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Communications on Pure & Applied Analysis

2015 , Volume 14 , Issue 6

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Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane
Piotr Biler, Ignacio Guerra and Grzegorz Karch
2015, 14(6): 2117-2126 doi: 10.3934/cpaa.2015.14.2117 +[Abstract](149) +[PDF](359.8KB)
As it is well known, the parabolic-elliptic Keller-Segel system of chemotaxis on the plane has global-in-time regular nonnegative solutions with total mass below the critical value $8\pi$. Solutions with mass above $8\pi$ blow up in a finite time. We show that the case of the parabolic-parabolic Keller-Segel is different: each mass may lead to a global-in-time-solution, even if the initial data is a finite signed measure. These solutions need not be unique, even if we limit ourselves to nonnegative solutions.
Cyclicity of some Liénard Systems
Na Li, Maoan Han and Valery G. Romanovski
2015, 14(6): 2127-2150 doi: 10.3934/cpaa.2015.14.2127 +[Abstract](118) +[PDF](490.1KB)
The Liénard system and its generalizations are important models of nonlinear oscillators. We study small-amplitude limit cycles of two families of Liénard systems and find exact number of such limit cycles bifurcating from a center or focus at the origin for these families, thus obtaining the precise bound for cyclicity of the families.
Homogenization of bending theory for plates; the case of oscillations in the direction of thickness
Maroje Marohnić and Igor Velčić
2015, 14(6): 2151-2168 doi: 10.3934/cpaa.2015.14.2151 +[Abstract](99) +[PDF](428.4KB)
In this paper we study the homogenization effects on the model of elastic plate in the bending regime, under the assumption that the energy density (material) oscillates in the direction of thickness. We study two different cases. First, we show, starting from 3D elasticity, by means of $\Gamma$-convergence and under general (not necessarily periodic) assumption, that the effective behavior of the limit is not influenced by oscillations in the direction of thickness. In the second case, we study periodic in-plane oscillations of the energy density coupled with periodic oscillations in the direction of thickness. In contrast to the first case we show that there are homogenization effects coming also from the oscillations in the direction of thickness.
Blow up threshold for a parabolic type equation involving space integral and variational structure
Baiyu Liu and Li Ma
2015, 14(6): 2169-2183 doi: 10.3934/cpaa.2015.14.2169 +[Abstract](104) +[PDF](376.9KB)
In this paper, we study a parabolic type equation involving space integrals on a bounded smooth domain. First, using the Banach fixed point theorem, we establish the well-posedness in Lebesgue spaces. Then, with the help of Nehari functional, we find the threshold of the initial data such that the solution either exists globally or blows up in finite time.
On some semilinear equation in $R^4$ containing a Laplacian term and involving nonlinearity with exponential growth
Sami Aouaoui
2015, 14(6): 2185-2201 doi: 10.3934/cpaa.2015.14.2185 +[Abstract](118) +[PDF](404.5KB)
In this paper, we prove a multiplicity result for some semilinear elliptic equation of biharmoninc type in $R^4$ containing a Laplacian term. The nonlinear term exhibits an exponential growth.
An improved result for the full justification of asymptotic models for the propagation of internal waves
Ralph Lteif, Samer Israwi and Raafat Talhouk
2015, 14(6): 2203-2230 doi: 10.3934/cpaa.2015.14.2203 +[Abstract](73) +[PDF](495.9KB)
We consider here asymptotic models that describe the propagation of one-dimensional internal waves at the interface between two layers of immiscible fluids of different densities, under the rigid lid assumption and with uneven bottoms. The aim of this paper is to show that the full justification result of the model obtained by Duchêne, Israwi and Talhouk [SIAM J. Math. Anal., 47(1), 240–-290], in the sense that it is consistent, well-posed, and that its solutions remain close to exact solutions of the full Euler system with corresponding initial data, can be improved in two directions. The first direction is taking into account medium amplitude topography variations and the second direction is allowing strong nonlinearity using a new pseudo-symmetrizer, thus canceling out the smallness assumptions of the Camassa-Holm regime for the well-posedness and stability results.
On Dirac equation with a potential and critical Sobolev exponent
Wenmin Gong and Guangcun Lu
2015, 14(6): 2231-2263 doi: 10.3934/cpaa.2015.14.2231 +[Abstract](130) +[PDF](571.4KB)
In this paper we consider a critical Dirac equation with a potential on a compact spin manifold. We prove the existence of solutions based on the analysis of the spectrum of Dirac operator with a potential and the dual variational method.
