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2013

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Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise
Abiti Adili and Bixiang Wang
2013(special): 1-10 doi: 10.3934/proc.2013.2013.1 +[Abstract](396) +[PDF](335.6KB)
Abstract:
In this paper, we prove the existence and uniqueness of random attractors for the FitzHugh-Nagumo system defined on $\mathbb{R}^n$ driven by both deterministic non-autonomous forcing and multiplicative noise. The periodicity of random attractors is established when the system is perturbed by time periodic forcing. We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero.
Asymptotic behavior of a ratio-dependent predator-prey system with disease in the prey
Inkyung Ahn, Wonlyul Ko and Kimun Ryu
2013(special): 11-19 doi: 10.3934/proc.2013.2013.11 +[Abstract](341) +[PDF](316.9KB)
Abstract:
In this paper, we consider a ratio-dependent predator-prey model with disease in the prey under Neumann boundary condition. we construct a global attractor region for all time-dependent non-negative solutions of the system and investigate the asymptotic behavior of positive constant solution. Furthermore, we also study the asymptotic behavior of the non-negative equilibria.
Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem
Inmaculada Antón and Julián López-Gómez
2013(special): 21-30 doi: 10.3934/proc.2013.2013.21 +[Abstract](287) +[PDF](329.0KB)
Abstract:
This paper studies the existence of coexistence states in a spatially heterogeneous reaction diffusion system arising in nuclear dynamics. Essentially, it establishes the existence of an unbounded component $\mathfrak{C}_+$ of the set of coexistence states of the system bifurcating from the trivial steady state solution, and it characterizes the values of the parameters where $\mathfrak{C}_+$ bifurcates from the trivial solution and from infinity. Throughout this paper, by a component it is meant a closed and connected subset which is maximal for the inclusion.
Classification of positive solutions of semilinear elliptic equations with Hardy term
Soohyun Bae
2013(special): 31-39 doi: 10.3934/proc.2013.2013.31 +[Abstract](458) +[PDF](393.7KB)
Abstract:
We study the elliptic equation $\Delta u+\mu/|x|^2+K(|x|)u^p=0$ in $\mathbb{R}^n \setminus \left \{ 0 \right \}$, where $n\geq1$ and $p>1$. In particular, when $K(|x|)=|x|^l$, a classification of radially symmetric solutions is presented in terms of $\mu$ and $l$. Moreover, we explain the separation structure for the equation, and study the stability of positive radial solutions as steady states.
Radial solutions of semilinear elliptic equations with broken symmetry on unbounded domains
Sara Barile and Addolorata Salvatore
2013(special): 41-49 doi: 10.3934/proc.2013.2013.41 +[Abstract](346) +[PDF](350.2KB)
Abstract:
We study a superlinear perturbed elliptic problem on $\mathbb R^N$ with rotational symmetry. Using variational and perturbative methods we find infinitely many radial solutions for any growth exponent $p$ of the nonlinearity greater than $2$ and less than $2^*$ if $N \geq 4$ and for any $p$ greater than $3$ and subcritical if $N =3$.
Infinitely many radial solutions of a non--homogeneous $p$--Laplacian problem
Rossella Bartolo, Anna Maria Candela and Addolorata Salvatore
2013(special): 51-59 doi: 10.3934/proc.2013.2013.51 +[Abstract](313) +[PDF](381.7KB)
Abstract:
In this paper we investigate the existence of infinitely many radial solutions for the elliptic Dirichlet problem \[ \left\{ \begin{array}{ll} \displaystyle{-\Delta_p u\ =|u|^{q-2}u + f(x)} & \mbox{ in } B_R,\\ \displaystyle{u=\xi} & \mbox{ on } \partial B_R,\\ \end{array} \right. \] where $B_R$ is the open ball centered in $0$ with radius $R$ in $\mathbb{R}^N$ ($N \geq 3$), $2 < p < N$, $p< q < p^*$ (with $p^* = \frac{pN}{N-p}$), $\xi\in\mathbb{R}$ and $f$ is a continuous radial function in $\overline B_R$. The lack of even symmetry for the related functional is overcome by using some perturbative methods and the radial assumptions allow us to improve some previous results.
R&d dynamics
J. Becker, M. Ferreira, B.M.P.M. Oliveira and A.A. Pinto
2013(special): 61-68 doi: 10.3934/proc.2013.2013.61 +[Abstract](286) +[PDF](651.8KB)
Abstract:
We study a Cournot duopoly model using Ferreira-Oliveira-Pinto's R&D investment function. We find the multiple perfect Nash equilibria and we analyse the economical relevant quantities like output levels, prices, consumer surplus, profits and welfare.
Modeling confinement in Étang de Thau: Numerical simulations and multi-scale aspects
Jean-Philippe Bernard, Emmanuel Frénod and Antoine Rousseau
2013(special): 69-76 doi: 10.3934/proc.2013.2013.69 +[Abstract](368) +[PDF](3942.6KB)
Abstract:
The main purpose of this paper is to implement the mathematical model of confinement proposed in [8] in an actual lagoon, here the Étang de Thau, in southern France.
An iterative method for the canard explosion in general planar systems
Morten Brøns
2013(special): 77-83 doi: 10.3934/proc.2013.2013.77 +[Abstract](228) +[PDF](329.3KB)
Abstract:
The canard explosion is the change of amplitude and period of a limit cycle born in a Hopf bifurcation in a very narrow parameter interval. The phenomenon is well understood in singular perturbation problems where a small parameter controls the slow/fast dynamics. However, canard explosions are also observed in systems where no such parameter can obviously be identified. Here we show how the iterative method of Roussel and Fraser, devised to construct regular slow manifolds, can be used to determine a canard point in a general planar system of nonlinear ODEs. We demonstrate the method on the van der Pol equation, showing that the asymptotics of the method is correct, and on a templator model for a self-replicating system.
The homogenization of the heat equation with mixed conditions on randomly subsets of the boundary
Carmen Calvo-Jurado, Juan Casado-Díaz and Manuel Luna-Laynez
2013(special): 85-94 doi: 10.3934/proc.2013.2013.85 +[Abstract](297) +[PDF](361.9KB)
Abstract:
We consider a domain in $\mathbb{R}^N$, $N\geq 3$, such that a portion of its boundary is plane. In this portion we fix a sequence $K_\epsilon$ of small subsets randomly distributed in such way that the distance between them is of order $\epsilon$ and their diameters are of order $\epsilon^\frac{N-1}{N-2}$. We study the asymptotic behavior of the heat equation with Dirichlet conditions on $K_\epsilon$ and Neumann conditions on the rest of the boundary. We prove the convergence to a limit problem with a Fourier-Robin boundary condition which has the physical interest of being deterministic.
Bifurcation to positive solutions in BVPs of logistic type with nonlinear indefinite mixed boundary conditions
Santiago Cano-Casanova
2013(special): 95-104 doi: 10.3934/proc.2013.2013.95 +[Abstract](290) +[PDF](309.4KB)
Abstract:
In this paper a nonlinear boundary value problem of logistic type is considered, with nonlinear mixed boundary conditions, and with spatial heterogeneities of arbitrary sign in the differential equation and on the boundary conditions. The main goal of this paper is analyzing the structure of the continuum of positive solutions emanating from the trivial state at a unique bifurcation value, depending on the size and sign of the different potentials and parameters of the problem. The results in this paper extend the previous ones obtained by R. Gómez-Reñasco and J. López-Gómez [5, Proposition 2.1], for a superlinear indefinite problem of logistic type under Dirichlet boundary conditions, to a wide class of superlinear indefinite problems with nonlinear indefinite mixed boundary conditions.
New class of exact solutions for the equations of motion of a chain of $n$ rigid bodies
Dmitriy Chebanov
2013(special): 105-113 doi: 10.3934/proc.2013.2013.105 +[Abstract](241) +[PDF](322.3KB)
Abstract:
In this paper we construct a new class of nonstationary exact solutions for the equations of motion of a classical model of multibody dynamics -- a chain of $n$ heavy rigid bodies that are sequentially coupled by ideal spherical hinges. We establish sufficient conditions for the existence of the solutions and show how the equations of motion can be reduced to quadratures in the case when these conditions are fulfilled.
