ISSN:

1534-0392

eISSN:

1553-5258

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## Communications on Pure & Applied Analysis

2016 , Volume 15 , Issue 2

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2016, 15(2): 299-317
doi: 10.3934/cpaa.2016.15.299

*+*[Abstract](529)*+*[PDF](488.4KB)**Abstract:**

The phase separation of an isothermal incompressible binary fluid in a porous medium can be described by the so-called Brinkman equation coupled with a convective Cahn-Hilliard (CH) equation. The former governs the average fluid velocity $\mathbf{u}$, while the latter rules evolution of $\varphi$, the difference of the (relative) concentrations of the two phases. The two equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular, the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg force) which is proportional to $\mu\nabla\varphi$, where $\mu$ is the chemical potential. When the viscosity vanishes, then the system becomes the Cahn-Hilliard-Hele-Shaw (CHHS) system. Both systems have been studied from the theoretical and the numerical viewpoints. However, theoretical results on the CHHS system are still rather incomplete. For instance, uniqueness of weak solutions is unknown even in 2D. Here we replace the usual CH equation with its physically more relevant nonlocal version. This choice allows us to prove more about the corresponding nonlocal CHHS system. More precisely, we first study well-posedness for the CHB system, endowed with no-slip and no-flux boundary conditions. Then, existence of a weak solution to the CHHS system is obtained as a limit of solutions to the CHB system. Stronger assumptions on the initial datum allow us to prove uniqueness for the CHHS system. Further regularity properties are obtained by assuming additional, though reasonable, assumptions on the interaction kernel. By exploiting these properties, we provide an estimate for the difference between the solution to the CHB system and the one to the CHHS system with respect to viscosity.

2016, 15(2): 319-334
doi: 10.3934/cpaa.2016.15.319

*+*[Abstract](485)*+*[PDF](386.5KB)**Abstract:**

This paper is concerned with the traveling wave solutions of a reaction-diffusion equation with state-dependent delay. When the birth function is monotone, the existence and nonexistence of monotone traveling wave solutions are established. When the birth function is not monotone, the minimal wave speed of nontrivial traveling wave solutions is obtained. The results are proved by the construction of upper and lower solutions and application of the fixed point theorem.

2016, 15(2): 335-340
doi: 10.3934/cpaa.2016.15.335

*+*[Abstract](518)*+*[PDF](352.7KB)**Abstract:**

In this note, we continue our study of weak solution $u_k$ to fractional elliptic equation $(-\Delta)^\alpha u+u^p=k\delta_0$ in $\Omega$ which vanishes in $\Omega^c$, where $\Omega\subset \mathbb{R}^N (N\ge2)$ is an open $C^2$ domain containing $0$, $(-\Delta)^\alpha$ with $\alpha\in(0,1)$ is the fractional Laplacian, $k>0$ and $\delta_0$ is the Dirac mass at $0$. We prove that the limit of $u_k$ as $k\to\infty$ blows up in whole $\Omega$ when $p=\min\{1+\frac{2\alpha}{N},\frac{N}{2\alpha}\}$ and $1+\frac{2\alpha}{N}\not=\frac{N}{2\alpha}$.

2016, 15(2): 341-365
doi: 10.3934/cpaa.2016.15.341

*+*[Abstract](475)*+*[PDF](529.6KB)**Abstract:**

We consider the nonlinear Schrödinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the formulation introduced by Bourgain [4]. This result reproves a dynamical consequence of the infinite dimensional Birkhoff normal form theorem by Bambusi and Grebert [2].

2016, 15(2): 367-383
doi: 10.3934/cpaa.2016.15.367

*+*[Abstract](490)*+*[PDF](456.3KB)**Abstract:**

Denote with $\mu _{1}(\Omega ;e^{h( |x|) })$ the first nontrivial eigenvalue of the Neumann problem \begin{eqnarray} &-div( e^{h( |x|) }\nabla u) =\mu e^{h(|x|) }u \quad in \ \Omega \\ &\frac{\partial u}{\partial \nu }=0 \quad on \ \partial \Omega, \end{eqnarray} where $\Omega $ is a bounded and Lipschitz domain in $\mathbb{R}^{N}$. Under suitable assumption on $h$ we prove that the ball centered at the origin is the unique set maximizing $\mu _{1}(\Omega ;e^{h( |x|)})$ among all Lipschitz bounded domains $\Omega $ of $\mathbb{R}^{N}$ of prescribed $ e^{h( |x|) }dx$-measure and symmetric about the origin. Moreover, an example in the model case $h( |x|) =|x|^{2},$ shows that, in general, the assumption on the symmetry of the domain cannot be dropped. In the one-dimensional case, i.e. when $\Omega $ reduces to an interval $(a,b), $ we consider a wide class of weights (including both Gaussian and anti-Gaussian). We then describe the behavior of the eigenvalue as the interval $(a,b)$ slides along the $x$-axis keeping fixed its weighted length.

