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Communications on Pure & Applied Analysis

2013 , Volume 12 , Issue 5

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Two-dimensional stability analysis in a HIV model with quadratic logistic growth term
Claude-Michel Brauner , Xinyue Fan and  Luca Lorenzi
2013, 12(5): 1813-1844 doi: 10.3934/cpaa.2013.12.1813 +[Abstract](34) +[PDF](880.2KB)
We consider a Human Immunode ciency Virus (HIV) model with a logistic growth term and continue the analysis of the previous article [6]. We now take the viral di usion in a two-dimensional environment. The model consists of two ODEs for the concentrations of the target T cells, the infected cells, and a parabolic PDE for the virus particles. We study the stability of the uninfected and infected equilibria, the occurrence of Hopf bifurcation and the stability of the periodic solutions.
An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type
Jean-François Couchouron , Mikhail Kamenskii and  Paolo Nistri
2013, 12(5): 1845-1859 doi: 10.3934/cpaa.2013.12.1845 +[Abstract](31) +[PDF](352.7KB)
In this paper we consider an infinite dimensional bifurcation equation depending on a parameter $ \varepsilon>0 . $ By means of the theory of condensing operators, we prove the existence of a branch of solutions, parametrized by $ \varepsilon , $ bifurcating from a curve of solutions of the bifurcation equation obtained for $\varepsilon =0 . $ We apply this result to a specific problem, namely to the existence of periodic solutions bifurcating from the limit cycle of an autonomous functional differential equation of neutral type when it is periodically perturbed by a nonlinear perturbation term of small amplitude.
Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space
Yūki Naito and  Takasi Senba
2013, 12(5): 1861-1880 doi: 10.3934/cpaa.2013.12.1861 +[Abstract](25) +[PDF](419.1KB)
In this paper, we consider solutions to a Cauchy problem for a parabolic-elliptic system in two dimensional space. This system is a simplified version of a chemotaxis model, and is also a model of self-interacting particles.
The behavior of solutions to the problem closely depends on the $L^1$-norm of the solutions. If the quantity is larger than $8\pi$, the solution blows up in finite time. If the quantity is smaller than the critical mass, the solution exists globally in time. In the critical case, infinite blowup solutions were found.
In the present paper, we direct our attention to radial solutions to the problem whose $L^1$-norm is equal to $8\pi$ and find bounded and unbounded oscillating solutions.
Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains
Mahamadi Warma
2013, 12(5): 1881-1905 doi: 10.3934/cpaa.2013.12.1881 +[Abstract](38) +[PDF](523.0KB)
We show that on a bounded domain $\Omega\subset R^N$ with Lipschitz continuous boundary $\partial \Omega$, weak solutions of the elliptic equation $\lambda u-Au=f$ in $\Omega$ with the boundary conditions $-\gamma\Delta_\Gamma u+\partial_\nu^a u+\beta u=g$ on $\partial \Omega$ are globally Hölder continuous on $\bar \Omega$. Here $A$ is a uniformly elliptic operator in divergence form with bounded measurable coefficients, $\Delta_\Gamma$ is the Laplace-Beltrami operator on $\partial \Omega$, $\partial_\nu^a u$ denotes the conormal derivative of $u$, $\lambda,\gamma>0$ are real numbers and $\beta$ is a bounded measurable function on $\partial Omega$. We also obtain that a realization of the operator $A$ in $C(\bar \Omega)$ with the general Wentzell boundary conditions $(Au)|_{\partial \Omega}-\gamma\Delta_\Gamma u+\partial_\nu^a u+\beta u=g$ on $\partial \Omega$ generates a strongly continuous compact semigroup. Some analyticity results of the semigroup are also discussed.
Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations
Zhilei Liang
2013, 12(5): 1907-1926 doi: 10.3934/cpaa.2013.12.1907 +[Abstract](30) +[PDF](411.5KB)
This paper deals with the zero dissipation limit problem for the Navier-Stokes equations when the viscosity and the heat-conductivity are of the same order. In the case when the Riemann solution of the Euler equations is piecewise constants with a contact discontinuity, we prove that there exist global solutions to the compressible Navier-Stokes equations, which converge to the in-viscid solution away from the contact discontinuity on any finite time interval, at some convergence rate as the dissipations tend towards zero. In addition, a faster convergence rate is obtained, so long as the strength of contact discontinuity $\delta=|\theta_+ -\theta_-|$ is taken suitably small.
