and time translations, as well as a dilation on the space of independent variables and another one with respect to the depend variable. From its symmetries, explicit solutions of the equation are obtained, some of them expressed in terms of the solutions of the Airy equation and Abel equation of the second kind. Additionally, by using the direct method we establish three conservation laws for the equation, one of them new.
Noether's theorem for higher-order variational problems of Herglotz type
Simão P. S. Santos, Natália Martins and Delfim F. M. Torres
2015(special): 990-999 doi: 10.3934/proc.2015.990 +[Abstract](3300) +[PDF](328.5KB)
We approach higher-order variational problems of Herglotz type from an optimal control point of view. Using optimal control theory, we derive a generalized Euler--Lagrange equation, transversality conditions, DuBois--Reymond necessary optimality condition and Noether's theorem for Herglotz's type higher-order variational problems, valid for piecewise smooth functions.
Absorbing boundary conditions for the Westervelt equation
Igor Shevchenko and Barbara Kaltenbacher
2015(special): 1000-1008 doi: 10.3934/proc.2015.1000 +[Abstract](3148) +[PDF](447.4KB)
The focus of this work is on the construction of a family of nonlinear absorbing boundary conditions for the Westervelt equation in one and two space dimensions. The principal ingredient used in the design of such conditions is pseudo-differential calculus. This approach enables to develop high order boundary conditions in a consistent way which are typically more accurate than their low order analogs. Under the hypothesis of small initial data, we establish local well-posedness for the Westervelt equation with the absorbing boundary conditions. The performed numerical experiments illustrate the efficiency of the proposed boundary conditions for different regimes of wave propagation.
Large-time behavior for a PDE model of isothermal grain boundary motion with a constraint
Ken Shirakawa and Hiroshi Watanabe
2015(special): 1009-1018 doi: 10.3934/proc.2015.1009 +[Abstract](3036) +[PDF](338.6KB)
In this paper, a system of parabolic initial-boundary value problems is considered as a possible PDE model of isothermal grain boundary motion. The solvability of this system was proved in [preprint, arXiv:1408.4204., by means of the notion of weighted total variation. In this light, we set our goal to prove two main theorems, which are concerned with the $ \Gamma $-convergence for time-dependent versions of the weighted total variations, and the large-time behavior of solution.
Nonlinear Schrödinger equations with inverse-square potentials in two dimensional space
Toshiyuki Suzuki
2015(special): 1019-1024 doi: 10.3934/proc.2015.1019 +[Abstract](3016) +[PDF](332.2KB)
Nonlinear Schrödinger equations with inverse-square potentials are considered in space dimension $N=2$. Stricharz estimates for (NLS)a are shown by Burq, Planchon, Stalker and Tahvildar-Zadeh [1] even when $N=2$. Here there seems not to be the study of solvability of (NLS)a when dimension is two. By virtue of the Hardy inequality the solvability is proved in Okazawa-Suzuki-Yokota, [3,4] if $N\ge 3$. Although strongly singular potential $a|x|^{-2}$ is available and the energy space is not exactly $H^{1}$ in (NLS)a, we can apply the energy methods established by Okazawa-Suzuki-Yokota [4].
Singular extremal solutions to a Liouville-Gelfand type problem with exponential nonlinearity
Futoshi Takahashi
2015(special): 1025-1033 doi: 10.3934/proc.2015.1025 +[Abstract](3136) +[PDF](334.6KB)
We consider a Liouville-Gelfand type problem \[ -\Delta u = e^u - 1 + \lambda f(x) \quad \text{in} \; \Omega, \quad u > 0 \quad \text{in} \; \Omega, \quad u = 0 \quad \text{on} \; \partial\Omega, \] where $\Omega \subset \mathbb{R}^N \; (N \ge 1)$ is a smooth bounded domain, $f \ge 0$, $f \not\equiv 0$ is a given smooth function, and $\lambda \ge 0$ is a parameter. We are concerned with the regularity property of extremal solutions to the problem, and prove that there exists a domain $\Omega$ and a smooth nonnegative function $f$ such that the extremal solution of the problem is singular when the dimension $N \ge 10$. This result is sharp in the sense that the extremal solution is always regular (bounded) for any $f$ and $\Omega$ when $1 \le N \le 9$.