2016, 15(2): 385-398
doi: 10.3934/cpaa.2016.15.385

*+*[Abstract](448)*+*[PDF](396.4KB)**Abstract:**

In this paper, we are concerned with the sufficient and necessary conditions for the existence and nonexistence of the positive solutions of the following system involving Wolff type potential: \begin{eqnarray} & u(x) =c_{1}(x)W_{\beta,\gamma}(v^{q})(x), \\ &v(x) =c_{2}(x)W_{\alpha,\tau}(u^{p})(x). \end{eqnarray} Here $x\in R^{n}$, $1 < \gamma,\tau \leq 2$, $\alpha,\beta > 0$, $0< \beta\gamma$, $\alpha \tau < n $, and the functions $c_{1}(x),c_{2}(x)$ are double bounded. This system is helpful to well understand some nonlinear PDEs and other nonlinear problems. Different from the case of $\alpha=\beta,\gamma=\tau$, it is more difficult to handle the critical condition. Fortunately, by applying the special iteration scheme and some critical asymptotic analysis, we establish the sharp criteria for existence and nonexistence of positive solutions to system (0.1). Then, we use the method of moving planes to prove the symmetry and monotonicity for the positive solutions of (0.1) when $c_{1}(x)\equiv c_{2}(x)\equiv1$ in the case \begin{eqnarray} \frac{\gamma-1}{p+\gamma-1}+\frac{\tau-1}{q+\tau-1}=\frac{n-\alpha\tau}{2n-\alpha\tau+\beta\gamma} +\frac{n-\beta\gamma}{2n-\beta\gamma+\alpha\tau}. \end{eqnarray}

2016, 15(2): 399-412
doi: 10.3934/cpaa.2016.15.399

*+*[Abstract](439)*+*[PDF](337.6KB)**Abstract:**

Structure of solutions of boundary value problems for a semilinear elliptic equation with singular nonlinearity is studied. It is seen that the structure of solutions relies on the boundary values. The global branches of solutions of the boundary value problems are established. Moreover, some Liouville type results for the entire solutions of the equation are also obtained.

2016, 15(2): 413-428
doi: 10.3934/cpaa.2016.15.413

*+*[Abstract](482)*+*[PDF](418.1KB)**Abstract:**

In this paper, we study a fractional nonlinear Schrödinger equation. Applying the finite reduction method, we prove that the equation has multi-bump positive solutions under some suitable conditions which are given in section 1.

2016, 15(2): 429-444
doi: 10.3934/cpaa.2016.15.429

*+*[Abstract](559)*+*[PDF](442.4KB)**Abstract:**

In this article, we are concerned with the boundary value problem \begin{equation} \left\{ \begin{array}{ll} \displaystyle -\left(a+b\int_{\Omega}|\nabla u|^{2}\right)\Delta u +\phi u= f(x, u) &\text{in }\Omega \hbox{} \\ -\Delta \phi= u^{2} &\text{in }\Omega \hbox{} \\ u=\phi=0&\text{on }\partial\Omega, \hbox{} \end{array} \right. \end{equation} where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$ ($N=1,2$ or $3$), $a>0$, $b\geq0$, and $f:\overline{\Omega}\times \mathbb{R}\to\mathbb{R}$ is a continuous function which is globally $3$-superlinear. By using some variants of the mountain pass theorem established in this paper, we show that this problem has at least three solutions: one positive, one negative, and one which changes its sign. Furthermore, in case $f$ is odd with respect to $u$ we obtain an unbounded sequence of sign-changing solutions.

2016, 15(2): 445-455
doi: 10.3934/cpaa.2016.15.445

*+*[Abstract](391)*+*[PDF](350.3KB)**Abstract:**

In this paper, the existence of a positive solution for the Kirchhoff-type equations in $\mathbb{R}^N$ is proved by using cut-off and monotonicity tricks, which unify and sharply improve the results of Li et al. [Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012) 2285--2294]. Our result cover the case where the nonlinearity satisfies asymptotically linear and superlinear at infinity.

