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Communications on Pure & Applied Analysis

2013 , Volume 12 , Issue 3

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Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients
Zhigang Wang , Lei Wang and  Yachun Li
2013, 12(3): 1163-1182 doi: 10.3934/cpaa.2013.12.1163 +[Abstract](33) +[PDF](435.2KB)
Abstract:
We study the well-posedness of renormalized entropy solutions to the Cauchy problem for general degenerate parabolic-hyperbolic equations of the form \begin{eqnarray*} \partial_{t}u+ \sum_{i=1}^{d}\partial_{x_{i}f_{i}(u,t,x)}= \sum_{i,j=1}^{d}\partial_{x_j}(a_{ij}(u,t,x)\partial_{x_i}u)+\gamma(t,x) \end{eqnarray*} with initial data $u(0,x)=u_{0}(x)$, where the convection flux function $f$, the diffusion function $a$, and the source term $\gamma$ depend explicitly on the independent variables $t$ and $x$. We prove the uniqueness by using Kružkov's device of doubling variables and the existence by using vanishing viscosity method.
The Katok-Spatzier conjecture, generalized symmetries, and equilibrium-free flows
L. Bakker
2013, 12(3): 1183-1200 doi: 10.3934/cpaa.2013.12.1183 +[Abstract](32) +[PDF](451.7KB)
Abstract:
The nature and the classification of equilibrium-free flows on compact manifolds without boundary that possess nontrivial generalized symmetries are investigated. Such flows are shown to be rare in the sense that the set of those flows not possessing a generalized symmetry is residual. An equilibrium-free flow on the $2$-torus that possesses nontrivial generalized symmetries is classified as topologically conjugate to a minimal flow. A generalized symmetry is shown to be nontrivial when its Lyapunov exponent in the direction of the flow is nonzero. Conditions are given by which the multiplier of a nontrivial generalized symmetry is a real algebraic number of norm $\pm 1$. A set of conditions, which includes the Katok-Spatzier conjecture, is given by which an equilibrium-free flow on $n$-torus that possesses nontrivial generalized symmetries is shown to be projectively conjugate to an irrational flow of Koch type.
Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data
Mostafa Bendahmane , Kenneth Hvistendahl Karlsen and  Mazen Saad
2013, 12(3): 1201-1220 doi: 10.3934/cpaa.2013.12.1201 +[Abstract](41) +[PDF](435.4KB)
Abstract:
We prove existence and regularity results for distributional solutions of nonlinear elliptic and parabolic equations with general anisotropic diffusivities with variable exponents. The data are assumed to be merely integrable.
Phragmén-Lindelöf alternative for an exact heat conduction equation with delay
M. Carme Leseduarte and  Ramon Quintanilla
2013, 12(3): 1221-1235 doi: 10.3934/cpaa.2013.12.1221 +[Abstract](39) +[PDF](331.4KB)
Abstract:
In this paper we investigate the spatial behavior of the solutions for a theory for the heat conduction with one delay term. We obtain a Phragmén-Lindelöf type alternative. That is, the solutions either decay in an exponential way or blow-up at infinity in an exponential way. We also show how to obtain an upper bound for the amplitude term. Later we point out how to extend the results to a thermoelastic problem. We finish the paper by considering the equation obtained by the Taylor approximation to the delay term. A Phragmén-Lindelöf type alternative is obtained for the forward and backward in time equations.
On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent
Futoshi Takahashi
2013, 12(3): 1237-1241 doi: 10.3934/cpaa.2013.12.1237 +[Abstract](54) +[PDF](303.8KB)
Abstract:
In this note, we prove that least energy solutions of the two-dimensional Hénon equation \begin{eqnarray*} -\Delta u = |x|^{2\alpha} u^p \quad x \in \Omega, \quad u 0 \quad x \in \Omega, \quad u = 0 \quad x \in \partial \Omega, \end{eqnarray*} where $\Omega$ is a smooth bounded domain in $R^2$ with $0 \in \Omega$, $\alpha \ge 0$ is a constant and $p >1$, have only one global maximum point when $\alpha > e-1$ and the nonlinear exponent $p$ is sufficiently large. This answers positively to a recent conjecture by C. Zhao (preprint, 2011).
