ISSN:

1534-0392

eISSN:

1553-5258

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## Communications on Pure & Applied Analysis

May 2012 , Volume 11 , Issue 3

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2012, 11(3): 861-883
doi: 10.3934/cpaa.2012.11.861

*+*[Abstract](1777)*+*[PDF](429.3KB)**Abstract:**

In this paper we extend the classical Bernstein estimates for systems of weakly coupled fully non-linear elliptic equations as well as scalar elliptic equations with non-local integral terms and singular kernels.

2012, 11(3): 885-909
doi: 10.3934/cpaa.2012.11.885

*+*[Abstract](1747)*+*[PDF](555.9KB)**Abstract:**

The purpose of this article is to establish regularity and pointwise upper bounds for the (relative) fundamental solution of the heat equation associated to the weighted $\bar\partial$-operator in $L^2(C^n)$ for a certain class of weights. The weights depend on a parameter, and we find pointwise bounds for heat kernel, as well as its derivatives in time, space, and the parameter. We also prove cancellation conditions for the heat semigroup. We reduce the $n$-dimensional case to the one-dimensional case, and the estimates in one-dimensional case are achieved by Duhamel's principle and commutator properties of the operators. As an application, we recover estimates of the □$_b$-heat kernel on polynomial models in $C^2$.

2012, 11(3): 911-934
doi: 10.3934/cpaa.2012.11.911

*+*[Abstract](1836)*+*[PDF](457.2KB)**Abstract:**

This paper is concerned with the existence of traveling waves for a class of degenerate reaction-diffusion systems with cross-diffusion. By applying the analytic singular perturbation method and the theorem of center manifold approximation, we prove the existence of traveling waves with transition layers for the more general degenerate systems with cross-diffusion. Especially for the degenerate S-K-T competition system with cross-diffusion we prove that some new wave patterns exhibiting competition exclusion are induced by the cross-diffusion. We also extend some of the existence results in [5] for the non-cross diffusion systems to the more general degenerate biological systems with cross-diffusion, however the detailed fast-slow structure of the waves for the systems with cross-diffusion is a little different from those for the systems without cross-diffusion.

2012, 11(3): 935-943
doi: 10.3934/cpaa.2012.11.935

*+*[Abstract](1609)*+*[PDF](352.7KB)**Abstract:**

The aim of this paper is to prove some nonexistence results for nonnegative weak solutions of the nonlinear differential inequalities with gradient nonlinearities in $R^N$. The proofs are based on the test function method developed by Bidaut-Véron, Mitidieri and Pohozaev in [3] and [14].

2012, 11(3): 945-958
doi: 10.3934/cpaa.2012.11.945

*+*[Abstract](1930)*+*[PDF](392.0KB)**Abstract:**

In this paper, we consider the the second-order ordinary differential equation with periodic boundary problem $ - \ddot{x}(t)=f(t,x(t))$, subject to $x(0)-x(2\pi)=\dot{x}(0)-\dot{x}(2\pi)=0$, where $f:C([0, 2\pi]\times R, R)$. The operator $K=(-\frac{d^2}{dt^2}+I)^{-1}$ plays an important role. By using Morse index, Leray-Schauder degree and Morse index theorem of the type Lazer-Solimini, we obtain that the equation has at least two or three nontrivial solutions without assuming nondegeneracy of critical points and has at least four nontrivial solutions assuming nondegeneracy of critical points.

2012, 11(3): 959-971
doi: 10.3934/cpaa.2012.11.959

*+*[Abstract](1773)*+*[PDF](380.5KB)**Abstract:**

In this paper we consider the magnetohydrodynamic flows giving rise to a variety of mathematical problems in many areas. We here study the issue of asymptotic analysis of the full magnetohydrodynamics flows and the main idea is based on Feireisl et al [6], [8], [9] for the Navier-Stokes-Fourier systems.

2012, 11(3): 973-980
doi: 10.3934/cpaa.2012.11.973

*+*[Abstract](2023)*+*[PDF](307.6KB)**Abstract:**

In this paper, we establish some improved regularity conditions for the 3D incompressible magnetohydrodynamic equations via only two components of the velocity and magnetic fields. This is an improvement of the result given by Ji and Lee [8].

2012, 11(3): 981-1002
doi: 10.3934/cpaa.2012.11.981

*+*[Abstract](1837)*+*[PDF](441.2KB)**Abstract:**

We present a one space dimensional model with finite speed of propagation for population dynamics, based both on the hyperbolic Cattaneo dynamics and the evolutionary game theory. We prove analytical properties of the model and global estimates for solutions, by using a hyperbolic nonlinear Trotter product formula.