Recurrence of multi-dimensional diffusion processes in Brownian environments
Hiroshi Takahashi and Yozo Tamura
2015(special): 1034-1040 doi: 10.3934/proc.2015.1034 +[Abstract](2734) +[PDF](292.5KB)
We consider limiting behavior of multi-dimensional diffusion processes in two types of Brownian environments. One is given values at different $d$ points of a one-dimensional Brownian motion, which is supposed to be a multi-parameter environment, and other is given by $d$ independent one-dimensional Brownian motions. We show recurrence of multi-dimensional diffusion processes in both Brownian environments above for any dimension and almost all environments. Their limiting behavior is quite different from that of ordinary multi-dimensional Brownian motion. We also consider cases of reflected Brownian environments.
Optimal portfolios based on weakly dependent data
Hiroshi Takahashi, Tatsuhiko Saigo, Shuya Kanagawa and Ken-ichi Yoshihara
2015(special): 1041-1049 doi: 10.3934/proc.2015.1041 +[Abstract](2409) +[PDF](335.7KB)
Let $\{\xi_k, k=1,2, \ldots\}$ be a strictly stationary sequence of centered $d$-dimensional random vectors satisfying the strong mixing condition. Using $\{\xi_k\}$, we consider a stochastic difference equation with a random volatility composed by $d$ stocks and a random trend and show a convergence theorem. In the one-dimensional case, the solution of this difference equation converges almost surely to a Black-Scholes type model. The purpose of this paper is to extend the results to multi-dimensional cases. Using the result, we obtain an approximations of $d$ stocks prices models with random volatilities. We also give examples of optimal portfolios for the models.
Imperfect bifurcations via topological methods in superlinear indefinite problems
Andrea Tellini
2015(special): 1050-1059 doi: 10.3934/proc.2015.1050 +[Abstract](3102) +[PDF](1272.8KB)
In [5] the structure of the bifurcation diagrams of a class of superlinear indefinite problems with a symmetric weight was ascertained, showing that they consist of a primary branch and secondary loops bifurcating from it. In [4] it has been proved that, when the weight is asymmetric, the bifurcation diagrams are no longer connected since parts of the primary branch and loops of the symmetric case form an arbitrarily high number of isolas. In this work we give a deeper insight on this phenomenon, studying how the secondary bifurcations break as the weight is perturbed from the symmetric situation. Our proofs rely on the approach of [5,4], i.e. on the construction of certain Poincaré maps and the study of how they vary as some of the parameters of the problems change, obtaining in this way the bifurcation diagrams.
Simulation of complex dynamics using POD 'on the fly' and residual estimates
Filippo Terragni and José M. Vega
2015(special): 1060-1069 doi: 10.3934/proc.2015.1060 +[Abstract](2611) +[PDF](1906.9KB)
Proper orthogonal decomposition (POD) is a very effective means to identify dynamical information contained in sets of snapshots that cover numerically computed trajectories of dissipative systems of partial differential equations. Such information is organized in a hierarchy of POD modes and the system can be Galerkin-projected onto the associated linear subspace. Quite frequently, the outcome is a low dimensional model of the problem. Flexibility and efficiency of the approximation can be enhanced if POD is applied `on the fly', adaptively combining a standard numerical solver (which provides the necessary snapshots) with the reduced system in interspersed intervals, as both time and a bifurcation parameter are varied. Residual estimates are introduced to make this adaptation accurate and robust, preventing possible mode truncation instabilities in the presence of complex dynamics. All ideas are illustrated in some bifurcation scenarios including quasi-periodic and chaotic attractors, which highlights a good computational efficiency.
A new way for decreasing of amplitude of wave reflected from artificial boundary condition for 1D nonlinear Schrödinger equation
Vyacheslav A. Trofimov and Evgeny M. Trykin
2015(special): 1070-1078 doi: 10.3934/proc.2015.1070 +[Abstract](3038) +[PDF](334.1KB)
In this report, we focus on computation performance enhancing at computer simulation of a laser pulse interaction with optical periodic structure (photonic crystal). Decreasing the domain before the photonic crystal one can essentially increases a computation performance. With this aim, firstly, we provide a computation in linear medium and storage the complex amplitude at chosen section of coordinate along which a laser light propagates. Then we use this time-dependent value of complex amplitude as left boundary condition for the 1D nonlinear Schrödinger equation in decreased domain containing a nonlinear photonic crystal. Because a part of a laser pulse reflects from faces of photonic crystal layers, we use artificial boundary condition. To decrease amplitude of the wave reflected from artificial boundary we introduce in consideration some additional number of waves related with the equation under consideration.
