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Communications on Pure & Applied Analysis

2015 , Volume 14 , Issue 3

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General types of spherical mean operators and $K$-functionals of fractional orders
Thaís Jordão and Xingping Sun
2015, 14(3): 743-757 doi: 10.3934/cpaa.2015.14.743 +[Abstract](609) +[PDF](398.5KB)
Abstract:
We design a general type of spherical mean operators and employ them to approximate $L_p$ class functions. We show that optimal orders of approximation are achieved via appropriately defined K-functionals of fractional orders.
Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities
Zhi-Qiang Shao
2015, 14(3): 759-792 doi: 10.3934/cpaa.2015.14.759 +[Abstract](579) +[PDF](548.5KB)
Abstract:
In the present paper the author investigates the generalized nonlinear initial-boundary Riemann problem with small BV data for general $n\times n$ quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions in a half space $\{(t,x)|t\geq 0,x\geq 0\}$, where the Riemann solution only contains shocks and contact discontinuities. Combining the techniques employed by Li-Kong with the modified Glimm's functional, the author obtains the almost global existence and lifespan of classical discontinuous solutions to a class of the generalized nonlinear initial-boundary Riemann problem, which can be regarded as a small BV perturbation of the corresponding nonlinear initial-boundary Riemann problem. This result is also applied to the system of traffic flow on a road network using the Aw-Rascle model.
Differential Harnack estimates for backward heat equations with potentials under geometric flows
Shouwen Fang and Peng Zhu
2015, 14(3): 793-809 doi: 10.3934/cpaa.2015.14.793 +[Abstract](659) +[PDF](409.3KB)
Abstract:
In the paper we consider a closed Riemannian manifold $M$ with a time-dependent Riemannian metric $g_{i j}(t)$ evolving by $\partial_{t}g_{i j}=-2S_{i j}$ where $S_{i j}$ is a symmetric two-tensor on $(M,g(t))$. We prove some differential Harnack inequalities for positive solutions of backward heat equations with potentials when the metric satisfies the geometric flow. Some applications of these inequalities will be obtained. In particular, we show that the shrinking breathers of the Lorentzian mean curvature flow are the shrinking gradient solitons when the ambient Lorentzian manifold has nonnegative sectional curvature.
Uniform stability of the Boltzmann equation with an external force near vacuum
Zhigang Wu and Wenjun Wang
2015, 14(3): 811-823 doi: 10.3934/cpaa.2015.14.811 +[Abstract](575) +[PDF](399.2KB)
Abstract:
The temporal uniform $L^1(x,v)$ stability of mild solutions for the Boltzmann equation with an external force is considered. We give a unified proof of the stability for two kinds of the forces. Firstly, we extend the soft potential case in [9] to both soft and hard cases. Secondly, we weaken the condition on the force in [13]. Furthermore, we give some new examples satisfying the constructive conditions on the force in [11].
Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations
Jaeyoung Byeon, Ohsang Kwon and Yoshihito Oshita
2015, 14(3): 825-842 doi: 10.3934/cpaa.2015.14.825 +[Abstract](612) +[PDF](463.4KB)
Abstract:
For $k =1,\cdots,K,$ let $M_k$ be a $q_k$-dimensional smooth compact framed manifold in $R^N$ with $q_k \in \{1,\cdots,N-1\} $. We consider the equation $-\varepsilon^2\Delta u + V(x)u - u^p = 0$ in $R^N$ where for each $k \in \{1,\cdots,K\}$ and some $m_k > 0,$ $V(x)=|\textrm{dist}(x,M_k)|^{m_k}+O(|\textrm{dist}(x,M_k)|^{m_k+1})$ as $\textrm{dist}(x,M_k) \to 0 $. For a sequence of $\varepsilon$ converging to zero, we will find a positive solution $u_{\varepsilon}$ of the equation which concentrates on $M_1\cup \dots \cup M_K$.
Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves
Jun-ichi Segata
2015, 14(3): 843-859 doi: 10.3934/cpaa.2015.14.843 +[Abstract](684) +[PDF](434.9KB)
Abstract:
We consider the fourth order nonlinear Schrödinger type equation (4NLS) which arises in context of the motion of vortex filament. The purposes of this paper are twofold. Firstly, we consider the initial value problem for (4NLS) under the periodic boundary condition. By refining the modified energy method used in our previous paper [23], we prove the unique existence of the global solution for (4NLS) in the energy space $H_{p e r}^2(0,2L)$ with $L>0$. Secondly, we study the stability property of periodic standing waves for (4NLS). Using the spectrum properties of the Schrödinger operators associated to the periodic standing wave developed by Angulo [1], we prove that standing wave of dnoidal type is orbitally stable under the time evolution by (4NLS).