2012, 11(3): 1003-1011
doi: 10.3934/cpaa.2012.11.1003

*+*[Abstract](2136)*+*[PDF](366.4KB)**Abstract:**

We study the uniqueness of positive solutions of the following coupled nonlinear Schrödinger equations: \begin{eqnarray*} \Delta u_1-\lambda_1 u_1+\mu_1u_1^3+\beta u_1u_2^2=0\quad in\quad R^N,\\ \Delta u_2-\lambda_2u_2+\mu_2u_2^3+\beta u_1^2u_2=0\quad in\quad R^N, \\ u_1>0, u_2>0, u_1, u_2 \in H^1 (R^N), \end{eqnarray*} where $N\leq3$, $\lambda_1,\lambda_2,\mu_1,\mu_2$ are positive constants and $\beta\geq 0$ is a coupling constant. We prove first the uniqueness of positive solution for sufficiently small $\beta > 0$. Secondly, assuming that $\lambda_1=\lambda_2$, we show that $u_1=u_2\sqrt{\beta-\mu_1}/\sqrt{\beta-\mu_2}$ when $\beta > \max\{\mu_1,\mu_2\}$ and thus obtain the uniqueness of positive solution using the corresponding result of scalar equation. Finally, for $N=1$ and $\lambda_1=\lambda_2$, we prove the uniqueness of positive solution when $0\leq \beta\notin [\min\{\mu_1,\mu_2\},\max\{\mu_1,\mu_2\}]$ and thus give a complete classification of positive solutions.

2012, 11(3): 1013-1036
doi: 10.3934/cpaa.2012.11.1013

*+*[Abstract](2160)*+*[PDF](545.4KB)**Abstract:**

We extend previously known Carleman estimates [18, 16, 11] for the (time-dependent) Schrödinger operator $i\partial_t+\Delta$ to a wider range for which inhomogeneous Strichartz estimates ([9, 27]) are known to hold. Then we apply them to obtain new results on unique continuation for the Schrödinger equation which include more general classes of potentials. Also, we obtain a unique continuation result for nonlinear Schrödinger equations.

2012, 11(3): 1037-1050
doi: 10.3934/cpaa.2012.11.1037

*+*[Abstract](1427)*+*[PDF](353.5KB)**Abstract:**

In this paper we obtain a Caccioppoli type estimate for the model of the linearized Mullins-Sekerka equations by a new technique, then we use this estimate to derive it's Schauder type estimates by polynomial approximation method.

2012, 11(3): 1051-1062
doi: 10.3934/cpaa.2012.11.1051

*+*[Abstract](1724)*+*[PDF](355.6KB)**Abstract:**

We construct a symbolic extension of an aperiodic zero-dimensional topological system in such a way that the bonding map is one-to-one on the set of invariant measures.

2012, 11(3): 1063-1079
doi: 10.3934/cpaa.2012.11.1063

*+*[Abstract](2150)*+*[PDF](451.6KB)**Abstract:**

The long-time asymptotics is analyzed for finite energy solutions of the 1D Schrödinger equation coupled to a nonlinear oscillator through a localized nonlinearity. The coupled system is $U(1)$ invariant. This article, which extends the results of a previous one, provides a proof of asymptotic stability of the solitary wave solutions in the case that the linearization contains a single discrete oscillatory mode satisfying a non-degeneracy assumption of the type known as the Fermi Golden Rule.

2012, 11(3): 1081-1096
doi: 10.3934/cpaa.2012.11.1081

*+*[Abstract](1945)*+*[PDF](433.9KB)**Abstract:**

The Klein-Gordon-Schrödinger system in 3D is shown to be locally well-posed for Schrödinger data in $H^s$ and wave data in $H^{\sigma}\times H^{\sigma -1}$ , if $ s > - \frac{1}{4},$ $\sigma > - \frac{1}{2}$ , $\sigma -2s > \frac{3}{2} $ and $\sigma -2 < s < \sigma +1$ . This result is optimal up to the endpoints in the sense that the local flow map is not $C^2$ otherwise. It is also shown that (unconditional) uniqueness holds for $s = \sigma = 0$ in the natural solution space $C^0([0,T],L^2) \times C^0([0,T],L^2) \times C^0([0,T],H^{-\frac{1}{2}}).$ This solution exists even globally by Colliander, Holmer and Tzirakis [6]. The proofs are based on new well-posedness results for the Zakharov system by Bejenaru, Herr, Holmer and Tataru [3], and Bejenaru and Herr [4].

2012, 11(3): 1097-1109
doi: 10.3934/cpaa.2012.11.1097

*+*[Abstract](1968)*+*[PDF](412.6KB)**Abstract:**

Motivated by the work of Grujić and Kalisch, [Z. Grujić and H. Kalisch,

*Local well-posedness of the generalized Korteweg-de Vries equation in spaces of analytic functions*, Differential and Integral Equations

**15**(2002) 1325--1334], we prove the local well-posedness for the periodic KdV equation in spaces of periodic functions analytic on a strip around the real axis without shrinking the width of the strip in time.

