# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

November 2016 , Volume 15 , Issue 6

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2016, 15(6): 1975-2005 doi: 10.3934/cpaa.2016024 +[Abstract](1503) +[PDF](554.5KB)
Abstract:
In this paper we analyze the following elliptic problem related to some Caffarelli-Kohn-Nirenberg inequalities: \begin{eqnarray} -div(|x|^{-2\gamma}\nabla u)-\lambda \frac{u}{|x|^{2(\gamma+1)}}=|\nabla u|^p|x|^{-\gamma p}+cf,\; u>0\; \mbox{ in }\; \Omega, \qquad u_{|\partial \Omega}\equiv0, \end{eqnarray} where $\Omega \subset R^N$ is a domain such that $0\in\Omega$, $N\geq 3$, and $c, \lambda, \gamma, p$ are positive constants verifying $0 < \lambda \leq \Lambda_{N,\gamma}=\left(\frac{N-2(\gamma+1)}{2}\right)^{2}$, $-\infty<\gamma<\frac{N-2} 2$ and $p>0$. Our study concerns to existence of solutions to the former problem. More precisely, first we determine a critical thereshold for the power $p$, in the sense that, beyond this value it does not exist any positive supersolution to our problem, not even in a very weak sense. In addition, we show existence of solutions for all the values $p>0$ below this threshold, with the restriction $\gamma>-\frac{N(1-p)+2}{2}$, whenever the righthand side verifies $f(x)\leq |x|^{-2(\gamma+1)}$ if $\gamma>-1$. When $-\frac{N(1-p)+2}{2}<\gamma\leq -1$ it suffices that $f\in L^{2/p}(\Omega)$. The existence of solutions for $0 < p < 1$ and $\gamma\leq -\frac{N(1-p)+2}{2}$ is an open question.
2016, 15(6): 2007-2021 doi: 10.3934/cpaa.2016025 +[Abstract](1981) +[PDF](426.0KB)
Abstract:
The Euler-Maxwell system with small parameters arises in the modeling of magnetized plasmas and semiconductors. For initial data close to constant equilibrium states, we prove uniform energy estimates with respect to the parameters, which imply the global existence of smooth solutions. Under reasonable assumptions on the convergence of initial conditions, this allows to show the global-in-time convergence of the Euler-Maxwell system as each of the parameters goes to zero.
2016, 15(6): 2023-2058 doi: 10.3934/cpaa.2016026 +[Abstract](2096) +[PDF](645.8KB)
Abstract:
Consider the focusing energy critical Schrödinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters. In analogy with [19], the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows to classify the solutions is the one-pass lemma. The main difference between [19] and this paper is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the one-pass lemma by comparing the contribution in the variational region and in the hyperbolic region.
2016, 15(6): 2059-2074 doi: 10.3934/cpaa.2016027 +[Abstract](1737) +[PDF](426.7KB)
Abstract:
In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS system, corresponds to the Euler-Lagrange equations of the fundamental HLS inequality. In each case, we give a brief survey on some important results and useful methods. Some simplifications and extensions based on somewhat more direct and intuitive ideas are presented. Also, a few new qualitative properties are obtained and several open problems are raised for future research.
2016, 15(6): 2075-2101 doi: 10.3934/cpaa.2016028 +[Abstract](1750) +[PDF](493.1KB)
Abstract:
We study in this article a class of robust control problems associated with a coupled Cahn-Hilliard-Navier-Stokes model in a two dimensional bounded domain. The model consists of the Navier-Stokes equations for the velocity, coupled with the Cahn-Hilliard model for the order (phase) parameter. We prove the existence and uniqueness of solutions and we derive a first-order necessary optimality condition for these robust control problems.