2016, 15(2): 457-475
doi: 10.3934/cpaa.2016.15.457

*+*[Abstract](369)*+*[PDF](438.6KB)**Abstract:**

The existence of traveling front solutions to bistable lattice differential equations in the absence of a comparison principle is studied. The results are in the spirit of those in Bates, Chen, and Chmaj [1], but are applicable to vector equations and to more general limiting systems. An abstract result on the persistence of traveling wave solutions is obtained and is then applied to lattice differential equations with repelling first and/or second neighbor interactions and to some problems with infinite range interactions.

2016, 15(2): 477-494
doi: 10.3934/cpaa.2016.15.477

*+*[Abstract](509)*+*[PDF](454.8KB)**Abstract:**

We study the initial and initial-boundary value problems for the $p$-th power Newtonian fluid in one space dimension with general large initial data. The existence and uniqueness of globally smooth non-vacuum solutions are established when the thermal conductivity is some non-negative power of the temperature. Our analysis is based on some detailed estimates on the bounds of both density and temperature.

2016, 15(2): 495-506
doi: 10.3934/cpaa.2016.15.495

*+*[Abstract](457)*+*[PDF](407.2KB)**Abstract:**

In this work we prove continuity of solutions with respect to initial conditions and couple parameters and we prove joint upper semicontinuity of a family of global attractors for the problem \begin{eqnarray} &\frac{\partial u_{s}}{\partial t}(t)-\textrm{div}(D_s|\nabla u_{s}|^{p_s(x)-2}\nabla u_{s})+|u_s|^{p_s(x)-2}u_s=B(u_{s}(t)),\;\; t>0,\\ &u_{s}(0)=u_{0s}, \end{eqnarray} under homogeneous Neumann boundary conditions, $u_{0s}\in H:=L^2(\Omega),$ $\Omega\subset\mathbb{R}^n$ ($n\geq 1$) is a smooth bounded domain, $B:H\rightarrow H$ is a globally Lipschitz map with Lipschitz constant $L\geq 0$, $D_s\in[1,\infty)$, $p_s(\cdot)\in C(\bar{\Omega})$, $p_s^-:=\textrm{ess inf}\;p_s\geq p,$ $p_s^+:=\textrm{ess sup}\;p_s\leq a,$ for all $s\in \mathbb{N},$ when $p_s(\cdot)\rightarrow p$ in $L^\infty(\Omega)$ and $D_s\rightarrow\infty$ as $s\rightarrow\infty,$ with $a,p>2$ positive constants.

2016, 15(2): 507-517
doi: 10.3934/cpaa.2016.15.507

*+*[Abstract](430)*+*[PDF](358.2KB)**Abstract:**

We study a local regularity condition for a suitable weak solutions of the magnetohydrodynamics equations near the curved boundary.

2016, 15(2): 519-533
doi: 10.3934/cpaa.2016.15.519

*+*[Abstract](437)*+*[PDF](397.5KB)**Abstract:**

We examine the equation $$ \Delta^2 u = \lambda f(u) \qquad \Omega, $$ with either Navier or Dirichlet boundary conditions. We show some uniqueness results under certain constraints on the parameter $ \lambda$. We obtain similar results for the sytem \begin{eqnarray} &-\Delta u = \lambda f(v) \qquad \Omega, \\ &-\Delta v = \gamma g(u) \qquad \Omega, \\ &u= v = 0 \qquad \partial \Omega. \end{eqnarray}

2016, 15(2): 535-547
doi: 10.3934/cpaa.2016.15.535

*+*[Abstract](446)*+*[PDF](406.1KB)**Abstract:**

In this paper we study the non-linear fractional Schrödinger equation with steep potential well \begin{eqnarray} (-\Delta)^{\alpha}u + \lambda V(x)u = f(x,u)\ in\ R^{n}, \ u\in H^{\alpha}(R^n), \end{eqnarray} where $(-\Delta)^\alpha$ ($\alpha \in (0,1)$) denotes the fractional Laplacian, $\lambda$ is a parameter, $V\in C(\mathbb{R}^n)$ and $V^{-1}(0)$ has nonempty interior. Under some suitable conditions, the existence of nontrivial solutions are obtained by using variational methods. Furthermore, the phenomenon of concentration of solutions is also explored.

2016, 15(2): 549-562
doi: 10.3934/cpaa.2016.15.549

*+*[Abstract](404)*+*[PDF](410.1KB)**Abstract:**

In this paper we consider the elliptic system $\Delta u = u^p -v^q$, $\Delta v= -u^r +v^s$ in $\Omega$, where the exponents verify $p,s>1$, $q,r>0$ and $ps>qr$, and $\Omega$ is a smooth bounded domain of $R^N$. First, we show existence and uniqueness of boundary blow-up solutions, that is, solutions $(u,v)$ verifying $u=v=+\infty$ on $\partial \Omega$. Then, we use them to analyze the removability of singularities of positive solutions of the system in the particular case $qr\leq 1$, where comparison is available.