2012, 11(3): 1111-1127
doi: 10.3934/cpaa.2012.11.1111

*+*[Abstract](2043)*+*[PDF](438.7KB)**Abstract:**

In this work, we study a two-dimensional version of the BBM equation. We prove that the Cauchy problem for this equation is globally well-posed in a natural space. We also show that the orbital stability of the solitary waves of the equation. Furthermore, we establish that if the solution of the Cauchy problem has a compact support for all times, then this solution vanishes identically.

2012, 11(3): 1129-1156
doi: 10.3934/cpaa.2012.11.1129

*+*[Abstract](1548)*+*[PDF](603.5KB)**Abstract:**

We study the dynamics of the one dimensional Swift-Hohenberg equation defined on a large interval $(-l,l)$ with Dirichlet-Neumann boundary conditions, where $l>0$ is large and lies outside of some small neighborhoods of the points $n\pi$ and $(n+1/2)\pi,n \in N$. The arguments are based on dynamical system formulation and bifurcation theory. We show that the system with Dirichlet-Neumann boundary conditions can be reduced to a two-dimensional center manifold for each bifurcation parameter $O(l^{-2})$-close to its critical values when $l$ is sufficiently large. On this invariant manifold, we find families of steady solutions and heteroclinic connections with each connecting two different steady solutions. Moreover, by comparing the above dynamics with that of the Swift-Hohenberg equation defined on $R$ and admitting $2\pi$-spatially periodic solutions in [4], we find that the dynamics in our case preserves the main features of the dynamics in the $2\pi$ spatially periodic case.

2012, 11(3): 1157-1166
doi: 10.3934/cpaa.2012.11.1157

*+*[Abstract](1523)*+*[PDF](354.4KB)**Abstract:**

We prove one-dimensional symmetry of monotone solutions for some anisotropic quasilinear elliptic equations in the plane.

2012, 11(3): 1167-1183
doi: 10.3934/cpaa.2012.11.1167

*+*[Abstract](2086)*+*[PDF](388.6KB)**Abstract:**

In this paper, we study the 3D compressible magnetohydrodynamic equations. We extend the well-known Serrin's blow-up criterion(see [32]) for the 3D incompressible Navier-Stokes equations to the 3D compressible magnetohydrodynamic equations. In addition, initial vacuum is allowed in our case.

2012, 11(3): 1185-1203
doi: 10.3934/cpaa.2012.11.1185

*+*[Abstract](1964)*+*[PDF](442.6KB)**Abstract:**

We consider a mathematical model which describes the quasistatic frictionless contact between a piezoelectric body and a foundation. The novelty of the model consists in the fact that the foundation is assumed to be electrically conductive, the material's behavior is described with an electro-elastic-visco-plastic constitutive law, the contact is modelled with normal compliance and finite penetration and the problem is studied in an unbounded interval of time. We derive a variational formulation of the problem and prove existence, uniqueness and regularity results. The proofs are based on recent results on history-dependent quasivariational inequalities obtained in [21].

2012, 11(3): 1205-1215
doi: 10.3934/cpaa.2012.11.1205

*+*[Abstract](1955)*+*[PDF](356.9KB)**Abstract:**

For Cauchy problems given by Hamilton-Jacobi evolutive type equations, we consider the variational solution proposed by Chaperon, Sikorav and Viterbo. This is a weak, Lipschitz solution constructed via a minimax procedure from the generating function quadratic at infinity of the Lagrangian manifold associated with the Cauchy problem. We state and prove a representational formula for the variational solution. This formula requires a condition on the nature of the minimax critical value of the generating function, but makes no assumption about the convexity or concavity of the Hamiltonian. We show that it generalises the well-known formula which applies when the Hamiltonian is convex or concave in the momentum variable. We then prove that the required conditions of the formula are satisfied by the non-convex Hamiltonian arising from the control-affine $H_{\infty }$ problem. Given results in the literature that the variational solution to this problem is equivalent to the lower value of the associated differential game, we therefore obtain a representational formula for this lower value.

2012, 11(3): 1217-1229
doi: 10.3934/cpaa.2012.11.1217

*+*[Abstract](1694)*+*[PDF](398.6KB)**Abstract:**

This paper analyzes the behavior of solutions for anisotropic problems of $(p_i)$-Laplacian type as the exponents go to infinity. We show that solutions converge uniformly to a function that solves, in the viscosity sense, a certain problem that we identify. The results are presented in a two-dimensional setting but can be extended to any dimension.