Stochastic geodesics and forward-backward stochastic differential equations on Lie groups
Xin Chen and Ana Bela Cruzeiro
2013(special): 115-121 doi: 10.3934/proc.2013.2013.115 +[Abstract](357) +[PDF](288.3KB)
Abstract:
We describe how to generalize to the stochastic case the notion of geodesic on a Lie group equipped with an invariant metric. As second order equations (in time), stochastic geodesics are characterized in terms of stochastic forward-backward differential systems.
    When the group is the diffeomorphisms group this corresponds to a probabilistic description of the Navier-Stokes equations.
On the uniqueness of singular solutions for a Hardy-Sobolev equation
Jann-Long Chern, Yong-Li Tang, Chuan-Jen Chyan and Yi-Jung Chen
2013(special): 123-128 doi: 10.3934/proc.2013.2013.123 +[Abstract](278) +[PDF](318.1KB)
Abstract:
In this paper, we consider the positive singular solutions for the following Hardy-Sobolev equation
                        $\Delta u+u^p+\frac{u^{2^*(s)-1}}{|x|^s}=0 $      in    $B_1 \setminus \left \{ 0 \right \},$
where $p>1, 0 < s < 2, 2^*(s)=\frac{2(n-s)}{n-2}$, $n\geq 3$ and $B_1$ is the unit ball in $ R^n$ centered at the origin. We prove that if $p>\frac{n+2}{n-2}$ then such solution is unique.
Numerical optimal unbounded control with a singular integro-differential equation as a constraint
Shihchung Chiang
2013(special): 129-137 doi: 10.3934/proc.2013.2013.129 +[Abstract](361) +[PDF](293.9KB)
Abstract:
This study presents a discussion of numerical methods for optimal control using an integro-differential equation of singular kernel as a constraint. The proposed scheme attempts to set the objective to minimize the gap between optimal state and target function for certain period of time. By assuming that control is unbounded, this study proposes a method of feedback correction that makes correction each step for optimal control. These corrections are proportional to the corresponding state-target distance until a certain accuracy criterion is satisfied. There are several advantages to this method, including user-decided accuracy, user-decided number of iterations, and time saving. This study presents a comparison of the numerical results with the results of other methods [4],[7].
Investigation of the long-time evolution of localized solutions of a dispersive wave system
C. I. Christov and M. D. Todorov
2013(special): 139-148 doi: 10.3934/proc.2013.2013.139 +[Abstract](321) +[PDF](1220.7KB)
Abstract:
We consider the long-time evolution of the solution of an energy-consistent system with dispersion and nonlinearity, which is the progenitor of the different Boussinesq equations. Unlike the classical Boussinesq models, the energy-consistent one possesses Galilean invariance. As initial condition we use the superposition of two analytical one-soliton solutions. We use a strongly dynamical implicit difference scheme with internal iterations, which allows us to follow the evolution of the solution at very long times. We focus on the behavior of traveling localized solutions developing from critical initial data. The main solitary waves appear virtually undeformed from the interaction, but additional oscillations are excited at the trailing edge of each one of them. We track their evolution for very long times when they tend to adopt a self-similar shape. We test a hypothesis about the dependence on time of the amplitude and the support of Airy-function shaped coherent structures. The investigation elucidates the mechanism of evolution of interacting solitary waves in the energy-consistent Boussinesq equation.
Optimal control of underactuated mechanical systems with symmetries
Leonardo Colombo and David Martín de Diego
2013(special): 149-158 doi: 10.3934/proc.2013.2013.149 +[Abstract](300) +[PDF](380.0KB)
Abstract:
The aim of this paper is to study optimal control problems for underactuated mechanical systems with symmetries using higher-order Lagrangian mechanics. We variationally derive the corresponding Lagrange -Poincaré equations for second-order Lagrangians with constraints defined on trivial principal bundles and apply them to study an optimal control problem for an underactuated vehicle.
On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space
Chiara Corsato, Franco Obersnel, Pierpaolo Omari and Sabrina Rivetti
2013(special): 159-169 doi: 10.3934/proc.2013.2013.159 +[Abstract](357) +[PDF](344.4KB)
Abstract:
We develop a lower and upper solution method for the Dirichlet problem associated with the prescribed mean curvature equation in Minkowski space \begin{equation*} \begin{cases} -{\rm div}\Big( \nabla u /\sqrt{1 - |\nabla u|^2}\Big)= f(x,u) & \hbox{ in } \Omega, \\ u=0& \hbox{ on } \partial \Omega. \end{cases} \end{equation*} Here $\Omega$ is a bounded regular domain in $\mathbb {R}^N$ and the function $f$ satisfies the Carathéodory conditions. The obtained results display various peculiarities due to the special features of the involved differential operator.
Analysis of the accelerated weighted ensemble methodology
Ronan Costaouec, Haoyun Feng, Jesús Izaguirre and Eric Darve
2013(special): 171-181 doi: 10.3934/proc.2013.2013.171 +[Abstract](295) +[PDF](508.6KB)
Abstract:
The main issue addressed in this note is the study of an algorithm to accelerate the computation of kinetic rates in the context of molecular dynamics (MD). It is based on parallel simulations of short-time trajectories and its main components are: a decomposition of phase space into macrostates or cells, a resampling procedure that ensures that the number of parallel replicas (MD simulations) in each macro-state remains constant, the use of multiple populations (colored replicas) to compute multiple rates (e.g., forward and backward rates) in one simulation. The method leads to enhancing the sampling of macro-states associated to the transition states, since in traditional MD these are likely to be depleted even after short-time simulations. By allowing parallel replicas to carry different probabilistic weights, the number of replicas within each macro-state can be maintained constant without introducing any bias. The empirical variance of the estimated reaction rate, defined as a probability flux, is expectedly diminished. This note is a first attempt towards a better mathematical and numerical understanding of this method. It provides first a mathematical formalization of the notion of colors. Then, the link between this algorithm and a set of closely related methods having been proposed in the literature within the past few years is discussed. Lastly, numerical results are provided that illustrate the efficiency of the method.
Small data solutions for semilinear wave equations with effective damping
Marcello D'Abbicco
2013(special): 183-191 doi: 10.3934/proc.2013.2013.183 +[Abstract](219) +[PDF](346.8KB)
Abstract:
We consider the Cauchy problem for the semi-linear damped wave equation
    $ u_{tt} - \Delta u + b(t)u_t = f(t,u),\qquad u(0,x) = u_0(x),\qquad u_t(0,x) = u_1(x). $
We prove the global existence of small data solution in low space dimension, and we derive $(L^m\cap L^2)-L^2$ decay estimates, for $m\in[1,2)$. We assume that the time-dependent damping term $b(t)>0$ is effective, that is, the equation inherits some properties of the parabolic equation $b(t)u_t - \Delta u = f(t,u)$.
A unique positive solution to a system of semilinear elliptic equations
Diane Denny
2013(special): 193-195 doi: 10.3934/proc.2013.2013.193 +[Abstract](409) +[PDF](222.5KB)
Abstract:
We study a system of semilinear elliptic equations that arises from a predator-prey model. Previous related work proved the existence of a unique positive solution to this system of equations in the special case in which the parameter $\alpha=0$ in this system of equations, provided that a positive parameter $\kappa$ in this system of equations is sufficiently large. We prove the existence of a unique positive solution to this system of equations for any $\alpha \geq 0$ and for any $\kappa>0$.
Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity
Priyanjana M. N. Dharmawardane
2013(special): 197-206 doi: 10.3934/proc.2013.2013.197 +[Abstract](340) +[PDF](334.2KB)
Abstract:
In this paper, we consider a quasi-linear hyperbolic systems of viscoelasticity. This system has dissipative properties of the memory type and the friction type. The decay property of this system is of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted energy method. Moreover, we combine this time-weighted energy method with the semigroup argument to obtain the global existence and sharp decay estimate of solutions under the smallness conditions and enough regularity assumptions on the initial data.