Admissibility, a general type of Lipschitz shadowing and structural stability
Davor Dragičević
2015, 14(3): 861-880 doi: 10.3934/cpaa.2015.14.861 +[Abstract](582) +[PDF](461.2KB)
Abstract:
For a general one-sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of a uniform exponential dichotomy and we characterize it completely in terms of the admissibility of a large class of function spaces. We apply those results to show that structural stability of a diffeomorphism is equivalent to a very general type of Lipschitz shadowing property. Our results extend those in [37] in various directions.
No--flux boundary value problems with anisotropic variable exponents
Maria-Magdalena Boureanu and Cristian Udrea
2015, 14(3): 881-896 doi: 10.3934/cpaa.2015.14.881 +[Abstract](575) +[PDF](440.6KB)
Abstract:
We are concerned with elliptic problems involving generalized anisotropic operators with variable exponents and a nonlinearity $f$. For such problems with no-flux boundary conditions we establish the existence, the uniqueness, or the multiplicity of weak solutions, under various hypotheses.
Gradient estimates and comparison principle for some nonlinear elliptic equations
Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo and Adamaria Perrotta
2015, 14(3): 897-922 doi: 10.3934/cpaa.2015.14.897 +[Abstract](660) +[PDF](506.1KB)
Abstract:
We consider a class of Dirichlet boundary problems for nonlinear elliptic equations with a first order term. We show how the summability of the gradient of a solution increases when the summability of the datum increases. We also prove comparison principle which gives in turn uniqueness results by strenghtening the assumptions on the operators.
Traveling wave phenomena of a diffusive and vector-bias malaria model
Zhiting Xu and Yiyi Zhang
2015, 14(3): 923-940 doi: 10.3934/cpaa.2015.14.923 +[Abstract](460) +[PDF](428.4KB)
Abstract:
This paper is devoted to the study of a diffusive and vector-bias malaria model. We first analyze the well-posedness of the initial value problem of the model. Then, according to the basic reproduction ratio $\mathcal{R}_0$, we establish the existence and non-existence of traveling wave solutions for the model. The proof of the main theorems is based on Schauder fixed point theorem and the variation of constants formula of ODEs.
KAM Tori for generalized Benjamin-Ono equation
Dongfeng Yan
2015, 14(3): 941-957 doi: 10.3934/cpaa.2015.14.941 +[Abstract](552) +[PDF](423.0KB)
Abstract:
In this paper, we investigate one-dimensional generalized Benjamin-Ono equation, \begin{eqnarray} u_t+\mathcal{H}u_{xx}+u^{4}u_x=0,x\in\mathbb{T}, \end{eqnarray} and prove the existence of quasi-periodic solutions with two frequencies. The proof is based on partial Birkhoff normal form and an unbounded KAM theorem developed by Liu-Yuan[Commun.Math.Phys.307(2011)629-673].
On the variational $p$-capacity problem in the plane
Jie Xiao
2015, 14(3): 959-968 doi: 10.3934/cpaa.2015.14.959 +[Abstract](590) +[PDF](364.8KB)
Abstract:
Under $1 < p \leq 2$, this note presents some new optimal isoperimetric type properties of the variational $p$-capacitary potentials on convex plane rings.
Time-dependent singularities in the heat equation
Jin Takahashi and Eiji Yanagida
2015, 14(3): 969-979 doi: 10.3934/cpaa.2015.14.969 +[Abstract](638) +[PDF](333.9KB)
Abstract:
We consider solutions of the heat equation with time-dependent singularities. It is shown that a singularity is removable if it is weaker than the order of the fundamental solution of the Laplace equation. Some examples of non-removable singularities are also given, which show the optimality of the condition for removability.
Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model
Haibo Cui, Lei Yao and Zheng-An Yao
2015, 14(3): 981-1000 doi: 10.3934/cpaa.2015.14.981 +[Abstract](592) +[PDF](420.2KB)
Abstract:
In this paper, we consider global existence and optimal time decay rates of global smooth solutions to three-dimensional reduced gravity two and a half layer model. Indeed we show that the upper and middle layer thicknesses and horizontal velocities converge to their equilibrium state at the $L^2$-rate $(1+t)^{-\frac{3}{4}}$ or $L^\infty$-rate $(1+t)^{-\frac{3}{2}}$, respectively. These convergence rates are also shown to be optimal. The proof is based on the detailed analysis of the Green's function to the linearized system and elaborate energy estimates to the nonlinear system.