On Fractional Schrödinger Equations in sobolev spaces
Younghun Hong and Yannick Sire
2015, 14(6): 2265-2282 doi: 10.3934/cpaa.2015.14.2265 +[Abstract](288) +[PDF](445.8KB)
Let $\sigma \in (0,1)$ with $\sigma \neq \frac{1}{2}$. We investigate the fractional nonlinear Schrödinger equation in $\mathbb R^d$: \begin{eqnarray} i\partial_tu+(-\Delta)^\sigma u+\mu|u|^{p-1}u=0, u(0)=u_0\in H^s, \end{eqnarray} where $(-\Delta)^\sigma$ is the Fourier multiplier of symbol $|\xi|^{2\sigma}$, and $\mu=\pm 1$. This model has been introduced by Laskin in quantum physics [23]. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.
Large time behavior of solution for the full compressible navier-stokes-maxwell system
Weike Wang and Xin Xu
2015, 14(6): 2283-2313 doi: 10.3934/cpaa.2015.14.2283 +[Abstract](148) +[PDF](524.6KB)
In this paper, the Cauchy problem for the compressible Navier-Stokes-Maxwell equation is studied in $R^3$, the $L^p$ time decay rate for the global smooth solution is established. Our method is mainly based on a detailed analysis to the Green's function of the linearized system and some elaborate energy estimates. To give the explicit representation of the Green's function, we use the Helmholtz decomposition by which we can decompose the solution into two parts and give the expression to each part. Our results show a sharp difference between the decay of solution for Navier-Stokes-Maxwell system and that for the Navier-Stokes equation.
Krasnosel'skii type formula and translation along trajectories method on the scale of fractional spaces
Piotr Kokocki
2015, 14(6): 2315-2334 doi: 10.3934/cpaa.2015.14.2315 +[Abstract](93) +[PDF](454.9KB)
We provide a global continuation principle of periodic solutions for the equation $\dot u = - Au + F(t,u)$, where $ A:D(A) \to X$ is a sectorial operator on a Banach space $X$ and $F:[0, +\infty) \times X^\alpha \to X$ is a nonlinear map defined on a fractional space $X^\alpha$. The approach that we use in this paper is based upon the theory of topological invariants that applies in the situation when the Poincaré operator associated with the equation is endowed with some form of compactness.
Nonlinear elliptic systems involving the fractional Laplacian in the unit ball and on a half space
Chenchen Mou
2015, 14(6): 2335-2362 doi: 10.3934/cpaa.2015.14.2335 +[Abstract](185) +[PDF](504.6KB)
In this paper, we study the following nonlinear elliptic system \begin{eqnarray} \left\{\begin{array}{ll} (-\Delta)^{\frac{\alpha}{2}}u_i=f_i(u),\ x\in \Omega,\quad i=1,...,m, \\ u_i(x)=0, \quad \quad\quad \ \ x\in \Omega^c,\quad i=1,...,m, \end{array} \right. \end{eqnarray} where $0 < \alpha < 2$ and $\Omega$ is either the unit ball $B_1(0)=\{x\in \mathbb R^n | \|x\| < 1 \}$ or the half space $\mathbb R_+^n = \{x=(x_1,...,x_n)\in \mathbb R^n | x_n > 0\}$. Instead of investigating the pseudo differential system directly, we study an equivalent integral system, i.e., \begin{eqnarray} u_i(x)=\int_{B_1(0)}G_1(x,y)f_i(u(y))dy,\quad x\in B_1(0),\quad i=1,...,m, \end{eqnarray} and \begin{eqnarray} u_i(x)=C_ix_n^{\frac{\alpha}{2}}+\int_{\mathbb{R}_+^n}G_{\infty}(x,y)f_i(u(y))dy,\quad x\in \mathbb{R}_+^n,\quad i=1,...,m, \end{eqnarray} where $C_i$ are non-negative constants, $G_1(x,y)$ is Green's function for $B_1(0)$ and $G_{\infty}(x,y)$ is Green function of $\mathbb R_+^n$. We use the method of moving planes in integral forms to prove the radial symmetry and monotonicity of positive solutions in $B_1(0)$ and non-existence of positive solutions in $\mathbb R_+^n$. Moreover, we also study regularity of positive solutions in $B_1(0)$.
Harnack inequality for degenerate elliptic equations and sum operators
Giuseppe Di Fazio, Maria Stella Fanciullo and Pietro Zamboni
2015, 14(6): 2363-2376 doi: 10.3934/cpaa.2015.14.2363 +[Abstract](109) +[PDF](382.7KB)
We define Stummel-Kato type classes in a quasimetric homogeneous setting using sum operators introduced in [13] by Franchi, Perez and Wheeden. Then we prove a Harnack inequality for positive solutions of some linear subelliptic equations.
