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Communications on Pure & Applied Analysis

2016 , Volume 15 , Issue 5

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Heat--structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers
Irena Lasiecka and Roberto Triggiani
2016, 15(5): 1515-1543 doi: 10.3934/cpaa.2016001 +[Abstract](430) +[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.
Positive solutions for parametric $p$-Laplacian equations
Nikolaos S. Papageorgiou and George Smyrlis
2016, 15(5): 1545-1570 doi: 10.3934/cpaa.2016002 +[Abstract](351) +[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.
Scattering for a nonlinear Schrödinger equation with a potential
Younghun Hong
2016, 15(5): 1571-1601 doi: 10.3934/cpaa.2016003 +[Abstract](405) +[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.
Decay of the compressible viscoelastic flows
W. Wei, Yin Li and Zheng-An Yao
2016, 15(5): 1603-1624 doi: 10.3934/cpaa.2016004 +[Abstract](406) +[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.
Multiplicity of subharmonic solutions and periodic solutions of a particular type of super-quadratic Hamiltonian systems
Xiao-Fei Zhang and Fei Guo
2016, 15(5): 1625-1642 doi: 10.3934/cpaa.2016005 +[Abstract](382) +[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.
On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain
Duchao Liu, Beibei Wang and Peihao Zhao
2016, 15(5): 1643-1659 doi: 10.3934/cpaa.2016018 +[Abstract](396) +[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.
Exponential stability for the compressible nematic liquid crystal flow with large initial data
Qiang Tao and Ying Yang
2016, 15(5): 1661-1669 doi: 10.3934/cpaa.2016007 +[Abstract](306) +[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.
Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$
Qingfang Wang
2016, 15(5): 1671-1688 doi: 10.3934/cpaa.2016008 +[Abstract](439) +[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.
Polyharmonic Kirchhoff type equations with singular exponential nonlinearities
Pawan Kumar Mishra, Sarika Goyal and K. Sreenadh
2016, 15(5): 1689-1717 doi: 10.3934/cpaa.2016009 +[Abstract](341) +[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.
A new proof of gradient estimates for mean curvature equations with oblique boundary conditions
Jinju Xu
2016, 15(5): 1719-1742 doi: 10.3934/cpaa.2016010 +[Abstract](374) +[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.
Long time behavior of solutions to a Schrödinger-Poisson system in $R^3$
Amna Dabaa and O. Goubet
2016, 15(5): 1743-1756 doi: 10.3934/cpaa.2016011 +[Abstract](419) +[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.
Global and blowup solutions for general Lotka-Volterra systems
Shaohua Chen, Runzhang Xu and Hongtao Yang
2016, 15(5): 1757-1768 doi: 10.3934/cpaa.2016012 +[Abstract](334) +[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.
A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds
Weisong Dong, Tingting Wang and Gejun Bao
2016, 15(5): 1769-1780 doi: 10.3934/cpaa.2016013 +[Abstract](356) +[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.
Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations
Dengfeng Lü
2016, 15(5): 1781-1795 doi: 10.3934/cpaa.2016014 +[Abstract](396) +[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 $\mu\rightarrow +\infty$.
A direct method of moving planes for fractional Laplacian equations in the unit ball
Meixia Dou
2016, 15(5): 1797-1807 doi: 10.3934/cpaa.2016015 +[Abstract](390) +[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.
On small data scattering of Hartree equations with short-range interaction
Yonggeun Cho, Gyeongha Hwang and Tohru Ozawa
2016, 15(5): 1809-1823 doi: 10.3934/cpaa.2016016 +[Abstract](357) +[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.
Liouville type theorems for singular integral equations and integral systems
Xiaohui Yu
2016, 15(5): 1825-1840 doi: 10.3934/cpaa.2016017 +[Abstract](337) +[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.
Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four
Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao and Xing-Ping Wu
2016, 15(5): 1841-1856 doi: 10.3934/cpaa.2016006 +[Abstract](351) +[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 $\gamma\in(0,\frac{1}{2})$, $2+2\gamma < \beta < 3$.
Smooth quasi-periodic solutions for the perturbed mKdV equation
Siqi Xu and Dongfeng Yan
2016, 15(5): 1857-1869 doi: 10.3934/cpaa.2016019 +[Abstract](361) +[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].
Improved higher order poincaré inequalities on the hyperbolic space via Hardy-type remainder terms
Elvise Berchio and Debdip Ganguly
2016, 15(5): 1871-1892 doi: 10.3934/cpaa.2016020 +[Abstract](300) +[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.
Global attractors for nonlinear viscoelastic equations with memory
Monica Conti, Elsa M. Marchini and V. Pata
2016, 15(5): 1893-1913 doi: 10.3934/cpaa.2016021 +[Abstract](376) +[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.
Distributionally chaotic families of operators on Fréchet spaces
J. Alberto Conejero, Marko Kostić, Pedro J. Miana and Marina Murillo-Arcila
2016, 15(5): 1915-1939 doi: 10.3934/cpaa.2016022 +[Abstract](346) +[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.
The Hele-Shaw problem with surface tension in the case of subdiffusion
Nataliya Vasylyeva and Vitalii Overko
2016, 15(5): 1941-1974 doi: 10.3934/cpaa.2016023 +[Abstract](399) +[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.

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