An approximation model for the density-dependent magnetohydrodynamic equations
Jishan Fan and Tohru Ozawa
2013(special): 207-216 doi: 10.3934/proc.2013.2013.207 +[Abstract](291) +[PDF](314.3KB)
Abstract:
The global Cauchy problem for an approximation model for the density-dependent MHD system is studied. The vanishing limit on $\alpha$ is also discussed.
The role of lower and upper solutions in the generalization of Lidstone problems
João Fialho and Feliz Minhós
2013(special): 217-226 doi: 10.3934/proc.2013.2013.217 +[Abstract](293) +[PDF](320.2KB)
Abstract:
In this the authors consider the nonlinear fully equation
          \begin{equation*} u^{(iv)} (x) + f( x,u(x) ,u^{\prime}(x) ,u^{\prime \prime}(x) ,u^{\prime \prime \prime}(x) ) = 0 \end{equation*} for $x\in [ 0,1] ,$ where $f:[ 0,1] \times \mathbb{R} ^{4} \to \mathbb{R}$ is a continuous functions, coupled with the Lidstone boundary conditions, \begin{equation*} u(0) = u(1) = u^{\prime \prime}(0) = u^{\prime \prime }(1) = 0. \end{equation*}
    They discuss how different definitions of lower and upper solutions can generalize existence and location results for boundary value problems with Lidstone boundary data. In addition, they replace the usual bilateral Nagumo condition by a one-sided condition, allowing the nonlinearity to be unbounded$.$ An example will show that this unilateral condition generalizes the usual one and stress the potentialities of the new definitions.
A reinjected cuspidal horseshoe
Marcus Fontaine, William D. Kalies and Vincent Naudot
2013(special): 227-236 doi: 10.3934/proc.2013.2013.227 +[Abstract](253) +[PDF](3066.0KB)
Abstract:
Horseshoes play a central role in dynamical systems and are observed in many chaotic systems. However most points in a neighborhood of the horseshoe escape after finite iterations. In this work we construct a model that possesses an attracting set that contains a cuspidal horseshoe with positive entropy. This model is obtained by reinjecting the points that escape the horseshoe and can be realized in a 3-dimensional vector field.
Abstract theory of variational inequalities and Lagrange multipliers
Takeshi Fukao and Nobuyuki Kenmochi
2013(special): 237-246 doi: 10.3934/proc.2013.2013.237 +[Abstract](294) +[PDF](324.2KB)
Abstract:
In this paper, the existence and uniqueness questions of abstract parabolic variational inequalities are considered in connection with Lagrange multipliers. The focus of authors' attention is the characterization of parabolic variational inequalities by means of Lagrange multipliers. It is well-known that various kinds of parabolic differential equations under convex constraints are represented by variational inequalities with time-dependent constraints, and the usage of Lagrange multipliers associated with constraints makes it possible to reformulate the variational inequalities as equations. In this paper, as a typical case, a parabolic problem with nonlocal time-dependent obstacle is treated in the framework of abstract evolution equations governed by time-dependent subdifferentials.
Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator
Charles Fulton, David Pearson and Steven Pruess
2013(special): 247-257 doi: 10.3934/proc.2013.2013.247 +[Abstract](278) +[PDF](165.9KB)
Abstract:
In this paper we give a first order system of difference equations which provides a useful companion system in the study of Jacobi matrix operators and make use of it to obtain a characterization of the spectral density function for a simple case involving absolutely continuous spectrum on the stability intervals.
Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space
Matthew A. Fury
2013(special): 259-272 doi: 10.3934/proc.2013.2013.259 +[Abstract](335) +[PDF](419.7KB)
Abstract:
We prove regularization for ill-posed evolution problems that are both inhomogeneous and nonautonomous in a Hilbert Space $H$. We consider the ill-posed problem $du/dt = A(t,D)u(t)+h(t)$, $u(s)=\chi$, $0\leq s \leq t< T$ where $A(t,D)=\sum_{j=1}^ka_j(t)D^j$ with $a_j\in C([0,T]:\mathbb{R}^+)$ for each $1\leq j\leq k$ and $D$ a positive, self-adjoint operator in $H$. Assuming there exists a solution $u$ of the problem with certain stabilizing conditions, we approximate $u$ by the solution $v_{\beta}$ of the approximate well-posed problem $dv/dt = f_{\beta}(t,D)v(t)+h(t)$, $v(s)=\chi$, $0\leq s \leq t< T$ where $0<\beta <1$. Our method implies the existence of a family of regularizing operators for the given ill-posed problem with applications to a wide class of ill-posed partial differential equations including the inhomogeneous backward heat equation in $L^2(\mathbb{R}^n)$ with a time-dependent diffusion coefficient.
Existence of nontrivial solutions to systems of multi-point boundary value problems
John R. Graef, Shapour Heidarkhani and Lingju Kong
2013(special): 273-281 doi: 10.3934/proc.2013.2013.273 +[Abstract](327) +[PDF](368.8KB)
Abstract:
In this paper, sufficient conditions are established for the existence of at least one nontrivial solution of the multi-point boundary value system $$ \left\{\begin{array}{ll} -(\phi_{p_i}(u'_{i}))'=\lambda F_{u_{i}}(x,u_{1},\ldots,u_{n}),\ x\in(0,1),\\ u_{i}(0)=\sum_{j=1}^m a_ju_i(x_j),\ u_{i}(1)=\sum_{j=1}^m b_ju_i(x_j), \end{array} \right. i=1,\ldots,n. $$ The approach is based on variational methods and critical point theory.
Positive solutions of nonlocal fractional boundary value problems
John R. Graef, Lingju Kong, Qingkai Kong and Min Wang
2013(special): 283-290 doi: 10.3934/proc.2013.2013.283 +[Abstract](339) +[PDF](306.2KB)
Abstract:
The authors study a type of nonlinear fractional boundary value problem with nonlocal boundary conditions. An associated Green's function is constructed. Then a criterion for the existence of at least one positive solution is obtained by using fixed point theory on cones.
Existence of multiple solutions to a discrete fourth order periodic boundary value problem
John R. Graef, Lingju Kong and Min Wang
2013(special): 291-299 doi: 10.3934/proc.2013.2013.291 +[Abstract](394) +[PDF](384.1KB)
Abstract:
Sufficient conditions are obtained for the existence of multiple solutions to the discrete fourth order periodic boundary value problem \begin{equation*} \begin{array}{l} \Delta^4 u(t-2)-\Delta(p(t-1)\Delta u(t-1))+q(t) u(t)=f(t,u(t)),\quad t\in [1,N]_{\mathbb{Z}},\\ \Delta^iu(-1)=\Delta^iu(N-1),\quad i=0, 1,2, 3. \end{array} \end{equation*} Our analysis is mainly based on the variational method and critical point theory. One example is included to illustrate the result.
Optimization problems for the energy integral of p-Laplace equations
Antonio Greco and Giovanni Porru
2013(special): 301-310 doi: 10.3934/proc.2013.2013.301 +[Abstract](307) +[PDF](350.0KB)
Abstract:
We study maximization and minimization problems for the energy integral of a sub-linear $p$-Laplace equation in a domain $\Omega$, with weight $\chi_D$, where $D\subset\Omega$ is a variable subset with a fixed measure $\alpha$. We prove Lipschitz continuity for the energy integral of a maximizer and differentiability for the energy integral of the minimizer with respect to $\alpha$.
An optimal control problem in HIV treatment
Ellina Grigorieva, Evgenii Khailov and Andrei Korobeinikov
2013(special): 311-322 doi: 10.3934/proc.2013.2013.311 +[Abstract](392) +[PDF](288.4KB)
Abstract:
We consider a three-dimensional nonlinear control model, which describes the dynamics of HIV infection with nonlytic immune response and possible effects of controllable medication intake on HIV-infected patients. This model has the following phase variables: populations of the infected and uninfected cells and the concentration of an antiviral drug. The medication intake rate is chosen to be a bounded control function. The optimal control problem of minimizing the infected cells population at the terminal time is stated and solved. The types of the optimal control for different model parameters are obtained analytically. This allowed us to reduce the two-point boundary value problem for the Pontryagin Maximum Principle to one of the finite dimensional optimization. Numerical results are presented to demonstrate the optimal solution.