Infinite multiplicity for an inhomogeneous supercritical problem in entire space
Liping Wang and  Juncheng Wei
2013, 12(3): 1243-1257 doi: 10.3934/cpaa.2013.12.1243 +[Abstract](36) +[PDF](412.0KB)
Abstract:
Let $K(x)$ be a positive function in $R^N, N \geq 3$ and satisfy $\lim\limits_{|x|\rightarrow \infty} K(x) = K_\infty$ where $K_\infty$ is a positive constant. When $p > \frac{N + 1}{N - 3}, N \geq 4$, we prove the existence of infinitely many positive solutions to the following supercritical problem: \begin{eqnarray*} \Delta u(x) + K(x)u^p = 0, u>0 \quad in \quad R^N, \lim_{|x|\rightarrow \infty} u(x) = 0. \end{eqnarray*} If in addition we have, for instance, $\lim\limits_{|x| \rightarrow \infty}|x|^\mu (K(x) - K_\infty ) = C_0 \neq 0, 0 < \mu \leq N - \frac{2p+2}{p-1}$, then this result still holds provided that $p > \frac{N + 2}{N - 2}$.
On vector solutions for coupled nonlinear Schrödinger equations with critical exponents
Seunghyeok Kim
2013, 12(3): 1259-1277 doi: 10.3934/cpaa.2013.12.1259 +[Abstract](48) +[PDF](481.6KB)
Abstract:
In this paper, we study the existence and asymptotic behavior of a solution with positive components (which we call a vector solution) for the coupled system of nonlinear Schrödinger equations with doubly critical exponents \begin{eqnarray*} \Delta u + \lambda_1 u + \mu_1 u^{\frac{N+2}{N-2}} + \beta u^{\frac{2}{N-2}}v^{\frac{N}{N-2}} = 0\\ \Delta v + \lambda_2 v + \mu_2 v^{\frac{N+2}{N-2}} + \beta u^{\frac{N}{N-2}}v^{\frac{2}{N-2}} = 0 \quad in \quad \Omega\\ u, v > 0 \quad in \quad \Omega, \quad u, v = 0 \quad on \quad \partial \Omega \end{eqnarray*} as the coupling coefficient $\beta \in R$ tends to 0 or $+\infty$, where the domain $\Omega \subset R^n (N \geq 3)$ is smooth bounded and certain conditions on $\lambda_1, \lambda_2 > 0$ and $\mu_1, \mu_2 > 0$ are imposed. This system naturally arises as a counterpart of the Brezis-Nirenberg problem (Comm. Pure Appl. Math. 36: 437-477, 1983).
Uniqueness of positive steady state solutions to the unstirred chemostat model with external inhibitor
Hua Nie , Wenhao Xie and  Jianhua Wu
2013, 12(3): 1279-1297 doi: 10.3934/cpaa.2013.12.1279 +[Abstract](31) +[PDF](437.2KB)
Abstract:
A competition model of two organisms is considered in the un-stirred chemostat-type system in the presence of an external inhibitor. Asymptotic stability properties of the trivial and semi-trivial steady state solutions are established by spectral analysis. The stability and uniqueness of positive steady state solutions are also given by Lyapunov Schmidt procedure and perturbation technique.
An anisotropic regularity criterion for the 3D Navier-Stokes equations
Xuanji Jia and  Zaihong Jiang
2013, 12(3): 1299-1306 doi: 10.3934/cpaa.2013.12.1299 +[Abstract](65) +[PDF](340.8KB)
Abstract:
In this paper, we establish an anisotropic regularity criterion for the 3D incompressible Navier-Stokes equations. It is proved that a weak solution $u$ is regular on $[0,T]$, provided $\frac{\partial u_3}{\partial x_3} \in L^{t_1}(0,T;L^{s_1}(R^3))$, with $\frac{2}{t_1}+\frac{3}{s_1}\leq 2$, $s_1\in(\frac{3}{2},+\infty]$ and $\nabla_h u_3 \in L^{t_2}(0, T; L^{s_2}(R^3))$, with either $\frac{2}{t_2}+\frac{3}{s_2}\leq \frac{19}{12}+\frac{1}{2s_2}$, $s_2\in(\frac{30}{19},3]$ or $ \frac{2}{t_2}+\frac{3}{s_2}\leq \frac{3}{2}+\frac{3}{4s_2}$, $s_2\in(3,+\infty]$. Our result in fact improves a regularity criterion of Zhou and Pokorný [Nonlinearity 23 (2010), 1097--1107].
The regularity for a class of singular differential equations
Huaiyu Jian , Xiaolin Liu and  Hongjie Ju
2013, 12(3): 1307-1319 doi: 10.3934/cpaa.2013.12.1307 +[Abstract](33) +[PDF](367.8KB)
Abstract:
We find an iteration technique and thus prove the optimal global regularity for the boundary value problem of a class of singular differential equations with strongly singular lower terms at the boundary. As applications, we obtain the regularity for the radial solutions of Ginzburg-Landau equations and harmonic maps.