On qualitative analysis for a two competing fish species model with a combined non-selective harvesting effort in the presence of toxicity
Yunfeng Jia , Jianhua Wu and  Hong-Kun Xu
2013, 12(5): 1927-1941 doi: 10.3934/cpaa.2013.12.1927 +[Abstract](24) +[PDF](376.6KB)
In this paper, a two competing fish species model with combined harvesting is concerned, both the species obey the law of logistic growth and release a toxic substance to the other. Use spectrum analysis and bifurcation theory, the stability of semi-trivial solution, positive constant solution and the bifurcation solutions of model are investigated. We discuss bifurcation solutions which emanate from positive constant solution and trivial solution by taking the growth rate as bifurcation parameter. By the monotonic method, the existence result of positive steady-state of the model is discussed. The possibility of existence of a bionomic equilibrium is also obtained by taking the economical factor into consideration. Finally, some numerical examples are given to illustrate the results.
Elliptic equations with cylindrical potential and multiple critical exponents
Xiaomei Sun and  Yimin Zhang
2013, 12(5): 1943-1957 doi: 10.3934/cpaa.2013.12.1943 +[Abstract](29) +[PDF](424.6KB)
In this paper, we deal with the following problem: \begin{eqnarray*} -\Delta u-\lambda |y|^{-2}u=|y|^{-s}u^{2^{*}(s)-1}+u^{2^{*}-1}\ \ \ in \ \ R^N , y\neq 0\\ u\geq 0 \end{eqnarray*} where $u(x)=u(y,z): R^m\times R^{N-m}\longrightarrow R$, $N\geq 4$, $2 < m < N$, $\lambda < (\frac{m-2}{2})^2$ and $0 < s < 2$, $2^*(s)=\frac{2(N-s)}{N-2}$, $2^*=\frac{2N}{N-2}$. Using the Variational method, we proved the existence of a ground state solution for the case $0 < \lambda < (\frac{m-2}{2})^2$ and the existence of a cylindrical weak solution under the case $\lambda<0$.
Tug-of-war games and the infinity Laplacian with spatial dependence
Ivana Gómez and  Julio D. Rossi
2013, 12(5): 1959-1983 doi: 10.3934/cpaa.2013.12.1959 +[Abstract](32) +[PDF](524.6KB)
In this paper we look for PDEs that arise as limits of values of tug-of-war games when the possible movements of the game are taken in a family of sets that are not necessarily Euclidean balls. In this way we find existence of viscosity solutions to the Dirichlet problem for an equation of the form $- \langle D^2 v\cdot J_x(D v) ; J_x(Dv)\rangle (x) =0$, that is, an infinity Laplacian with spatial dependence. Here $J_x (Dv(x))$ is a vector that depends on the spatial location and the gradient of the solution.
Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential
Leszek Gasiński and  Nikolaos S. Papageorgiou
2013, 12(5): 1985-1999 doi: 10.3934/cpaa.2013.12.1985 +[Abstract](63) +[PDF](406.4KB)
We consider a semilinear Neumann problem driven by the negative Laplacian plus an indefinite and unbounded potential and with a Carathéodory reaction term. Using variational methods based on the critical point theory, combined with Morse theory (critical groups), we prove two multiplicity theorems.
Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions
Le Thi Phuong Ngoc and  Nguyen Thanh Long
2013, 12(5): 2001-2029 doi: 10.3934/cpaa.2013.12.2001 +[Abstract](42) +[PDF](511.9KB)
The paper is devoted to the study of a nonlinear wave equation with nonlocal boundary conditions of integral forms. First, we establish two local existence theorems by using Faedo-Galerkin method. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions.
Asymptotically periodic solutions of neutral partial differential equations with infinite delay
Hernán R. Henríquez , Claudio Cuevas and  Alejandro Caicedo
2013, 12(5): 2031-2068 doi: 10.3934/cpaa.2013.12.2031 +[Abstract](133) +[PDF](662.8KB)
In this paper we discuss the existence and uniqueness of asymptotically almost automorphic and $S$-asymptotically $\omega$-periodic mild solutions to some abstract nonlinear integro-differential equation of neutral type with infinite delay. We apply our results to neutral partial differential equations with infinite delay.
Distributional chaos for strongly continuous semigroups of operators
Angela A. Albanese , Xavier Barrachina , Elisabetta M. Mangino and  Alfredo Peris
2013, 12(5): 2069-2082 doi: 10.3934/cpaa.2013.12.2069 +[Abstract](30) +[PDF](395.5KB)
Distributional chaos for strongly continuous semigroups is studied and characterized. It is shown to be equivalent to the existence of a distributionally irregular vector. Finally, a sufficient condition for distributional chaos on the point spectrum of the generator of the semigroup is presented. An application to the semigroup generated in $L^2(R)$ by a translation of the Ornstein-Uhlenbeck operator is also given.