Fast iteration of cocycles over rotations and computation of hyperbolic bundles
Gemma Huguet, Rafael de la Llave and Yannick Sire
2013(special): 323-333 doi: 10.3934/proc.2013.2013.323 +[Abstract](290) +[PDF](788.9KB)
Abstract:
We present numerical algorithms that use small requirements of storage and operations to compute the iteration of cocycles over a rotation. We also show that these algorithms can be used to compute efficiently the stable and unstable bundles and the Lyapunov exponents of the cocycle.
$L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems
Sachiko Ishida
2013(special): 335-344 doi: 10.3934/proc.2013.2013.335 +[Abstract](326) +[PDF](411.9KB)
Abstract:
This paper deals with quasilinear degenerate Keller-Segel systems of parabolic-elliptic type. In this type, Sugiyama-Kunii [10] established the $L^r$-decay ($1\leq r<\infty$) of solutions with small initial data when $q\geq m+\frac{2}{N}$ ($m$ denotes the intensity of diffusion and $q$ denotes the nonlinearity). However, the $L^\infty$-decay property was not obtained yet. This paper gives the $L^\infty$-decay property in the super-critical case with small initial data.
Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems
Sachiko Ishida and Tomomi Yokota
2013(special): 345-354 doi: 10.3934/proc.2013.2013.345 +[Abstract](274) +[PDF](358.7KB)
Abstract:
The global existence of weak solutions to quasilinear ``degenerate'' Keller-Segel systems is shown in the recent papers [3], [4]. This paper gives some improvements and supplements of these. More precisely, the differentiability and the smallness of initial data are weakened when the spatial dimension $N$ satisfies $N\geq2$. Moreover, the global existence is established in the case $N=1$ which is unsolved in [4].
Nonpolynomial spline finite difference scheme for nonlinear singuiar boundary value problems with singular perturbation and its mechanization
Navnit Jha
2013(special): 355-363 doi: 10.3934/proc.2013.2013.355 +[Abstract](429) +[PDF](138.9KB)
Abstract:
A general scheme for the numerical solution of nonlinear singular perturbation problems using nonpolynomial spline basis is proposed in the paper. The special non-equidistant formulation of mesh takes into account the boundary and interior layer structures. The proposed scheme is almost fourth order accurate and applicable to both singular and nonsingular cases. Convergence analysis of the scheme is briefly discussed. Maple program for the generation of difference scheme is presented. Computational illustrations characterized by boundary and interior layers show that the practical order of accuracy is close to the theoretical order of the method.
Regularity of a vector valued two phase free boundary problems
Huiqiang Jiang
2013(special): 365-374 doi: 10.3934/proc.2013.2013.365 +[Abstract](345) +[PDF](326.3KB)
Abstract:
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$, $n\geq2$ and $\Sigma$ be a $q$ dimensional smooth submanifold of $\mathbb{R}^{m}$ with $0 \leq q < m$. We use $\mathcal{M}_{\Omega,\Sigma}$ to denote the collection of all pairs of $(A,u) $ such that $A\subset\Omega$ is a set of finite perimeter and $u\in H^{1}\left( \Omega,\mathbb{R}^{m}\right) $ satisfies \[ u\left( x\right) \in\Sigma\text{ a.e. }x\in A. \] We consider the energy functional \[ E_{\Omega}\left( A,u\right) =\int_{\Omega}\left\vert \nabla u\right\vert ^{2}+P_{\Omega}\left( A\right) , \] defined on $\mathcal{M}_{\Omega,\Sigma}$, where $P_{\Omega}\left( A\right) $ denotes the perimeter of $A$ inside $\Omega$. Let $\left( A,u\right) $ be a local energy minimizer. Our main result is that when $n\leq7$, $u$ is locally Lipschitz and the free boundary $\partial A$ is smooth in $\Omega$.
Finite-dimensional behavior in a thermosyphon with a viscoelastic fluid
A. Jiménez-Casas, Mario Castro and Justine Yassapan
2013(special): 375-384 doi: 10.3934/proc.2013.2013.375 +[Abstract](315) +[PDF](345.4KB)
Abstract:
We analyse the motion of a viscoelastic fluid in the interior of a closed loop thermosyphon under the effects of natural convection. We consider a viscoelastic fluid described by the Maxwell constitutive equation. This fluid presents elastic-like behavior and memory effects. We study the asymptotic properties of the fluid inside the thermosyphon and derive the exact equations of motion in the inertial manifold that characterize the asymptotic behavior. Our work is a generalization of some previous results on standard (Newtonian) fluids.
A unified approach to Matukuma type equations on the hyperbolic space or on a sphere
Yoshitsugu Kabeya
2013(special): 385-391 doi: 10.3934/proc.2013.2013.385 +[Abstract](267) +[PDF](324.6KB)
Abstract:
In this note, we consider the following Matukuma type equation on the hyperbolic space or on a spherical cap of the unit sphere under the homogeneous Dirichlet boundary condition \begin{eqnarray*} \Lambda u+K(x)u_+^{p}=0, \end{eqnarray*} where $\Lambda$ is the Laplace-Beltrami operator on the hyperbolic space or on the unit sphere. Under suitable assumptions on $K$, we determine the structure of positive solutions.
The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems
Byungik Kahng and Miguel Mendes
2013(special): 393-406 doi: 10.3934/proc.2013.2013.393 +[Abstract](308) +[PDF](397.1KB)
Abstract:
The main topic of this paper is the controllability/reachability problems of the maximal invariant sets of non-linear discrete-time multiple-valued iterative dynamical systems. We prove that the controllability/reachability problems of the maximal full-invariant sets of classical control dynamical systems are equivalent to those of the maximal quasi-invariant sets of disturbed control dynamical systems, when modeled by the iterative dynamics of multiple-valued self-maps. Also, we prove that the afore-mentioned maximal full-invariant sets and maximal quasi-invariant sets are countably infinite step controllable under some appropriate conditions. We take an abstract set theoretical approach, so that our main theorems remain valid regardless of the topological structure of the space or the analytical structure of the dynamics.
Asymptotic behavior of solutions to a coupled system of Maxwell's equations and a controlled differential inclusion
Dina Kalinichenko, Volker Reitmann and Sergey Skopinov
2013(special): 407-414 doi: 10.3934/proc.2013.2013.407 +[Abstract](473) +[PDF](321.1KB)
Abstract:
The present article consists of two parts. In the first part we consider evolutionary variational inequalities with a nonlinearity which is described by a differential inclusion. Using the frequency-domain method we prove, under certain assumptions, the dissipativity of our variational inequality which is important for the asymptotic behavior of the system. In the second part a coupled system of Maxwell's equation and the heat equation is considered. For this system we introduce the notion of stability on a finite-time interval and present a theorem on this type of stability.
The nonlinear Schrödinger equation created by the vibrations of an elastic plate and its dimensional expansion
Shuya Kanagawa and Ben T. Nohara
2013(special): 415-426 doi: 10.3934/proc.2013.2013.415 +[Abstract](353) +[PDF](341.0KB)
Abstract:
We first survey the two-dimensional governing equation that describes the propagation of a wave packet on an elastic plate using the method of multiple scales by [13]. We next expand the governing equation to the multi-dimensional case not only in the sense of mathematical science but also engineering.
Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system
Yukio Kan-On
2013(special): 427-436 doi: 10.3934/proc.2013.2013.427 +[Abstract](334) +[PDF](291.4KB)
Abstract:
In this paper, we consider a reaction-diffusion system which describes the dynamics of population density for a two competing species community, and discuss the structure on the set of radially symmetric positive stationary solutions for the system by assuming the habitat of the community to be a ball. To do this, we shall treat the dimension of the habitat and the diffusion rates of the system as bifurcation parameters, and employ the comparison principle and the implicit function theorem.