Global well-posedness for the Kawahara equation with low regularity
Takamori Kato
2013, 12(3): 1321-1339 doi: 10.3934/cpaa.2013.12.1321 +[Abstract](50) +[PDF](443.6KB)
Abstract:
We consider the global well-posedness for the Cauchy problem of the Kawahara equation which is one of fifth order KdV type equations. We first establish the local well-posedness in a more suitable function space for the global well-posedness by a variant of the Fourier restriction norm method introduced by Bourgain. Next, we extend this local solution globally in time by the I-method. In the present paper, we can apply the I-method to the modified Bourgain space in which the structure of the nonlinear term is reflected.
On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state
Juan Calvo
2013, 12(3): 1341-1347 doi: 10.3934/cpaa.2013.12.1341 +[Abstract](29) +[PDF](351.3KB)
Abstract:
We show that a pair of conjectures raised in [11] concerning the construction of normal solutions to the relativistic Boltzmann equation are valid. This ensures that the results in [11] hold for any range of positive temperatures and that the relativistic Euler system under the kinetic equation of state is hyperbolic and the speed of sound cannot overcome $c/\sqrt{3}$.
Spatial decay bounds in a linearized magnetohydrodynamic channel flow
Julie Lee and  J. C. Song
2013, 12(3): 1349-1361 doi: 10.3934/cpaa.2013.12.1349 +[Abstract](29) +[PDF](320.3KB)
Abstract:
This paper establishes exponential decay bounds for a transient magnetohydrodynamic flow in a semi-infinite channel. If net entrance flows into the channel are nonzero, then the solutions will not tend to zero as the distance from the entrance end tends to infinity when homogeneous lateral surface boundary conditions and homogenous initial conditions are applied. Assuming that the entrance data are small enough so that flows converge to transient laminar flows as the distance from the entrance section tends to infinity, we linearize the magnetohydrodynamic equations and derive an integro-differential inequality that leads to an exponential decay estimate. This paper also indicates how to bound the total energy in the spirit of earlier work of Lin and Payne [11].
Quasilinear elliptic problem with Hardy potential and singular term
Boumediene Abdellaoui and  Ahmed Attar
2013, 12(3): 1363-1380 doi: 10.3934/cpaa.2013.12.1363 +[Abstract](42) +[PDF](437.5KB)
Abstract:
We consider the following quasilinear elliptic problem \begin{eqnarray*} -\Delta_pu =\lambda\frac{u^{p-1}}{|x|^p}+\frac{h}{u^\gamma} \quad in \quad\Omega, \end{eqnarray*} where $1 < p < N, \Omega\subset R^N$ is a bounded regular domain such that $0\in \Omega, \gamma>0$ and $h$ is a nonnegative measurable function with suitable hypotheses.
The main goal of this work is to analyze the interaction between the Hardy potential and the singular term $u^{-\gamma}$ in order to get a solution for the largest possible class of the datum $h$. The regularity of the solution is also analyzed.
Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior
Zhijun Zhang
2013, 12(3): 1381-1392 doi: 10.3934/cpaa.2013.12.1381 +[Abstract](44) +[PDF](399.5KB)
Abstract:
In this paper, we study the existence and boundary behavior of solutions to boundary blow-up elliptic problems \begin{eqnarray*} \triangle u =b(x)f(u)(1+|\nabla u|^q), u\geq 0, \ x\in \Omega,\ u|_{\partial \Omega}=\infty, \end{eqnarray*} where $\Omega$ is a bounded domain with smooth boundary in $\mathbb R^N$, $q\in (0, 2]$, $b \in C^{\alpha}(\bar{\Omega})$ which is positive in $\Omega$, may be vanishing on the boundary, and $f$ is normalised regularly varying at infinity with positive index $p$ and $p+q>1$.
Multiple solutions for a class of $(p_1, \ldots, p_n)$-biharmonic systems
John R. Graef , Shapour Heidarkhani and  Lingju Kong
2013, 12(3): 1393-1406 doi: 10.3934/cpaa.2013.12.1393 +[Abstract](39) +[PDF](428.4KB)
Abstract:
In this paper, the authors prove the existence of at least three weak solutions for the $(p_{1},\ldots,p_{n})$-- biharmonic system $$\begin{cases} \Delta(|\Delta u_{i}|^{p_i-2}\Delta u_{i}) = \lambda F_{u_{i}}(x,u_{1},\ldots,u_{n}), & \mbox{in} \ \Omega,\\ u_{i}=\Delta u_i=0, & \mbox{on} \ \partial\Omega. \end{cases} $$ The main tool is a recent three critical points theorem of Averna and Bonanno ({\it A three critical points theorem and its applications to the ordinary Dirichlet problem}, Topol. Methods Nonlinear Anal. 22 (2003), 93-104).