2016, 15(2): 563-576
doi: 10.3934/cpaa.2016.15.563

*+*[Abstract](432)*+*[PDF](415.2KB)**Abstract:**

The aim of this paper is to show the existence of infinitely many concentration solutions for the fractional Nirenberg problem under the condition that $Q_s$ curvature has a sequence of strictly local maximum points moving to infinity.

2016, 15(2): 577-598
doi: 10.3934/cpaa.2016.15.577

*+*[Abstract](382)*+*[PDF](493.3KB)**Abstract:**

We study the center problem for planar systems with a linear center at the origin that in complex coordinates have a nonlinearity formed by the sum of two monomials. Our first result lists several centers inside this family. To the best of our knowledge this list includes a new class of Darboux centers that are also persistent centers. The rest of the paper is dedicated to try to prove that the given list is exhaustive. We get several partial results that seem to indicate that this is the case. In particular, we solve the question for several general families with arbitrary high degree and for all cases of degree less or equal than 19. As a byproduct of our study we also obtain the highest known order for weak-foci of planar polynomial systems of some given degrees.

2016, 15(2): 599-622
doi: 10.3934/cpaa.2016.15.599

*+*[Abstract](421)*+*[PDF](502.0KB)**Abstract:**

In this paper, we study the following Hamiltonian elliptic system with gradient term \begin{eqnarray} &-\epsilon^{2}\Delta \psi +\epsilon b\cdot \nabla \psi +\psi+V(x)\varphi=K(x)f(|\eta|)\varphi \ \ \hbox{in}~\mathbb{R}^{N},\\ &-\epsilon^{2}\Delta \varphi -\epsilon b\cdot \nabla \varphi +\varphi+V(x)\psi=K(x)f(|\eta|)\psi \ \ \hbox{in}~\mathbb{R}^{N}, \end{eqnarray} where $\eta=(\psi,\varphi):\mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$, $V, K\in C(\mathbb{R}^{N}, \mathbb{R})$, $\epsilon$ is a small positive parameter and $b$ is a constant vector. Suppose that $V(x)$ is sign-changing and has at least one global minimum, and $K(x)$ has at least one global maximum, we prove the existence, exponential decay and concentration phenomena of semiclassical ground state solutions for all sufficiently small $\epsilon>0$.

2016, 15(2): 623-636
doi: 10.3934/cpaa.2016.15.623

*+*[Abstract](438)*+*[PDF](510.3KB)**Abstract:**

We discuss non-sharp travelling waves of a dual porous medium equation with monostable source and bistable source respectively. We show the existence of non-sharp travelling waves and find that though the equation is degenerate, the travelling waves are classical ones. Furthermore, for the monostable source, we show that the non-sharp travelling waves are infinite, while for the bistable source, the non-sharp travelling waves are semi-finite, which is in contrast with the case of the heat equation.

2016, 15(2): 637-655
doi: 10.3934/cpaa.2016.15.637

*+*[Abstract](433)*+*[PDF](467.5KB)**Abstract:**

This paper is devoted to the study of a spatially and size-structured population dynamics model with delayed birth process. Our focus is on the asymptotic behavior of the system, in particular on the effect of the spatial location and the time lag on the long-term dynamics. To this end, within a semigroup framework, we derive the locally asymptotic stability and asynchrony results respectively for the considered population system under some conditions. For our discussion, we use the approaches concerning operator matrices, Hille-Yosida operators, spectral analysis as well as Perron-Frobenius theory. We also do two numerical simulations to illustrate the obtained stability and asynchrony results.

2016, 15(2): 657-699
doi: 10.3934/cpaa.2016.15.657

*+*[Abstract](527)*+*[PDF](678.1KB)**Abstract:**

We consider a fractional Laplace equation and we give a self-contained elementary exposition of the representation formula for the Green function on the ball. In this exposition, only elementary calculus techniques will be used, in particular, no probabilistic methods or computer assisted algebraic manipulations are needed. The main result in itself is not new (see for instance [2, 9]), however we believe that the exposition is original and easy to follow, hence we hope that this paper will be accessible to a wide audience of young researchers and graduate students that want to approach the subject, and even to professors that would like to present a complete proof in a PhD or Master Degree course.

2017 Impact Factor: 0.884

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