Optimal control of a linear stochastic Schrödinger equation
Diana Keller
2013(special): 437-446 doi: 10.3934/proc.2013.2013.437 +[Abstract](337) +[PDF](320.7KB)
Abstract:
This paper concerns a linear controlled Schrödinger equation with additive noise and corresponding initial and Neumann boundary conditions. The existence and uniqueness of the variational solution of this Schrödinger problem and some of its properties will be discussed. Furthermore, a given objective functional shall be minimized by an optimal control. Though, instead of the control only the solution of the controlled Schrödinger problem appears explicitly in the objective functional. Based on the adjoint problem of the stochastic Schrödinger problem, a gradient formula is developed.
Quasi-subdifferential operators and evolution equations
Masahiro Kubo
2013(special): 447-456 doi: 10.3934/proc.2013.2013.447 +[Abstract](313) +[PDF](316.1KB)
Abstract:
We introduce the concept of a quasi-subdifferential operator and that of a quasi-subdifferential evolution equation. We prove the existence of solutions to related problems and give applications to variational and quasi-variational inequalities.
Analytical approach of one-dimensional solute transport through inhomogeneous semi-infinite porous domain for unsteady flow: Dispersion being proportional to square of velocity
Atul Kumar and R. R. Yadav
2013(special): 457-466 doi: 10.3934/proc.2013.2013.457 +[Abstract](391) +[PDF](394.1KB)
Abstract:
In this study, we present an analytical solution for solute transport in a semi-infinite inhomogeneous porous domain and a time-varying boundary condition. Dispersion is considered directly proportional to the square of velocity whereas the velocity is time and spatially dependent function. It is expressed in degenerate form. Initially the domain is solute free. The input condition is considered pulse type at the origin and flux type at the other end of the domain. Certain new independent variables are introduced through separate transformation to eliminate the variable coefficients of Advection Diffusion Equation (ADE) into constant coefficient. Laplace transform technique (LTT) is used to get the analytical solution of ADE concentration values are illustrated graphically.
Bifurcation structure of steady-states for bistable equations with nonlocal constraint
Kousuke Kuto and Tohru Tsujikawa
2013(special): 467-476 doi: 10.3934/proc.2013.2013.467 +[Abstract](273) +[PDF](184.7KB)
Abstract:
This paper studies the 1D Neumann problem of bistable equations with nonlocal constraint. We obtain the global bifurcation structure of solutions by a level set analysis for the associate integral mapping. This structure implies that solutions can form a saddle-node bifurcation curve connecting boundary-layer states with internal-layer states. Furthermore, we exhibit the applications of our result to a couple of shadow systems arising in surface chemistry and physiology.
Existence of sliding motions for nonlinear evolution equations in Banach spaces
Laura Levaggi
2013(special): 477-487 doi: 10.3934/proc.2013.2013.477 +[Abstract](267) +[PDF](335.6KB)
Abstract:
In this paper the issue of existence of sliding motions for a class of control systems of parabolic type is considered. The operator satisfies standard hemicontinuity, monotonicity and coercivity assumptions; the control law is finite-dimensional and enters linearly in the equation. By using a Faedo-Galerkin approach, a family of finite-dimensional ODEs is constructed and an approximating sequence of sliding motions is obtained using classical variable structure control techniques. Previous results on the convergence of the approximations are extended, by taking into consideration more general growth assumptions on the feedbacks. A detailed description of the approach for semilinear partial differential equations with Neumann boundary control is discussed.
A discontinuous Galerkin least-squares finite element method for solving Fisher's equation
Runchang Lin and Huiqing Zhu
2013(special): 489-497 doi: 10.3934/proc.2013.2013.489 +[Abstract](341) +[PDF](266.6KB)
Abstract:
In the present study, a discontinuous Galerkin least-squares finite element algorithm is developed to solve Fisher's equation. The present method is effective and can be successfully applied to problems with strong reaction, to which obtaining stable and accurate numerical traveling wave solutions is challenging. Numerical results are given to demonstrate the convergence rates of the method and the performance of the algorithm in long-time integrations.
Discretizing spherical integrals and its applications
Shaobo Lin, Xingping Sun and Zongben Xu
2013(special): 499-514 doi: 10.3934/proc.2013.2013.499 +[Abstract](273) +[PDF](398.4KB)
Abstract:
Efficient discretization of spherical integrals is required in many numerical methods associated with solving differential and integral equations on spherical domains. In this paper, we discuss a discretization method that works particularly well with convolutions of spherical integrals. We utilize this method to construct spherical basis function networks, which are subsequently employed to approximate the solutions of a variety of differential and integral equations on spherical domains. We show that, to a large extend, the approximation errors depend only on the smoothness of the spherical basis function. We also derive error estimates of the pertinent approximation schemes. As an application, we discuss a Galerkin type solutions for spherical Fredholm integral equations of the first kind, and obtain rates of convergence of the spherical basis function networks to the solutions of these equations.
Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics
Julián López-Gómez, Marcela Molina-Meyer and Andrea Tellini
2013(special): 515-524 doi: 10.3934/proc.2013.2013.515 +[Abstract](291) +[PDF](1375.4KB)
Abstract:
It has been recently shown in [10] that Problem (1), for the special choice (2), admits an arbitrarily large number of positive solutions, provided that $\lambda$ is sufficiently negative. Moreover, using $b$ as the main bifurcation parameter, some fundamental qualitative properties of the associated global bifurcation diagrams have been established. Based on them, the authors computed such bifurcation diagrams by coupling some adaptation of the classical path-following solvers with spectral methods and collocation (see [9]). In this paper, we complete our original program by computing the global bifurcation diagrams for a wider relevant class of weight functions $a(x)$'s. The numerics suggests that the analytical results of [10] should be true for general symmetric weight functions, whereas some of them can fail if $a(x)$ becomes asymmetric around $0.5$. In any of these circumstances, the more negative $\lambda$, the larger the number of positive solutions of Problem (1). As an astonishing ecological consequence, facilitation in competitive environments within polluted habitat patches causes complex dynamics.
Attractors for weakly damped beam equations with $p$-Laplacian
T. F. Ma and M. L. Pelicer
2013(special): 525-534 doi: 10.3934/proc.2013.2013.525 +[Abstract](277) +[PDF](359.8KB)
Abstract:
This paper is concerned with a class of weakly damped one-dimensional beam equations with lower order perturbation of $p$-Laplacian type $$ u_{tt} + u_{xxxx} - (\sigma(u_x))_x + ku_t + f(u)= h \quad \hbox{in} \quad (0,L) \times \mathbb{R}^{+} , $$ where $\sigma(z)=|z|^{p-2}z$, $p \ge 2$, $k>0$ and $f(u)$ and $h(x)$ are forcing terms. Well-posedness, exponential stability and existence of a finite-dimensional attractor are proved.
On global existence and bounds for blow-up time in nonlinear parabolic problems with time dependent coefficients
Monica Marras and Stella Vernier Piro
2013(special): 535-544 doi: 10.3934/proc.2013.2013.535 +[Abstract](295) +[PDF](338.7KB)
Abstract:
This paper deals with the blow-up of the solutions to a class of nonlinear parabolic equations with Dirichlet boundary condition and time dependent coefficients. Under some conditions on the data and geometry of the spatial domain, explicit upper and lower bounds for the blow-up time are derived. Moreover, the influence of the data on the behaviour of the solution is investigated to obtain global existence.
A note on optimal control problem for a hemivariational inequality modeling fluid flow
Stanisław Migórski
2013(special): 545-554 doi: 10.3934/proc.2013.2013.545 +[Abstract](301) +[PDF](319.6KB)
Abstract:
We consider a class of distributed parameter optimal control problems for the boundary value problem for the stationary Navier--Stokes equation with a subdifferential boundary condition in a bounded domain. The weak formulation of the boundary value problem is a hemivariational inequality associated with a nonconvex nonsmooth locally Lipschitz superpotential. We establish the existence of solutions to the optimal control problem. We also address an open problem of potential identification in the hemivariational inequality.