2012, 11(3): 1231-1252
doi: 10.3934/cpaa.2012.11.1231

*+*[Abstract](1739)*+*[PDF](448.3KB)**Abstract:**

We consider the Cauchy problem for a non-autonomous nonclassical diffusion equation of the form $u_t-\varepsilon\Delta u_t - \Delta u+f(u)+\lambda u=g(t)$ on $R^n$. Under an arbitrary polynomial growth order of the nonlinearity $f$ and a suitable exponent growth of the external force $g$, using the method of tail-estimates and the asymptotic

*a priori*estimate method, we prove the existence of an $(H^{1}(R^n) L^{p}(R^n), H^{1}(R^n) L^{p}(R^n))$ - pullback attractor $\hat{A}_{\varepsilon}$ for the process associated to the problem. We also prove the upper semicontinuity of $\{\hat{A}_{\varepsilon}: \varepsilon\in [0,1]\}$ at $\varepsilon = 0$.

2012, 11(3): 1253-1267
doi: 10.3934/cpaa.2012.11.1253

*+*[Abstract](1742)*+*[PDF](415.8KB)**Abstract:**

It is well-known that the Ginzburg-Landau equation on $R$ has a global attractor [15] that attracts in $L^\infty_{l o c}(R)$ all the trajectories. This attractor contains bounded trajectories that are analytical functions in space. A famous theorem due to P. Collet and JP. Eckmann asserts that the $\varepsilon$-entropy per unit length in $L^\infty$ of this global attractor is finite and is smaller than the corresponding complexity for the space of functions which are analytical in a strip. This means that the global attractor is flatter than expected. We explain in this article how to establish the Collet-Eckmann Theorem in a Hilbert space framework.

2012, 11(3): 1269-1283
doi: 10.3934/cpaa.2012.11.1269

*+*[Abstract](1603)*+*[PDF](550.1KB)**Abstract:**

In this paper, we study the bifurcation of limit cycles of a class of planar quadratic reversible system $\dot{x}=y+4x^2$, $\dot{y}=-x+2xy$ under quadratic perturbations. It is proved that the cyclicity of the period annulus is equal to two.

2012, 11(3): 1285-1301
doi: 10.3934/cpaa.2012.11.1285

*+*[Abstract](2344)*+*[PDF](434.3KB)**Abstract:**

We study the large time behavior of positive solutions for the Laplace equation on the half-space with a nonlinear dynamical boundary condition. We show the convergence to the Poisson kernel in a suitable sense provided initial data are sufficiently small.

2012, 11(3): 1303-1337
doi: 10.3934/cpaa.2012.11.1303

*+*[Abstract](2122)*+*[PDF](596.9KB)**Abstract:**

We study

*small energy*solutions within the Landau-de Gennes theory for nematic liquid crystals, subject to Dirichlet boundary conditions. We consider two-dimensional and three-dimensional domains separately. In the two-dimensional case, we establish the equivalence of the Landau-de Gennes and Ginzburg-Landau theory. In the three-dimensional case, we give a new definition of the

*defect set*based on the normalized energy. In the three-dimensional uniaxial case, we demonstrate the equivalence between the defect set and the isotropic set and prove the $C^{1,\alpha}$-convergence of uniaxial small energy solutions to a limiting harmonic map, away from the defect set, for some $0 < \alpha < 1$, in the

*vanishing core limit*. Generalizations for biaxial small energy solutions are also discussed, which include physically relevant estimates for the solution and its scalar order parameters. This work is motivated by the study of defects in liquid crystalline systems and their applications.

2012, 11(3): 1339-1361
doi: 10.3934/cpaa.2012.11.1339

*+*[Abstract](2030)*+*[PDF](449.3KB)**Abstract:**

We construct a zero-entropy weakly mixing finite-valued process with the exponential limit law for return resp. hitting times. This limit law is obtained in almost every point, taking the limit along the full sequence of cylinders around the point. Till now, the exponential limit law for return resp. hitting times, taking the limit along the full sequence of cylinders, have been obtained only in positive-entropy processes satisfying some strong mixing conditions of Rosenblatt type.

2012, 11(3): 1363-1386
doi: 10.3934/cpaa.2012.11.1363

*+*[Abstract](1731)*+*[PDF](512.3KB)**Abstract:**

We are concerned with Sobolev type inequalities in $W^{1,p}_0(\Omega )$, $\Omega \subset R^n$, with optimal target norms and sharp constants. Admissible remainder terms depending on the gradient are characterized. As a consequence, the strongest possible remainder norm of the gradient is exhibited. Both the case when $p< n$ and the borderline case when $p = n$ are considered. Related Hardy inequalities with remainders are also derived.

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