# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

All Issues

## Communications on Pure & Applied Analysis

September 2016 , Volume 15 , Issue 5

Select all articles

Export/Reference:

2016, 15(5): 1515-1543 doi: 10.3934/cpaa.2016001 +[Abstract](2395) +[PDF](544.7KB)
Abstract:
We consider a heat--structure interaction model where the structure is subject to viscoelastic (strong) damping. This is a preliminary step toward the study of a fluid--structure interaction model where the heat equation is replaced by the linear version of the Navier--Stokes equation as it arises in applications. We prove four main results: analyticity of the corresponding contraction semigroup (which cannot follow by perturbation); sharp location of the spectrum of its generator, which does not have compact resolvent, and has the point $\lambda=-1$ in its continuous spectrum; exponential decay of the semigroup with sharp decay rate; finally, a characterization of the domains of fractional power related to the generator.
2016, 15(5): 1545-1570 doi: 10.3934/cpaa.2016002 +[Abstract](2221) +[PDF](496.1KB)
Abstract:
We consider parametric equations driven by the $p$ - Laplacian and with a reaction which has a $p$ - logistic form or is the sum of two competing nonlinearities (concave-convex nonlinearities). We look for positive solutions and how their solution set depends on the parameter $\lambda>0$. For the $p$ - logistic equation, we examine the subdiffusive, equidiffusive and superdiffusive cases. For the equations with competing nonlinearities, we consider the case of the sum of a concave'' (i.e., $(p-1)$ - sublinear) term and of a convex'' (i.e., $(p-1)$ - superlinear) term. For the latter, we do not assume the usual in such cases Ambrosetti-Rabinowitz condition. Our approach is variational based on the critical point theory combined with suitable truncation and comparison techniques.
2016, 15(5): 1571-1601 doi: 10.3934/cpaa.2016003 +[Abstract](2875) +[PDF](520.3KB)
Abstract:
We consider a 3d cubic focusing nonlinear Schrödinger equation with a potential $$i\partial_t u+\Delta u-Vu+|u|^2u=0,$$ where $V$ is a real-valued short-range potential having a small negative part. We find criteria for global well-posedness analogous to the homogeneous case $V=0$ [10, 5]. Moreover, by the concentration-compactness approach, we prove that if $V$ is repulsive, such global solutions scatter.
2016, 15(5): 1603-1624 doi: 10.3934/cpaa.2016004 +[Abstract](2165) +[PDF](466.2KB)
Abstract:
In this paper we study the time decay rates of the solution to the Cauchy problem for the compressible viscoelastic flows via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained, and the $\dot{H}^{-s}(0\leq s<\frac{3}{2})$ negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.
2016, 15(5): 1625-1642 doi: 10.3934/cpaa.2016005 +[Abstract](2119) +[PDF](464.6KB)
Abstract:
This paper considers the Hamiltonian systems with new generalized super-quadratic conditions. Using the variational principle and critical point theory, we obtain infinitely many distinct subharmonic solutions and an unbounded sequence of periodic solutions.
2016, 15(5): 1643-1659 doi: 10.3934/cpaa.2016018 +[Abstract](2203) +[PDF](438.6KB)
Abstract:
Under some reasonable conditions, some trace embedding properties of Musielak-Sobolev spaces in a bounded domain are given, including the trace on the inner lower dimensional hyperplane and the trace on the boundary. Furthermore, a compact trace embedding on the boundary is given.
2016, 15(5): 1661-1669 doi: 10.3934/cpaa.2016007 +[Abstract](1970) +[PDF](328.0KB)
Abstract:
In this paper, we consider the asymptotic behavior of the global spherically or cylindrically symmetric solution for the compressible nematic liquid crystal flow in multi-dimension with large initial data. Using the uniform point-wise positive lower and upper bounds of the density, we obtain the exponential stability of the global strong solution.
2016, 15(5): 1671-1688 doi: 10.3934/cpaa.2016008 +[Abstract](2311) +[PDF](448.2KB)
Abstract:
In this paper, we investigate the existence of positive solutions of the following equation \begin{eqnarray} (-\Delta)^s v +\lambda v=f(x) v^{p-1}+h(x)v^{q-1}, \ x\in R^N,\\ v\in H^s(R^N), \end{eqnarray} where $1\leq q< 2 < p < 2_s^*=\frac{2N}{N-2s}$, $0 < s < 1$, $N>2s$ and $\lambda>0$ is a parameter. Since the concave and convex nonlinearities are involved, the variational functional of the equation has different properties. Via variational method, we show that the equation admits a positive ground state solution for all $\lambda>0$ strictly larger than a threshold value. Moreover, under certain conditions on $f$ and for sufficiently large $\lambda>0$, we also prove that there are at least $k+1$ ($k$ is a positive integer) positive solutions of the equation.