Existence and multiplicity of solutions in fourth order BVPs with unbounded nonlinearities
Feliz Minhós and João Fialho
2013(special): 555-564 doi: 10.3934/proc.2013.2013.555 +[Abstract](298) +[PDF](363.2KB)
Abstract:
In this work the authors present some existence, non-existence and location results of the problem composed of the fourth order fully nonlinear equation \begin{equation*} u^{\left( 4\right) }\left( x\right) +f( x,u\left( x\right) ,u^{\prime }\left( x\right) ,u^{\prime \prime }\left( x\right) ,u^{\prime \prime \prime }\left( x\right) ) =s\text{ }p(x) \end{equation*} for $x\in \left[ a,b\right] ,$ where $f:\left[ a,b\right] \times \mathbb{R} ^{4}\rightarrow \mathbb{R},$ $p:\left[ a,b\right] \rightarrow \mathbb{R}^{+}$ are continuous functions and $s$ a real parameter, with the boundary conditions \begin{equation*} u\left( a\right) =A,\text{ }u^{\prime }\left( a\right) =B,\text{ }u^{\prime \prime \prime }\left( a\right) =C,\text{ }u^{\prime \prime \prime }\left( b\right) =D,\text{ } \end{equation*} for $A,B,C,D\in \mathbb{R}.$ In this work they use an Ambrosetti-Prodi type approach, with some new features: the existence part is obtained in presence of nonlinearities not necessarily bounded, and in the multiplicity result it is not assumed a speed growth condition or an asymptotic condition, as it is usual in the literature for these type of higher order problems.
    The arguments used apply lower and upper solutions technique and topological degree theory.
    An application is made to a continuous model of the human spine, used in aircraft ejections, vehicle crash situations, and some forms of scoliosis.
Representation formula for the plane closed elastic curves
Minoru Murai, Waichiro Matsumoto and Shoji Yotsutani
2013(special): 565-585 doi: 10.3934/proc.2013.2013.565 +[Abstract](254) +[PDF](738.8KB)
Abstract:
Let $\Gamma$ be a plane closed elastic curve with length $L>0.$ Let $M$ be the signed area of the domain bounded by $\Gamma$. We are interested in the following variational problem. Find a curve $\Gamma$ (the curvature $\kappa(s)$) which minimizes the elastic energy subject to $L^{2}-4 \pi M >0$ and $ L^{2} \neq 4 \pi \omega M$, where $\omega$ is the winding number. This variational problem was first studied in the case $\omega=1$ and the Euler-Lagrange equation was derived. The existence of the minimizer was showed and the profile near the disk was investigated by using the Euler-Lagrange equation. As the first step to investigate the structure of solutions of this equation, we show all the solutions to an auxiliary second order boundary value problem. Moreover, we obtain the representation of the integral of $\kappa(s)$.
Efficient recurrence relations for univariate and multivariate Taylor series coefficients
Richard D. Neidinger
2013(special): 587-596 doi: 10.3934/proc.2013.2013.587 +[Abstract](373) +[PDF](331.4KB)
Abstract:
The efficient use of Taylor series depends, not on symbolic differentiation, but on a standard set of recurrence formulas for each of the elementary functions and operations. These relationships are often rediscovered and restated, usually in a piecemeal fashion. We seek to provide a fairly thorough and unified exposition of efficient recurrence relations in both univariate and multivariate settings. Explicit formulas all stem from the fact that multiplication of functions corresponds to a Cauchy product of series coefficients, which is more efficient than the Leibniz rule for nth-order derivatives. This principle is applied to function relationships of the form h'=v*u', where the prime indicates a derivative or partial derivative. Each standard (calculator button) function corresponds to an equation, or pair of equations, of this form. A geometric description of the multivariate operation helps clarify and streamline the computation for each desired multi-indexed coefficient. Several research communities use such recurrences including the Differential Transform Method to solve differential equations with initial conditions.
Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response
Kazuhiro Oeda
2013(special): 597-603 doi: 10.3934/proc.2013.2013.597 +[Abstract](300) +[PDF](282.2KB)
Abstract:
This paper is concerned with the steady state problem of a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. A sufficient condition for the existence of positive steady state solutions is given. Our proof is based on the bifurcation theory and some a priori estimates.
Orthogonal polynomials on the unit circle with quasiperiodic Verblunsky coefficients have generic purely singular continuous spectrum
Darren C. Ong
2013(special): 605-609 doi: 10.3934/proc.2013.2013.605 +[Abstract](251) +[PDF](253.4KB)
Abstract:
As an application of the Gordon lemma for orthogonal polynomials on the unit circle, we prove that for a generic set of quasiperiodic Verblunsky coefficients the corresponding two-sided CMV operator has purely singular continuous spectrum. We use a similar argument to that of the Boshernitzan-Damanik result that establishes the corresponding theorem for the discrete Schrödinger operator.
Spatial stability of horizontally sheared flow
Iordanka N. Panayotova, Pai Song and John P. McHugh
2013(special): 611-618 doi: 10.3934/proc.2013.2013.611 +[Abstract](210) +[PDF](265.9KB)
Abstract:
We investigate the stability of a shear flow in a stratified fluid. The flow is assumed to be inviscid and Boussinesq and the base state density gradient is vertical with constant Brunt-Vaisala frequency. The shear is taken as horizontal, where the base-state velocity has uniform direction and it's magnitude depends on the transverse horizontal coordinate, U(y). Unlike vertical shear flows, this combination of horizontal shear with vertical stratification is inherently three-dimensional and Squire's theorem is inapplicable. Spatial stability characteristics are obtained using the normal-mode approach and the Riccati transform. Sensitivity of the stability characteristics and their qualitative features are investigated by numerical methods for free-shear flow approximated by the hyperbolic tangent.
Fuzzy system of linear equations
Purnima Pandit
2013(special): 619-627 doi: 10.3934/proc.2013.2013.619 +[Abstract](267) +[PDF](288.6KB)
Abstract:
Real life applications arising in various fields of Engineering and Sciences like Electrical, Civil, Economics, Dietary etc. can be modeled using system of linear equations. In such models it may happen that the values of the parameters are not known or they cannot be stated precisely only their estimation due to experimental data or experts knowledge is available. In such situation it is convenient to represent such parameters by fuzzy numbers (refer [22]). Klir, [15] gave the results for the existence of solution of linear algebraic equation involving fuzzy numbers. The method to obtain solution of system of linear equations with all the involved parameters being fuzzy is proposed here. The $\alpha$-cut technique is well known in obtaining weak solutions, (refer [7]) for fully fuzzy systems of linear equations (FFSL). In this paper, the conditions for the existence and uniqueness of the fuzzy solution are proved.
Liapunov-type integral inequalities for higher order dynamic equations on time scales
Saroj Panigrahi
2013(special): 629-641 doi: 10.3934/proc.2013.2013.629 +[Abstract](284) +[PDF](357.1KB)
Abstract:
In this paper, we obtain Liapunov-type integral inequalities for certain nonlinear, nonhomogeneous dynamic equations of higher order without any restriction on the zeros of their higher-order delta derivatives of solutions by using time scale analysis. As an applications of our results, we show that oscillatory solutions of the equation converge to zero as $t\to \infty$. Using these inequalities, it is also shown that $(t_{m+ k} - t_{m}) \to \infty $ as $m \to \infty$, where $1 \le k \le n-1$ and $\langle t_m \rangle $ is an increasing sequence of generalized zeros of an oscillatory solution of $ D^n y + y f(t, y(\sigma(t)))|y(\sigma(t))|^{p-2} = 0$, $t \ge 0$, provided that $W(., \lambda) \in L^{\mu}([0, \infty)_{\mathbb{T}}, \mathbb{R}^{+})$, $1 \le \mu \le \infty$ and for all $\lambda > 0$. A criterion for disconjugacy of nonlinear homogeneous dynamic equation is obtained in an interval $[a, \sigma(b)]_{\mathbb{T}}$.