Traveling waves of a delayed diffusive SIR epidemic model
Yan Li, Wan-Tong Li and Guo Lin
2015, 14(3): 1001-1022 doi: 10.3934/cpaa.2015.14.1001 +[Abstract](716) +[PDF](425.2KB)
Abstract:
This paper is concerned with the minimal wave speed of a delayed diffusive SIR epidemic model with Holling-II incidence rate and constant external supplies. By presenting the existence and nonexistence of traveling wave solutions for any positive wave speed, the minimal wave speed is established. In particular, the minimal wave speed decreases when the latency of infection increases. Biologically speaking, the longer the latency of infection in a vector is, the slower the disease spreads.
Existence results for compressible radiation hydrodynamic equations with vacuum
Yachun Li and Shengguo Zhu
2015, 14(3): 1023-1052 doi: 10.3934/cpaa.2015.14.1023 +[Abstract](490) +[PDF](516.7KB)
Abstract:
In this paper, we consider the three-dimensional compressible isentropic radiation hydrodynamic (RHD) equations. The existence of unique local strong solutions is firstly proved when the initial data are arbitrarily large, contain vacuum and satisfy some initial layer compatibility condition. The initial mass density does not need to be bounded away from zero and may vanish in some open set. We also prove that if the initial vacuum is not so irregular, then the initial layer compatibility condition is necessary and sufficient to guarantee the existence of a unique strong solution. Finally, we establish a blow-up criterion for the strong solution that we obtained. The similar results also hold for the barotropic flow with general pressure law $p_m=p_m(\rho)\in C^1(\mathbb{\overline{R}}^+)$.
Asymptotic behavior for the unique positive solution to a singular elliptic problem
Ling Mi
2015, 14(3): 1053-1072 doi: 10.3934/cpaa.2015.14.1053 +[Abstract](613) +[PDF](442.0KB)
Abstract:
In this paper, by means of sub-supersolution method, we are concerned with the exact asymptotic behavior for the unique solution near the boundary to the following singular Dirichlet problem $ -\triangle u=b(x)g(u), u>0, x \in \Omega, u|_{\partial \Omega}=0$, where $\Omega$ is a bounded domain with smooth boundary in $\mathbb R^N$, $b \in C^{\alpha}_{l o c}({\Omega})$ ($0 < \alpha < 1$), is positive in $\Omega,$ may be vanishing or singular on the boundary, $g\in C^1((0,\infty), (0,\infty))$, $g$ is decreasing on $(0,\infty)$ with $\lim\limits_{s \rightarrow 0^+}g(s)=\infty$ and satisfies some appropriate assumptions related to Karamata regular variation theory.
Essential perturbations of polynomial vector fields with a period annulus
Adriana Buică, Jaume Giné and Maite Grau
2015, 14(3): 1073-1095 doi: 10.3934/cpaa.2015.14.1073 +[Abstract](481) +[PDF](431.9KB)
Abstract:
Chicone--Jacobs and Iliev found the essential perturbations of quadratic systems when considering the problem of finding the cyclicity of a period annulus. Given a perturbation of a particular family of centers of polynomial differential systems of arbitrary degree for which the expressions of its Poincaré--Liapunov constants are known, we give the structure of its $k$-th Melnikov function. This allows to find the essential perturbations in concrete cases. We study here in detail the essential perturbations for all the centers of the differential systems \begin{eqnarray} \dot{x} = -y + P_{\rm d}(x,y), \quad \dot{y} = x + Q_{d}(x,y), \end{eqnarray} where $P_d$ and $Q_d$ are homogeneous polynomials of degree $d$, for $ d=2$ and $ d=3$.
Klein-Gordon-Maxwell equations in high dimensions
Pierre-Damien Thizy
2015, 14(3): 1097-1125 doi: 10.3934/cpaa.2015.14.1097 +[Abstract](599) +[PDF](596.9KB)
Abstract:
We prove the existence of a mountain-pass solution and the a priori bound property for the electrostatic Klein-Gordon-Maxwell equations in high dimensions.
Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses
Jun Zhou
2015, 14(3): 1127-1145 doi: 10.3934/cpaa.2015.14.1127 +[Abstract](689) +[PDF](754.3KB)
Abstract:
In this paper, we study a modified Leslie-Gower predator-prey model with Crowley-Martin functional response. We show the existence of a bounded positive invariant attracting set and establish the permanence conditions. The parameter regions for the stability and instability of the unique constant steady state solution are derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method.