Solvability of generalized nonlinear heat equations with constraints coupled with Navier--Stokes equations in 2D domains
Yutaka Tsuzuki
2015(special): 1079-1088 doi: 10.3934/proc.2015.1079 +[Abstract](2487) +[PDF](394.4KB)
This paper is concerned with a system of nonlinear heat equations with constraints coupled with Navier--Stokes equations in two-dimensional domains. In 2012, Sobajima, the author and Yokota proved existence and uniqueness of solutions to the system with heat equations with the linear diffusion term $\Delta\theta$ and the nonlinear term $|\theta|^{q-1}\theta$. Recently, the author generalized the result for the equation with the $p$-Laplace operator $\Delta p$ and the logistic nonlinear term $|\theta|^{q-1}\theta - \alpha\theta$. This paper gives an existence result for the equation with $\Delta p$ and the more general nonlinear term $h(x,\theta)-\alpha\theta$ depending on the spacial variable $x$.
Direct scattering of AKNS systems with $L^2$ potentials
Cornelis van der Mee
2015(special): 1089-1097 doi: 10.3934/proc.2015.1089 +[Abstract](2470) +[PDF](341.9KB)
In this article the Jost solutions of the AKNS system with suitably weighted $L^2$ potential are constructed as Hardy space perturbations of their space-infinity asymptotics. The reflection coefficients are proven to be $L^2$-functions when the transmission coefficients are $L^\infty$-functions.
The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori
Dmitriy Yu. Volkov
2015(special): 1098-1104 doi: 10.3934/proc.2015.1098 +[Abstract](3677) +[PDF](287.4KB)
A dissipative Hopf -- Hopf bifurcation with 2 :1 resonance are studied. A parameter dependent polynomial truncated normal form is derived. We study this truncated normal form. This system displays a large variety of behaviour both regular and chaotic solution. Existence of the periodic solutions and invariant tori of full system are proved. Analogy between dissipative Hopf - Hopf bifurcation with 2:1 resonance, generations of second harmonics in non-linear optics and resonant interaction of waves in a plasma is presented.
Blow-up of solutions to semilinear wave equations with non-zero initial data
Kyouhei Wakasa
2015(special): 1105-1114 doi: 10.3934/proc.2015.1105 +[Abstract](2892) +[PDF](333.4KB)
In this paper, we are concerned with the initial value problem for $u_{tt}-\Delta u=|u_t|^p$ in $\mathbb{R}^n\times[0,\infty)$ with the initial data $u(x,0)=f(x)$, $u_t(x,0)=g(x)$, where $(f,g)$ are slowly decaying.
    H. Takamura [13] obtained the blow-up result for the case where $f\equiv0$ and $g\not\equiv0$. Our purpose in this paper is to show the blow-up result for the case where the both initial data do not vanish identically.
Classification of periodic orbits in the planar equal-mass four-body problem
Duokui Yan, Tiancheng Ouyang and Zhifu Xie
2015(special): 1115-1124 doi: 10.3934/proc.2015.1115 +[Abstract](3279) +[PDF](705.7KB)
In the N-body problem, many periodic orbits are found as local Lagrangian action minimizers. In this work, we classify such periodic orbits in the planar equal-mass four-body problem. Specific planar configurations are considered: line, rectangle, diamond, isosceles trapezoid, double isosceles, kite, etc. Periodic orbits are classified into 8 categories and each category corresponds to a pair of specific configurations. Furthermore, it helps discover several new sets of periodic orbits.
Existence of solutions to chemotaxis dynamics with logistic source
Tomomi Yokota and Noriaki Yoshino
2015(special): 1125-1133 doi: 10.3934/proc.2015.1125 +[Abstract](3356) +[PDF](350.3KB)
This paper is concerned with a chemotaxis system with nonlinear diffusion and logistic growth term $f(b) = \kappa b-\mu |b|^{\alpha-1}b$ with $\kappa>0$, $\mu>0$ and $\alpha > 1$ under the no-flux boundary condition. It is shown that there exists a local solution to this system for any $L^2$-initial data and that under a stronger assumption on the chemotactic sensitivity there exists a global solution for any $L^2$-initial data. The proof is based on the method built by Marinoschi [8].
Pullback uniform dissipativity of stochastic reversible Schnackenberg equations
Yuncheng You
2015(special): 1134-1142 doi: 10.3934/proc.2015.1134 +[Abstract](2635) +[PDF](349.5KB)
Asymptotic dynamics of stochastic reversible Schnackenberg equations with multiplicative white noise on a three-dimensional bounded domain is investigated in this paper. The pullback uniform dissipativity in terms of the existence of a common pullback absorbing set with respect to the reverse reaction rate of this typical autocatalytic reaction-diffusion system is proved through decomposed grouping estimates.

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