Parameter dependent stability/instability in a human respiratory control system model
Saroj P. Pradhan and Janos Turi
2013(special): 643-652 doi: 10.3934/proc.2013.2013.643 +[Abstract](258) +[PDF](344.8KB)
Abstract:
In this paper a computational procedure is presented to study the development of stable/unstable patterns in a system of three nonlinear, parameter dependent delay differential equations with two transport delays representing a simplified model of human respiration. It is demonstrated using simulations how sequences of changes in internal and external parameter values can lead to complex dynamic behavior due to forced transitions between stable/unstable equilibrium positions determined by particular parameter combinations. Since changes in the transport delays only influence the stability/instability of an equilibrium position a stability chart is constructed in that case by finding the roots of the characteristic equation of the corresponding linear variational system. Illustrative examples are included.
Dynamically consistent discrete-time SI and SIS epidemic models
Lih-Ing W. Roeger
2013(special): 653-662 doi: 10.3934/proc.2013.2013.653 +[Abstract](347) +[PDF](349.2KB)
Abstract:
Discrete-time $SI$ and $SIS$ epidemic models are constructed by applying the nonstandard finite difference (NSFD) schemes to the differential equation models. The difference equation systems are dynamically consistent with their analog continuous-time models. The basic standard incidence $SI$ and $SIS$ models without births and deaths, with births and deaths, and with immigrations, are considered. The continuous models possess either the conservation law that the total population is a constant or the total population $N$ satisfies $N'(t)=\lambda-\mu N$ and so that $N$ approaches a constant $\lambda/\mu$ as $t$ approaches infinity. The difference equation systems via NSFD schemes preserve all properties including the positivity of solutions, the conservation law, and the local and some of the global stability of the equilibria. They are said to be dynamically consistent with the continuous models with respect to these properties. We show that a simple criterion for choosing a certain NSFD scheme such that the positivity solutions are preserved is usually an indication of an appropriate NSFD scheme.
Analysis of a mathematical model for jellyfish blooms and the cambric fish invasion
Florian Rupp and Jürgen Scheurle
2013(special): 663-672 doi: 10.3934/proc.2013.2013.663 +[Abstract](346) +[PDF](368.0KB)
Abstract:
Dramatic increases in jellyfish populations which lead to the collapse of formerly healthy ecosystems are repeatedly reported from many different sites, cf. [6,8,14]. Due to their devastating effects on fishery the understanding of the causes for such a blooming are of major ecological as well as economical importance. Assuming fish as the dominant predator species we model a combined two species system subject to constant environmental conditions. By totally analytic means we completely classify all biologically relevant equilibria in terms of existence and Lyapunov stability, and give a complete description of this system's non-linear global dynamics supported by numerical simulations. This approach complements, from a systematic point of view, the studies given in the literature to better understand jellyfish blooms.
Stochastic heat equations with cubic nonlinearity and additive space-time noise in 2D
Henri Schurz
2013(special): 673-684 doi: 10.3934/proc.2013.2013.673 +[Abstract](285) +[PDF](357.9KB)
Abstract:
Semilinear heat equations on rectangular domains in $\mathbb{R}^2$ (conduction through plates) with cubic-type nonlinearities and perturbed by an additive Q-regular space-time white noise are considered analytically. These models as 2nd order SPDEs (stochastic partial differential equations) with non-random Dirichlet-type boundary conditions describe the temperature- or substance-distribution on rectangular domains as met in engineering and biochemistry. We discuss their analysis by the eigenfunction approach allowing us to truncate the infinite-dimensional stochastic systems (i.e. the SDEs of Fourier coefficients related to semilinear SPDEs), to control its energy, existence, uniqueness, continuity and stability. The functional of expected energy is estimated at time $t$ in terms of system-parameters.
Control of attractors in nonlinear dynamical systems using external noise: Effects of noise on synchronization phenomena
Masatoshi Shiino and Keiji Okumura
2013(special): 685-694 doi: 10.3934/proc.2013.2013.685 +[Abstract](452) +[PDF](529.4KB)
Abstract:
Synchronization phenomena occurring as a result of cooperative ones are ubiquitous in nonequilibrium physical and biological systems and also are considered to be of vital importance in information processing in the brain. Those systems, in general, are subjected to various kinds of noise. While in the case of equilibrium thermodynamic systems external Langevin noise is well-known to play the role of heat bath, few systematic studies have been conducted to explore effects of noise on nonlinear dynamical systems with many degrees of freedom exhibiting limit cycle oscillations and chaotic motions, due to their complexity. Considering simple nonlinear dynamical models that allow rigorous analyses based on use of nonlinear Fokker-Planck equations, we conduct systematic studies to observe effects of noise on oscillatory behavior with changes in several kinds of parameters characterising mean-field coupled oscillator ensembles and excitable element ones. Phase diagrams representing the dependence of the largest and the second largest Lyapunov exponents on the noise strength are studied to show the appearance and disappearance of synchronization of limit cycle oscillations.
Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents
Inbo Sim and Yun-Ho Kim
2013(special): 695-707 doi: 10.3934/proc.2013.2013.695 +[Abstract](298) +[PDF](384.9KB)
Abstract:
We study the following nonlinear problem \begin{equation*} -div(w(x)|\nabla u|^{p(x)-2}\nabla u)=\lambda f(x,u)\quad in \Omega \end{equation*} which is subject to Dirichlet boundary condition. Under suitable conditions on $w$ and $f$, employing the variational methods, we show the existence of solutions for the above problem in the weighted variable exponent Lebesgue-Sobolev spaces. Also we obtain the positivity of the infimum eigenvalue for the problem.
Initial boundary value problem for the singularly perturbed Boussinesq-type equation
Changming Song, Hong Li and Jina Li
2013(special): 709-717 doi: 10.3934/proc.2013.2013.709 +[Abstract](317) +[PDF](352.5KB)
Abstract:
We are concerned with the singularly perturbed Boussinesq-type equation including the singularly perturbed sixth-order Boussinesq equation, which describes the bi-directional propagation of small amplitude and long capillary-gravity waves on the surface of shallow water for bond number (surface tension parameter) less than but very close to $1/3$. The existence and uniqueness of the global generalized solution and the global classical solution of the initial boundary value problem for the singularly perturbed Boussinesq-type equation are proved.
Validity and dynamics in the nonlinearly excited 6th-order phase equation
Dmitry Strunin and Mayada Mohammed
2013(special): 719-728 doi: 10.3934/proc.2013.2013.719 +[Abstract](209) +[PDF](1474.2KB)
Abstract:
A slowly varying phase of oscillators coupled by diffusion is generally described by a partial differential equation comprising infinitely many terms. We consider a particular case when the coupling is nonlocal and, as a result, the equation can be reduced to a finite form with nonlinear excitation and 6th-order dissipation. We fulfilled two tasks: (1) evaluated the range of independent parameters rendering the form valid, and (2) developed and tested the numerical code for solving the equation; some numerical solutions are presented.
Morse indices and the number of blow up points of blowing-up solutions for a Liouville equation with singular data
Futoshi Takahashi
2013(special): 729-736 doi: 10.3934/proc.2013.2013.729 +[Abstract](277) +[PDF](361.0KB)
Abstract:
Let $\Omega \subset \mathbb{R}^2$ be a smooth bounded domain and let $\Gamma = \left \{ p_1, \cdots, p_N \right \} \subset \Omega$ be the set of prescribed points. Consider the Liouville type equation \[ -\delta u = \lambda \Pi_{j = 1}^{N} |x - p_j|^{2\alpha_j} V(x) e^u \quad \mbox{in} \; \Omega, \quad u = 0 \quad \mbox{on} \; \partial \Omega, \] where $\alpha_j \; (j=1,\cdots, N)$ are positive numbers, $V(x) > 0$ is a given smooth function on $\bar{\Omega}$, and $\lambda > 0$ is a parameter. Let $\{ u_n \}$ be a blowing up solution sequence for $\lambda = \lambda_n \downarrow 0$ having the $m$-points blow up set $S = \{ q_1, \cdots, q_m \} \subset \Omega$, i.e., \[ \lambda_n \prod_{j = 1}^N |x - p_j|^{2 \alpha_j} V(x) e^{u_n} dx \rightharpoonup \sum_{i=1}^m b_i \delta_{q_i} \] in the sense of measures, where $b_i = 8\pi$ if $q_i \notin \Gamma$, $b_i = 8\pi(1 + \alpha_j)$ if $q_i = p_j$ for some $p_j \in \Gamma$. We show that the number of blow up points $m$ is less than or equal to the Morse index of $u_n$ for $n$ sufficiently large, provided $\alpha_j \in (0,+\infty) \setminus \mathbb{N}$ for all $j = 1, \cdots, N$. This is a generalization of the result [13] in which nonsingular case ($\alpha_j = 0$ for all $j$) was studied.