Steady-state solutions and stability for a cubic autocatalysis model
Mei-hua Wei, Jianhua Wu and Yinnian He
2015, 14(3): 1147-1167 doi: 10.3934/cpaa.2015.14.1147 +[Abstract](636) +[PDF](639.3KB)
Abstract:
A reaction-diffusion system, based on the cubic autocatalytic reaction scheme, with the prescribed concentration boundary conditions is considered. The linear stability of the unique spatially homogeneous steady state solution is discussed in detail to reveal a necessary condition for the bifurcation of this solution. The spatially non-uniform stationary structures, especially bifurcating from the double eigenvalue, are studied by the use of Lyapunov-Schmidt technique and singularity theory. Further information about the multiplicity and stability of the bifurcation solutions are obtained. Numerical examples are presented to support our theoretical results.
An elliptic system and the critical hyperbola
Lucas C. F. Ferreira, Everaldo Medeiros and Marcelo Montenegro
2015, 14(3): 1169-1182 doi: 10.3934/cpaa.2015.14.1169 +[Abstract](517) +[PDF](453.1KB)
Abstract:
We consider a nonlinear elliptic system of Lane-Emden type in the whole space $\mathbb{R}^{n}$, namely \begin{eqnarray} \Delta u+v| v| ^{p-1}=0, \quad x\in\mathbb{R}^{n},\\ \Delta v+u| u| ^{q-1}+f=0, \quad x\in\mathbb{R}^{n}. \end{eqnarray} Our region for $(p,q)$ covers in particular the critical and supercritical cases with respect to the critical hyperbola $\frac{1}{p+1}+\frac{1} {q+1}=\frac{n-2}{n}.$ We prove existence of solutions for $f\in L^d (\mathbb{R}^n)$, by means of a fixed point technique in the Lebesgue space $L^{r_1}\times L^{r_2}$. Our results allow unbounded solutions without $H^{s}$-regularity. The solutions are shown to be classical and positive when $f$ is smooth enough and positive. Moreover, if $f$ is radial or odd (or even), we prove that the solutions preserve these properties. Also, it is shown that the solutions $(u,v)$ are nonradial when $f$ is nonradial.
Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate
Wen-Bin Yang, Jianhua Wu and Hua Nie
2015, 14(3): 1183-1204 doi: 10.3934/cpaa.2015.14.1183 +[Abstract](620) +[PDF](2209.1KB)
Abstract:
The paper is concerned with a predator-prey diffusive dynamics subject to homogeneous Dirichlet boundary conditions, in which the growth rate of the the predator is nonlinear. Taking $m$ as the main parameter, we show the existence, stability and exact number of positive solution when $m$ is large, and some numerical simulations are done to complement the analytical results. The main tools used here include the fixed point index theory, the super-sub solution method, the bifurcation theory and the perturbation technique.
Global stability and repulsion in autonomous Kolmogorov systems
Zhanyuan Hou and Stephen Baigent
2015, 14(3): 1205-1238 doi: 10.3934/cpaa.2015.14.1205 +[Abstract](571) +[PDF](673.5KB)
Abstract:
Criteria are established for the global attraction, or global repulsion on a compact invariant set, of interior and boundary fixed points of Kolmogorov systems. In particular, the notions of diagonal stability and Split Lyapunov stability that have found wide success for Lotka-Volterra systems are extended for Kolmogorov systems. Several examples from theoretical ecology and evolutionary game theory are discussed to illustrate the results.
Spectral asymptotics of the Dirichlet Laplacian in a conical layer
Monique Dauge, Thomas Ourmières-Bonafos and Nicolas Raymond
2015, 14(3): 1239-1258 doi: 10.3934/cpaa.2015.14.1239 +[Abstract](513) +[PDF](906.4KB)
Abstract:
The spectrum of the Dirichlet Laplacian on conical layers is analysed through two aspects: the infiniteness of the discrete eigenvalues and their expansions in the small aperture limit.

On the one hand, we prove that, for any aperture, the eigenvalues accumulate below the threshold of the essential spectrum: For a small distance from the essential spectrum, the number of eigenvalues farther from the threshold than this distance behaves like the logarithm of the distance.

On the other hand, in the small aperture regime, we provide a two-term asymptotics of the first eigenvalues thanks to a priori localization estimates for the associated eigenfunctions. We prove that these eigenfunctions are localized in the conical cap at a scale of order the cubic root of the aperture angle anthat they get into the other part of the layer at a scale involving the logarithm of the aperture angle.

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