The Liouville theorems for elliptic equations with nonstandard growth
Tomasz Adamowicz and Przemysław Górka
2015, 14(6): 2377-2392 doi: 10.3934/cpaa.2015.14.2377 +[Abstract](190) +[PDF](436.4KB)
We study solutions and supersolutions of homogeneous and nonhomogeneous $A$-harmonic equations with nonstandard growth in $\mathbb{R}^n$. Various Liouville-type theorems and nonexistence results are proved. The discussion is illustrated by a number of examples.
Symmetry of solutions to semilinear equations involving the fractional laplacian
Lizhi Zhang
2015, 14(6): 2393-2409 doi: 10.3934/cpaa.2015.14.2393 +[Abstract](175) +[PDF](430.9KB)
Let $0<\alpha<2$ be any real number. Let $\Omega\subset\mathbb{R}^n$ be a bounded or an unbounded domain which is convex and symmetric in $x_1$ direction. We investigate the following Dirichlet problem involving the fractional Laplacian: \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x)=f(x,u), & \qquad x\in\Omega,                                              (1)\\ u(x)\equiv0, & \qquad x\notin\Omega. \end{array}\right. \end{equation}
    Applying a direct method of moving planes for the fractional Laplacian, with the help of several maximum principles for anti-symmetric functions, we prove the monotonicity and symmetry of positive solutions in $x_1$ direction as well as nonexistence of positive solutions under various conditions on $f$ and on the solutions $u$. We also extend the results to some more complicated cases.
Least energy solutions for an elliptic problem involving sublinear term and peaking phenomenon
Qiuping Lu and Zhi Ling
2015, 14(6): 2411-2429 doi: 10.3934/cpaa.2015.14.2411 +[Abstract](158) +[PDF](484.6KB)
For a general elliptic problem $-\triangle{u} = g(u)$ in $R^N$ with $N \ge 3$, we show that all solutions have compact support and there exists a least energy solution, which is radially symmetric and decreases with respect to $|x| = r$. With this result we study a singularly perturbed elliptic problem $ -\epsilon^{2} \triangle{u} + |u|^{q-1}u = \lambda u + f(u)$ in a bounded domain $\Omega$ with $0 < q < 1 $, $\lambda \ge 0$ and $ u \in H^1_0(\Omega) $. For any $y \in \Omega$, we show that there exists a least energy solution $u_{\epsilon}$, which concentrates around this point $y$ as $\epsilon \to 0$. Conversely when $\epsilon $ is small, the boundary of the set $\{ x \in \Omega | u_{\epsilon}(x)>0 \}$ is a free boundary, where $u_{\epsilon}$ is any nonnegative least energy solution.
Regularity and nonexistence of solutions for a system involving the fractional Laplacian
De Tang and Yanqin Fang
2015, 14(6): 2431-2451 doi: 10.3934/cpaa.2015.14.2431 +[Abstract](123) +[PDF](475.9KB)
We consider a system involving the fractional Laplacian \begin{eqnarray} \left\{ \begin{array}{ll} (-\Delta)^{\alpha_{1}/2}u=u^{p_{1}}v^{q_{1}} & \mbox{in}\ \mathbb{R}^N_+,\\ (-\Delta)^{\alpha_{2}/2}v=u^{p_{2}}v^{q_{2}} &\mbox{in}\ \mathbb{R}^N_+,\\ u=v=0,&\mbox{in}\ \mathbb{R}^N\backslash\mathbb{R}^N_+, \end{array} \right. \end{eqnarray} where $\alpha_{i}\in (0,2)$, $p_{i},q_{i}>0$, $i=1,2$. Based on the uniqueness of $\alpha$-harmonic function [9] on half space, the equivalence between (1) and integral equations \begin{eqnarray} \left\{ \begin{array}{ll} u(x)=C_{1}x_{N}^{\frac{\alpha_{1}}{2}}+\displaystyle\int_{\mathbb{R}_{+}^{N}}G^{1}_{\infty}(x,y)u^{p_{1}}(y)v^{q_{1}}(y)dy,\\ v(x)=C_{2}x_{N}^{\frac{\alpha_{2}}{2}}+\displaystyle\int_{\mathbb{R}_{+}^{N}}G^{2}_{\infty}(x,y)u^{p_{2}}(y)v^{q_{2}}(y)dy. \end{array} \right. \end{eqnarray} are derived. Based on this result we deal with integral equations (2) instead of (1) and obtain the regularity. Especially, by the method of moving planes in integral forms which is established by Chen-Li-Ou [12], we obtain the nonexistence of positive solutions of integral equations (2) under only local integrability assumptions.
Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces
Hi Jun Choe, Bataa Lkhagvasuren and Minsuk Yang
2015, 14(6): 2453-2464 doi: 10.3934/cpaa.2015.14.2453 +[Abstract](149) +[PDF](353.7KB)
We consider the Keller-Segel model coupled with the incompressible Navier-Stokes equations in dimension three. We prove the local in time existence of the solution for large initial data and the global in time existence of the solution for small initial data plus some smallness condition on the gravitational potential in the critical Besov spaces, which are new results for the model.
Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source
Xiaoli Zhu, Fuyi Li and Ting Rong
2015, 14(6): 2465-2485 doi: 10.3934/cpaa.2015.14.2465 +[Abstract](164) +[PDF](475.7KB)
In this paper, a class of pseudo-parabolic equations with an exponential source is concerned. First, by the elliptic regularity theory, we establish the local existence and uniqueness of solutions. Sequently, the global existence and blow up of solutions with lower initial energy is considered via the potential wells method. Finally, we find a sufficient condition under which the solution blows up without any limit of initial energy by constructing a new functional which falls between the energy functional and Nehari functional. During this process, the properties of global solutions are also studied.
Nodal solutions for a quasilinear Schrödinger equation with critical nonlinearity and non-square diffusion
Yinbin Deng, Yi Li and Xiujuan Yan
2015, 14(6): 2487-2508 doi: 10.3934/cpaa.2015.14.2487 +[Abstract](176) +[PDF](477.6KB)
This paper is concerned with a type of quasilinear Schrödinger equations of the form \begin{eqnarray} -\Delta u+V(x)u-p\Delta(|u|^{2p})|u|^{2p-2}u=\lambda|u|^{q-2}u+|u|^{2p2^{*}-2}u, \end{eqnarray} where $\lambda>0, N\ge3, 4p < q < 2p2^*, 2^*=\frac{2N}{N-2}, 1< p < +\infty$. For any given $k \ge 0$, by using a change of variables and Nehari minimization, we obtain a sign-changing minimizer with $k$ nodes.
Gaps in the spectrum of the Laplacian on $3N$-Gaskets
D. Kelleher, N. Gupta, M. Margenot, J. Marsh, W. Oakley and A. Teplyaev
2015, 14(6): 2509-2533 doi: 10.3934/cpaa.2015.14.2509 +[Abstract](123) +[PDF](1015.4KB)
This article develops analysis on fractal $3N$-gaskets, a class of post-critically finite fractals which include the Sierpinski triangle for $N=1$, specifically properties of the Laplacian $\Delta$ on these gaskets. We first prove the existence of a self-similar geodesic metric on these gaskets, and prove heat kernel estimates for this Laplacian with respect to the geodesic metric. We also compute the elements of the method of spectral decimation, a technique used to determine the spectrum of post-critically finite fractals. Spectral decimation on these gaskets arises from more complicated dynamics than in previous examples, i.e. the functions involved are rational rather than polynomial. Due to the nature of these dynamics, we are able to show that there are gaps in the spectrum.
Dynamics of a host-pathogen system on a bounded spatial domain
Feng-Bin Wang, Junping Shi and Xingfu Zou
2015, 14(6): 2535-2560 doi: 10.3934/cpaa.2015.14.2535 +[Abstract](128) +[PDF](532.4KB)
We study a host-pathogen system in a bounded spatial habitat where the environment is closed. Extinction and persistence of the disease are investigated by appealing to theories of monotone dynamical systems and uniform persistence. We also carry out a bifurcation analysis for steady state solutions, and the results suggest that a backward bifurcation may occur when the parameters in the system are spatially dependent.
Nonlinear Neumann problems with indefinite potential and concave terms
Shouchuan Hu and Nikolaos S. Papageorgiou
2015, 14(6): 2561-2616 doi: 10.3934/cpaa.2015.14.2561 +[Abstract](103) +[PDF](659.0KB)
In this paper we conduct a detailed study of Neumann problems driven by a nonhomogeneous differential operator plus an indefinite potential and with concave contribution in the reaction. We deal with both superlinear and sublinear (possibly resonant) problems and we produce constant sign and nodal solutions. We also examine semilinear equations resonant at higher parts of the spectrum and equations with a negative concavity.

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