Modeling the thermal conductance of phononic crystal plates
Stefanie Thiem and Jörg Lässig
2013(special): 737-746 doi: 10.3934/proc.2013.2013.737 +[Abstract](260) +[PDF](529.4KB)
Abstract:
The paper presents a model to compute the phonon thermal conductance of phononic crystal plates. The goal is the optimization of the figure of merit for these materials, which is the primary criterion for the efficiency of a thermoelectric device. Values of about three or higher allow for the construction of thermoelectric generators based on the Seebeck effect, which are more efficient than conventional electrical generators. The paper introduces a numerical method to optimize the phonon thermal conductance of a given phononic material by varying the geometrical structure with respect to the width and thickness of a sample as well as pore size, shape, and mass density.
Analyzing the infection dynamics and control strategies of cholera
Jianjun Paul Tian, Shu Liao and Jin Wang
2013(special): 747-757 doi: 10.3934/proc.2013.2013.747 +[Abstract](284) +[PDF](619.2KB)
Abstract:
We conduct a rigorous analysis for the differential equation-based cholera model proposed in [3]. Unlike traditional infectious disease SIR-type models, this model explicitly includes cholerae bacteria from the environments, and the incidence rate is a dose-dependent Michaelis-Menten type functional response. By extending the theory of monotone dynamical systems, we prove that the endemic equilibrium, when it exists, of the model is globally asymptotically stable, implying the persistence of the disease in the absence of interventions. We then modify the model by incorporating various control strategies, and study the subsequent dynamics. We find that with strong control measures, the basic reproduction number will be reduced below 1 so that the disease-free equilibrium is globally asymptotically stable. With weak controls, instead, epidemicity still occurs and a unique and globally stable endemic equilibrium state exists, though at a lower infection level and with a reduced disease outbreak growth rate. The analytical predictions are confirmed by numerical results.
Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle
Yu Tian, John R. Graef, Lingju Kong and Min Wang
2013(special): 759-769 doi: 10.3934/proc.2013.2013.759 +[Abstract](317) +[PDF](329.9KB)
Abstract:
In this paper, by using the dual least action principle, the authors investigate the existence of solutions to a multi-point boundary value problem for a second-order differential system with $p$-Laplacian.
On the uniqueness of blow-up solutions of fully nonlinear elliptic equations
Antonio Vitolo, Maria E. Amendola and Giulio Galise
2013(special): 771-780 doi: 10.3934/proc.2013.2013.771 +[Abstract](332) +[PDF](374.9KB)
Abstract:
This paper contains new uniqueness results of the boundary blow-up viscosity solutions of second order elliptic equations, generalizing a well known result of Marcus-Veron for the Laplace operator.
Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients
Hiroshi Watanabe
2013(special): 781-790 doi: 10.3934/proc.2013.2013.781 +[Abstract](264) +[PDF](318.4KB)
Abstract:
In this paper, we consider the initial value problem for strongly degenerate parabolic equations with discontinuous coefficients. This equation has the both properties of parabolic equation and hyperbolic equation. Therefore, we should choose entropy solutions as generalized solutions to the equation. Moreover, entropy solutions to the equation may not belong to $BV$ in our setting. These are difficult points for this type of equations.
    In particular, we consider the case that coefficients are the functions of bounded variation with respect to the space variable $x$. Then, we prove the existence of Kružkov type entropy solutions. Moreover, we prove the uniqueness of the solution under additional conditions.
Schrödinger equation with noise on the boundary
Frank Wusterhausen
2013(special): 791-796 doi: 10.3934/proc.2013.2013.791 +[Abstract](291) +[PDF](259.7KB)
Abstract:
We treat the question of existence and uniqueness of distributional solutions for the linear Schrödinger equation in a bounded domain with boundary noise. We cover both Dirichlet and Neumann noise. For the proof we make use of spectral decomposition of the Laplacian with homogeneous Neumann/Direchlet boundary condition.
Longtime dynamics for an elastic waveguide model
Zhijian Yang and Ke Li
2013(special): 797-806 doi: 10.3934/proc.2013.2013.797 +[Abstract](278) +[PDF](368.7KB)
Abstract:
The paper studies the longtime dynamics for a nonlinear wave equation arising in elastic waveguide model: $u_{tt}- \Delta u-\Delta u_{tt}+\Delta^2 u- \Delta u_t -\Delta g(u)=f(x)$. It proves that the equation possesses in trajectory phase space a global trajectory attractor $\mathcal{A}^{tr}$ and the full trajectory of the equation in $\mathcal{A}^{tr}$ is of backward regularity provided that the growth exponent of nonlinearity $g(u)$ is supercritical.
Stochastic deformation of classical mechanics
Jean-Claude Zambrini
2013(special): 807-813 doi: 10.3934/proc.2013.2013.807 +[Abstract](281) +[PDF](281.7KB)
Abstract:
We describe a method of stochastic deformation of classical mechanics preserving the time symmetry of this theory. It provides a new general strategy to deform stochastically Geometric Mechanics.
Traveling wave solutions with mixed dispersal for spatially periodic Fisher-KPP equations
Aijun Zhang
2013(special): 815-824 doi: 10.3934/proc.2013.2013.815 +[Abstract](292) +[PDF](329.7KB)
Abstract:
Traveling wave solutions to a spatially periodic nonlocal/random mixed dispersal equation with KPP nonlinearity are studied. By constructions of super/sub solutions and comparison principle, we establish the existence of traveling wave solutions with all propagating speeds greater than or equal to the spreading speed in every direction. For speeds greater than the spreading speed, we further investigate their uniqueness and stability.
Foam cell formation in atherosclerosis: HDL and macrophage reverse cholesterol transport
Shuai Zhang, L.R. Ritter and A.I. Ibragimov
2013(special): 825-835 doi: 10.3934/proc.2013.2013.825 +[Abstract](300) +[PDF](460.0KB)
Abstract:
Macrophage derived foam cells are a major constituent of the fatty deposits characterizing the disease atherosclerosis. Foam cells are formed when certain immune cells (macrophages) take on oxidized low density lipoproteins through failed phagocytosis. High density lipoproteins (HDL) are known to have a number of anti-atherogenic effects. One of these stems from their ability to remove excess cellular cholesterol for processing in the liver---a process called reverse cholesterol transport (RCT). HDL perform macrophage RCT by binding to forming foam cells and removing excess lipids by efflux transporters.
    We propose a model of foam cell formation accounting for macrophage RCT. This model is presented as a system of non-linear ordinary differential equations. Motivated by experimental observations regarding time scales for oxidation of lipids and MRCT, we impose a quasi-steady state assumption and analyze the resulting systems of equations. We focus on the existence and stability of equilibrium solutions as determined by the governing parameters with the results interpreted in terms of their potential bio-medical implications.
Anosov diffeomorphisms
João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto and David A. Rand
2013(special): 837-845 doi: 10.3934/proc.2013.2013.837 +[Abstract](270) +[PDF](467.9KB)
Abstract:
We use Adler, Tresser and Worfolk decomposition of Anosov automorphisms to give an explicit construction of the stable and unstable $C^{1+}$ self-renormalizable sequences.

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