2016, 15(5): 1689-1717 doi: 10.3934/cpaa.2016009 +[Abstract](2255) +[PDF](625.1KB)
Abstract:
In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity \begin{eqnarray} -M\left(\int_\Omega |\nabla^m u|^{\frac{n}{m}}dx\right)\Delta_{\frac{n}{m}}^{m} u = \frac{f(x,u)}{|x|^\alpha} \; \text{in}\; \Omega{,} \\ \quad u = \nabla u=\cdots= {\nabla}^{m-1} u=0 \quad \text{on} \quad \partial \Omega{,} \end{eqnarray} where $\Omega\subset R^n$ is a bounded domain with smooth boundary, $0 < \alpha < n$, $n\geq 2m\geq 2$ and $f(x,u)$ behaves like $e^{|u|^{\frac{n}{n-m}}}$ as $|u|\to\infty$. Using mountain pass structure and {the} concentration compactness principle, we show the existence of a nontrivial solution.
In the later part of the paper, we also discuss the above problem with convex-concave type sign changing nonlinearity. Using {the} Nehari manifold technique, we show the existence and multiplicity of non-negative solutions.
2016, 15(5): 1719-1742 doi: 10.3934/cpaa.2016010 +[Abstract](2114) +[PDF](458.8KB)
Abstract:
In this paper, we will use the maximum principle to give a new proof of the gradient estimates for mean curvature equations with some oblique derivative problems. In particular, we shall give a new proof for the capillary problem with zero gravity.
2016, 15(5): 1743-1756 doi: 10.3934/cpaa.2016011 +[Abstract](2195) +[PDF](410.3KB)
Abstract:
We consider a damped forced nonlinear Schrödinger-Poisson system in $R^3$. This provides us with an infinite-dimensional dynamical system in the energy space $H^1(R^3)$. We prove the existence of a finite dimensional global attractor for this dynamical system.
2016, 15(5): 1757-1768 doi: 10.3934/cpaa.2016012 +[Abstract](2755) +[PDF](347.3KB)
Abstract:
This paper deals with global and blowup solutions of the degenerate parabolic system $u_t=\alpha(v)\nabla\cdot(u^{p}\nabla u)+f(u,v)$ and $v_t=\beta(u)\nabla\cdot(v^{q}\nabla v)+g(u,v)$ with homogeneous Dirichlet boundary conditions. We will give sufficient conditions such that the solutions either exist globally or blow up in a finite time. In special cases, a necessary and sufficient condition for the global existence is given.
2016, 15(5): 1769-1780 doi: 10.3934/cpaa.2016013 +[Abstract](2383) +[PDF](388.9KB)
Abstract:
We are concerned with A priori estimates for the obstacle problem of a wide class of fully nonlinear equations on Riemannian manifolds. We use new techniques introduced by Bo Guan and derive new results for A priori second order estimates of its singular perturbation problem under fairly general conditions. By approximation, the existence of a $C^{1,1}$ viscosity solution is proved.
2016, 15(5): 1781-1795 doi: 10.3934/cpaa.2016014 +[Abstract](2364) +[PDF](435.9KB)
Abstract:
In the present paper, we consider the following magnetic nonlinear Choquard equation \begin{eqnarray} \big(-i\nabla+A(x)\big)^{2}u+\big( g_{0}(x)+\mu g(x)\big)u=\big(|x|^{-\alpha}*|u|^{p}\big)|u|^{p-2}u,\\ u\in H^{1}(\mathbb{R}^{N},\mathbb{C}), \end{eqnarray} where $N\geq 3$, $\alpha\in (0,N)$, $p\in(\frac{2N-\alpha}{N}, \frac{2N-\alpha}{N-2})$, $A(x): {\mathbb{R}}^{N}\rightarrow {\mathbb{R}}^{N}$ is a magnetic vector potential, $\mu>0$ is a parameter, $g_{0}(x)$ and $g(x)$ are real valued electric potential functions on ${\mathbb{R}}^{N}$. Under some suitable conditions, we show that there exists $\mu^{*}>0$ such that the above equation has at least one ground state solution for $\mu\geq\mu^{*}$. Moreover, the concentration behavior of solutions is also studied as $\mu\rightarrow +\infty$.
2016, 15(5): 1797-1807 doi: 10.3934/cpaa.2016015 +[Abstract](2574) +[PDF](341.7KB)
Abstract:
In this paper, we employ a direct method of moving planes for the fractional Laplacian equation in the unit ball. Instead of using the conventional extension method introduced by Caffarelli and Silvestre [6], Chen, Li and Li developed a direct method of moving planes for the fractional Laplacian [8]. Inspired by this new method, in this paper we deal with the semilinear pseudo -differential equation in the unit ball directly. We first review key ingredients needed in the method of moving planes in a bounded domain, such as the narrow region principle for the fractional Laplacian. Then, by using this new method, we obtain the radial symmetry and monotonicity of positive solutions for some interesting semi-linear equations.
2016, 15(5): 1809-1823 doi: 10.3934/cpaa.2016016 +[Abstract](2031) +[PDF](445.3KB)
Abstract:
In this note we study Hartree type equations with $|\nabla|^\alpha (1 < \alpha \le 2)$ and potential whose Fourier transform behaves like $|\xi|^{-(d-\gamma_1)}$ at the origin and $|\xi|^{-(d-\gamma_2)}$ at infinity. We show non-existence of scattering when $0 < \gamma_1 \le 1$ and small data scattering in $H^s$ for $s > \frac{2-\alpha}2$ when $2 < \gamma_1 \le d$ and $0 < \gamma_2 \le 2$. For this we use $U^p-V^p$ space argument and Strichartz estimates.
2016, 15(5): 1825-1840 doi: 10.3934/cpaa.2016017 +[Abstract](2433) +[PDF](392.4KB)
Abstract:
In this paper, we establish some Liouville type theorems for positive solutions of some integral equations and integral systems in $R^N$. The main technique we use is the method of moving planes in an integral form.
2016, 15(5): 1841-1856 doi: 10.3934/cpaa.2016006 +[Abstract](2205) +[PDF](442.5KB)
Abstract:
In this article, we study the existence and multiplicity of positive solutions for the Kirchhoff type problem with singular and critical nonlinearities \begin{eqnarray} \begin{cases} -\left(a+b\displaystyle\int_\Omega|\nabla u|^2dx\right)\Delta u=\mu u^{3}+\frac{\lambda}{|x|^{\beta}u^{\gamma}}, &\rm \mathrm{in}\ \ \Omega, \\ u>0, &\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{eqnarray} where $\Omega\subset \mathbb{R}^{4}$ is a bounded smooth domain and $a, b>0$, $\lambda>0$. For all $\mu>0$ and $\gamma\in(0,1)$, $0\leq\beta < 3$, we obtain one positive solution. Particularly, we prove that this problem has at least two positive solutions for all $\mu> bS^{2}$ and $\gamma\in(0,\frac{1}{2})$, $2+2\gamma < \beta < 3$.
2016, 15(5): 1857-1869 doi: 10.3934/cpaa.2016019 +[Abstract](2229) +[PDF](441.7KB)
Abstract:
This paper aims to study the time quasi-periodic solutions for one dimensional modified KdV (mKdV, for short) equation with perturbation \begin{eqnarray} u_t=-u_{x x x}-6 u^{2}u_x+perturbation ,x\in \mathbb{T}. \end{eqnarray} We show that, for any $n \in \mathbb{N}$ and a subset of $\mathbb{Z} \backslash \{0\}$ like $\{j_1 < j_2 < \cdots < j_n\}$, this equation admits a large amount of smooth n-dimensional invariant tori, along which exists a quantity of smooth quasi-periodic solutions. The proof is based on partial Birkhoff normal form and an unbounded KAM theorem established by Liu-Yuan in [Commun. Math. Phys., 307 (2011), 629-673].
2016, 15(5): 1871-1892 doi: 10.3934/cpaa.2016020 +[Abstract](1965) +[PDF](522.9KB)
Abstract:
The paper deals about Hardy-type inequalities associated with the following higher order Poincaré inequality: $$\left( \frac{N-1}{2} \right)^{2(k -l)} :=\displaystyle \inf_{ u \in C^{\infty}_{0}(\mathbb{H}^{N} ) \setminus \{0\}} \qquad \frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^2 \ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}} },$$ where $0 \leq l < k$ are integers and $\mathbb{H}^{N}$ denotes the hyperbolic space. More precisely, we improve the Poincaré inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. Furthermore, when $k = 2$ and $l = 1$ the existence of further remainder terms are provided and the sharpness of some constants is also discussed. As an application, we derive improved Rellich type inequalities on upper half space of the Euclidean space with non-standard remainder terms.
2016, 15(5): 1893-1913 doi: 10.3934/cpaa.2016021 +[Abstract](2078) +[PDF](484.1KB)
Abstract:
We study the asymptotic properties of the semigroup $S(t)$ arising from the nonlinear viscoelastic equation with hereditary memory on a bounded three-dimensional domain \begin{eqnarray} |\partial_t u|^\rho \partial_{t t} u-\Delta \partial_{t t} u-\Delta \partial_t u\\ -\Big(1+\int_0^\infty \mu(s)\Delta s \Big)\Delta u +\int_0^\infty \mu(s)\Delta u(t-s)\Delta s +f(u)=h \end{eqnarray} written in the past history framework of Dafermos [10]. We establish the existence of the global attractor of optimal regularity for $S(t)$ when $\rho\in [0,4)$ and $f$ has polynomial growth of (at most) critical order 5.
2016, 15(5): 1915-1939 doi: 10.3934/cpaa.2016022 +[Abstract](2029) +[PDF](556.9KB)
Abstract:
The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and $C_0$-semigroups. In this paper we extend some previous results on both notions to sequences of operators, $C_0$-semigroups, $C$-regularized semigroups, and $\alpha$-times integrated semigroups on Fréchet spaces. We also add a study of rescaled distributionally chaotic $C_0$-semigroups. Some examples are provided to illustrate all these results.
2016, 15(5): 1941-1974 doi: 10.3934/cpaa.2016023 +[Abstract](2125) +[PDF](684.0KB)
Abstract:
In this paper we analyze anomalous diffusion version of the multidimensional Hele-Shaw problem with a nonzero surface tension of a free boundary. We prove the one-valued solvability of this moving boundary problem in the Hölder classes. In the two-dimensional case some numerical solutions are constructed.

2019  Impact Factor: 1.105