Optimal regularity for parabolic Schrödinger operators
Fengping Yao
2013, 12(3): 1407-1414 doi: 10.3934/cpaa.2013.12.1407 +[Abstract](54) +[PDF](343.2KB)
Abstract:
In this paper we study the regularity theory for the parabolic Schrödinger operator $P=\frac{\partial}{\partial t}-\triangle+V$ under optimal conditions. As a corollary we obtain $L^p$-type regularity estimates for such operator.
Controllability results for a class of one dimensional degenerate/singular parabolic equations
Morteza Fotouhi and  Leila Salimi
2013, 12(3): 1415-1430 doi: 10.3934/cpaa.2013.12.1415 +[Abstract](31) +[PDF](388.5KB)
Abstract:
We study the null controllability properties of some degenerate/singular parabolic equations in a bounded interval of R. For this reason we derive a new Carleman estimate whose proof is based on Hardy inequalities.
Convexity of the free boundary for an exterior free boundary problem involving the perimeter
Hayk Mikayelyan and  Henrik Shahgholian
2013, 12(3): 1431-1443 doi: 10.3934/cpaa.2013.12.1431 +[Abstract](45) +[PDF](494.5KB)
Abstract:
We prove that if the given compact set $K$ is convex then a minimizer of the functional \begin{eqnarray*} I(v)=\int_{B_R} |\nabla v|^p dx+ Per(\{v>0\}), 1 < p < \infty, \end{eqnarray*} over the set $\{v\in W^{1,p}_0 (B_R)| v\equiv 1 \ \text{on} \ K\subset B_R\}$ has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.
Energy conservative solutions to a nonlinear wave system of nematic liquid crystals
Geng Chen , Ping Zhang and  Yuxi Zheng
2013, 12(3): 1445-1468 doi: 10.3934/cpaa.2013.12.1445 +[Abstract](32) +[PDF](467.3KB)
Abstract:
We establish the global existence of solutions to the Cauchy problem for a system of hyperbolic partial differential equations in one space dimension modeling a type of nematic liquid crystals that has equal splay and twist coefficients. Our results have no restrictions on the angles of the director, as we use the director in its natural three-component form, rather than the two-component form of spherical angles.
Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$
Jiří Benedikt
2013, 12(3): 1469-1486 doi: 10.3934/cpaa.2013.12.1469 +[Abstract](43) +[PDF](518.3KB)
Abstract:
We are concerned with the Dirichlet and Neumann eigenvalue problem for the ordinary quasilinear fourth-order ($p$-biharmonic) equation \begin{eqnarray} (|u''|^{p-2}u'')''=\lambda|u|^{p-2}u, in \quad [0,1], \quad p>1. \end{eqnarray} It is known that the eigenvalues of the Dirichlet and Neumann $p$-biharmonic problem are positive and nonnegative, respectively, isolated, and form an increasing unbounded sequence. We prove that the eigenvalues depend continuously on $p$, and that they interlace with the eigenvalues of the Navier $p$-biharmonic problem.
Asymptotic analysis of continuous opinion dynamics models under bounded confidence
Domenica Borra and  Tommaso Lorenzi
2013, 12(3): 1487-1499 doi: 10.3934/cpaa.2013.12.1487 +[Abstract](40) +[PDF](402.3KB)
Abstract:
This paper deals with the asymptotic behavior of mathematical models for opinion dynamics under bounded confidence of Deffuant-Weisbuch type. Focusing on the Cauchy Problem related to compromise models with homogeneous bound of confidence, a general well-posedness result is provided and a systematic study of the asymptotic behavior in time of the solution is developed. More in detail, we prove a theorem that establishes the weak convergence of the solution to a sum of Dirac masses and characterizes the concentration points for different values of the model parameters. Analytical results are illustrated by means of numerical simulations.
Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models
John M. Hong , Cheng-Hsiung Hsu , Bo-Chih Huang and  Tzi-Sheng Yang
2013, 12(3): 1501-1526 doi: 10.3934/cpaa.2013.12.1501 +[Abstract](53) +[PDF](667.9KB)
Abstract:
The purpose of this work is to study the existence and stability of stationary waves for viscous traffic flow models. From the viewpoint of dynamical systems, the steady-state problem of the systems can be formulated as a singularly perturbed problem. Using the geometric singular perturbation method, we establish the existence of stationary waves for both the inviscid and viscous systems. The inviscid stationary waves contain smooth waves and discontinuous transonic waves. Both waves admit viscous profiles for the viscous systems. Then we consider the linearized eigenvalue problem of the systems along smooth stationary waves. Applying the technique of center manifold reduction, we show that any one of the supersonic smooth stationary waves is spectrally unstable.

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