The sign of the wave speed for the Lotka-Volterra competition-diffusion system
Jong-Shenq Guo and  Ying-Chih Lin
2013, 12(5): 2083-2090 doi: 10.3934/cpaa.2013.12.2083 +[Abstract](29) +[PDF](278.3KB)
In this paper, we study the traveling front solutions of the Lotka-Volterra competition-diffusion system with bistable nonlinearity. It is well-known that the wave speed of traveling front is unique. Although little is known for the sign of the wave speed. In this paper, we first study the standing wave which gives some criteria when the speed is zero. Then, by the monotone dependence on parameters, we obtain some criteria about the sign of the wave speed under some parameter restrictions.
Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves
K. T. Joseph and  Manas R. Sahoo
2013, 12(5): 2091-2118 doi: 10.3934/cpaa.2013.12.2091 +[Abstract](39) +[PDF](458.2KB)
We construct solution of Riemann problem for a system of four conservation laws admitting $\delta$, $\delta'$ and $\delta''$-waves, using vanishing viscosity method. The system considered here is an extension of a system studied in [9] and [12] and admits more singular solutions. We extend the weak formulation of [12] to the present case. For the rarefaction case, the limit is not yet fully understood, the limit given in [12] is not correct and it does not satisfy the inviscid system. In fact we show that the limit of the third component contains $\delta$ measure and the fourth component contains the measure $\delta$ and its derivative, for a special Riemann data. We also solve Riemann type initial-boundary value problem in the quarter plane.
Approximation of the trajectory attractor of the 3D MHD System
Gabriel Deugoue
2013, 12(5): 2119-2144 doi: 10.3934/cpaa.2013.12.2119 +[Abstract](30) +[PDF](521.6KB)
We study the connection between the long-time dynamics of the 3D magnetohydrodynamic-$\alpha$ model and the exact 3D magnetohydrodynamic system. We prove that the trajectory attractor $U_\alpha$ of the 3D magnetohydrodynamic-$\alpha$ model converges to the trajectory attractor $U_0$ of the 3D magnetohydrodynamic system (in an appropriate topology) when $\alpha$ approaches zero.
Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: superposition of rarefaction and contact waves
Christian Rohde , Wenjun Wang and  Feng Xie
2013, 12(5): 2145-2171 doi: 10.3934/cpaa.2013.12.2145 +[Abstract](28) +[PDF](478.9KB)
In this paper we consider a hyperbolic-hyperbolic relaxation limit problem for a 1D compressible radiation hydrodynamics (RHD) system. The RHD system consists of the full Euler system coupled with an elliptic equation for the radiation flux. The singular relaxation limit process we consider corresponds to the physical problem of letting the Bouguer number become infinite. We prove for appropriate initial datum that the solution of the initial value problem for the RHD system converges for vanishing reciprocal Bouguer number to a weak solution of the limit system which is the Euler system. The initial data are chosen such that the limit solution is composed by a $1$-rarefaction wave, a contact discontinuity and a $3$-rarefaction wave. Moreover we give the convergence rate in terms of the physical parameter.
The Fractional Ginzburg-Landau equation with distributional initial data
Jingna Li and  Li Xia
2013, 12(5): 2173-2187 doi: 10.3934/cpaa.2013.12.2173 +[Abstract](37) +[PDF](390.8KB)
The paper is concerned with real fractional Ginzburg-Landau equation. Existence and uniqueness of local and global mild solution with distributional initial data are obtained by contraction mapping principle and carefully choosing the working space, and Gevrey regularity of mild solution for flat torus case is discussed.
Existence of positive steady states for a predator-prey model with diffusion
Wenshu Zhou , Hongxing Zhao , Xiaodan Wei and  Guokai Xu
2013, 12(5): 2189-2201 doi: 10.3934/cpaa.2013.12.2189 +[Abstract](36) +[PDF](371.6KB)
In this paper, we are concerned with the existence of positive steady states for a diffusive predator-prey model in a spatially heterogeneous environment. We completely determine the intervals of certain parameter of the model in which a positive steady state exists.
One-dimensional symmetry for semilinear equations with unbounded drift
Annalisa Cesaroni , Matteo Novaga and  Andrea Pinamonti
2013, 12(5): 2203-2211 doi: 10.3934/cpaa.2013.12.2203 +[Abstract](27) +[PDF](380.5KB)
We consider semilinear equations with unbounded drift in the whole of $R^n$ and we show that monotone solutions with finite energy are one-dimensional.
Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation
Dominique Blanchard , Nicolas Bruyère and  Olivier Guibé
2013, 12(5): 2213-2227 doi: 10.3934/cpaa.2013.12.2213 +[Abstract](35) +[PDF](423.3KB)
We give existence and uniqueness results of the weak-renormalized solution for a class of nonlinear Boussinesq's systems. The main tools rely on regularity results for the Navier-Stokes equations with precise estimates on the solution with respect to the data in dimension $2$ and on the techniques involved for renormalized solutions of parabolic problems.
Energy decay for Maxwell's equations with Ohm's law in partially cubic domains
Kim Dang Phung
2013, 12(5): 2229-2266 doi: 10.3934/cpaa.2013.12.2229 +[Abstract](30) +[PDF](548.3KB)
We prove a polynomial energy decay for the Maxwell's equations with Ohm's law in partially cubic domains with trapped rays. We extend the results of polynomial decay for the scalar damped wave equation in partially rectangular or cubic domain. Our approach have some similitude with the construction of reflected gaussian beams.
Convexity of solutions to boundary blow-up problems
Petri Juutinen
2013, 12(5): 2267-2275 doi: 10.3934/cpaa.2013.12.2267 +[Abstract](35) +[PDF](343.5KB)
We prove convexity of solutions to boundary blow-up problems for the singular infinity Laplacian and the $p$-Laplacian for $p\ge 2$. The proof is based on an extension of the results of Alvarez, Lasry and Lions [2] and on estimates of the boundary blow-up rate.
On behavior of signs for the heat equation and a diffusion method for data separation
Mi-Ho Giga , Yoshikazu Giga , Takeshi Ohtsuka and  Noriaki Umeda
2013, 12(5): 2277-2296 doi: 10.3934/cpaa.2013.12.2277 +[Abstract](32) +[PDF](479.3KB)
Consider the solution $u(x,t)$ of the heat equation with initial data $u_0$. The diffusive sign $S_D[u_0](x)$ is defined by the limit of sign of $u(x,t)$ as $t\to 0$. A sufficient condition for $x\in R^d$ and $u_0$ such that $S_D[u_0](x)$ is well-defined is given. A few examples of $u_0$ violating and fulfilling this condition are given. It turns out that this diffusive sign is also related to variational problem whose energy is the Dirichlet energy with a fidelity term. If initial data is a difference of characteristic functions of two disjoint sets, it turns out that the boundary of the set $S_D[u_0](x) = 1$ (or $-1$) is roughly an equi-distance hypersurface from $A$ and $B$ and this gives a separation of two data sets.
Classification of bifurcation diagrams of a $P$-Laplacian nonpositone problem
Po-Chun Huang , Shin-Hwa Wang and  Tzung-Shin Yeh
2013, 12(5): 2297-2318 doi: 10.3934/cpaa.2013.12.2297 +[Abstract](39) +[PDF](470.0KB)
We study the bifurcation diagrams of positive solutions of the $p$-Laplacian Dirichlet problem \begin{eqnarray*} (\varphi_p(u'(x)))'+f_\lambda(u(x))=0, -1 < x < 1, \\ u(-1)=u(1)=0, \end{eqnarray*} where $\varphi_p(y)=|y|^{p-2}y$, $(\varphi_p(u'))'$ is the one-dimensional $p$-Laplacian, $p>1$, the nonlinearity $f_\lambda(u)=\lambda g(u)-h(u),$ $g,h\in C[0,\infty)\cap C^2(0,\infty )$, and $\lambda >0$ is a bifurcation parameter. Under certain hypotheses on functions $g$ and $h$, we give a complete classification of bifurcation diagrams. We prove that, on the $(\lambda, |u|_\infty)$-plane, each bifurcation diagram consists of exactly one curve which has exactly one turning point where the curve turns to the right. Hence we are able to determine the exact multiplicity of positive solutions for each $\lambda >0.$ In addition, we show the evolution phenomena of bifurcation diagrams of polynomial nonlinearities with positive coefficients.
The expansion of gas from a wedge with small angle into a vacuum
Weixia Zhao
2013, 12(5): 2319-2330 doi: 10.3934/cpaa.2013.12.2319 +[Abstract](26) +[PDF](424.3KB)
In this paper, the problem of the expansion of a wedge of gas into vacuum is investigated. Let $\theta$ be the half angle of the wedge. For a given $\bar{\theta}$ determined by the adiabatic exponent $\gamma$, we prove the global existence of the solution through the direct approach in the case $\theta\leq\bar{\theta}$, extending the previous result obtained by Li, Yang and Zheng. Our analysis relies on the special symmetric structure of the characteristic form as well as characteristic decompositions.

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