2016, 15(6): 2103-2116 doi: 10.3934/cpaa.2016029 +[Abstract](1624) +[PDF](432.5KB)
Abstract:
For an exterior domain $\Omega\subset R^d$ with smooth boundary, we study the existence and stability of bounded mild solutions in time $t$ to the abstract semi-linear evolution equation $u_t + Au = Pdiv (G(u)+F(t))$ where $-A$ generates a $C_0$-semigroup on the solenoidal space $L^d_{\sigma,w}(\Omega)$ (known as weak-$L^d$), $P$ is Helmholtz projection; $G$ is a nonlinear operator acting from $L^d_{\sigma,w}(\Omega)$ into $L^{d/2}_{\sigma,w}(\Omega)^{d^2}$, and $F(t)$ is a second-order tensor in $L^{d/2}_{\sigma,w}(\Omega)^{d^2}$. Our obtained abstract results can be applied not only to reestablish the known results on Navier-Stokes flows on exterior domains and/or around rotating obstacles, but also to obtain a new result on existence and polynomial stability of bounded solutions to Navier-Stokes-Oseen equations on exterior domains.
2016, 15(6): 2117-2134 doi: 10.3934/cpaa.2016030 +[Abstract](1437) +[PDF](412.9KB)
Abstract:
In this paper, we study some important properties of positive solutions for a nonlinear integral system. With the help of the method of moving planes in an integral form, we show that under certain integrable conditions, all of positive solutions to this system are radially symmetric and decreasing with respect to the origin. Meanwhile, using the regularity lifting lemma, which was recently introduced by Chen and Li in [1], we obtain the optimal integrable intervals and sharp asymptotic behaviors for such positive solutions, which characterize the closeness of system to some extent.
2016, 15(6): 2135-2160 doi: 10.3934/cpaa.2016031 +[Abstract](1838) +[PDF](561.0KB)
Abstract:
Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$ whose heat kernels have the Gaussian upper bound estimates. Assume that the growth function $\varphi:\mathbb{R}^n\times[0,\infty) \to[0,\infty)$ satisfies that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ (the class of uniformly Muckenhoupt weights). Let $H_{\varphi,L}(\mathbb{R}^n)$ be the Musielak-Orlicz-Hardy space introduced via the Lusin area function associated with the heat semigroup of $L$. In this article, the authors obtain several maximal function characterizations of the space $H_{\varphi,L}(\mathbb{R}^n)$, which, especially, answer an open question of L. Song and L. Yan under an additional mild assumption satisfied by Schrödinger operators on $\mathbb{R}^n$ with non-negative potentials belonging to the reverse Hölder class, and second-order divergence form elliptic operators on $\mathbb{R}^n$ with bounded measurable real coefficients.
2016, 15(6): 2161-2177 doi: 10.3934/cpaa.2016032 +[Abstract](2297) +[PDF](445.2KB)
Abstract:
In this paper, we study the following fourth-order elliptic equation with Kirchhoff-type \begin{eqnarray} \left\{\begin{array}{l} \Delta^{2}u-(a+b\int_{\mathbb{R}^N}|\nabla u|^{2}dx)\Delta u+V(x)u=f(u),\ \ \ x\in \mathbb{R}^{N},\\ u\in H^{2}(\mathbb{R}^{N}), \end{array}\right. \end{eqnarray} where the constants $a>0, b\geq 0$. By constraint variational method and quantitative deformation lemma, we obtain that the problem possesses one least energy sign-changing solution $u_{b}$. Moreover, we also prove that the energy of $u_{b}$ is strictly larger than two times the ground state energy. Finally, we give a convergence property of $u_{b}$ when $b$ as a parameter and $b\rightarrow 0$.
2016, 15(6): 2179-2202 doi: 10.3934/cpaa.2016033 +[Abstract](1639) +[PDF](530.2KB)
Abstract:
We study the initial value problem associated to a perturbation of the Benjamin-Ono equation or Chen-Lee equation. We prove that results about local and global well-posedness for initial data in $H^s(\mathbb{R})$, with $s>-1/2$, are sharp in the sense that the flow-map data-solution fails to be $C^3$ in $H^s(\mathbb{R})$ when $s<-\frac{1}{2}$. Also, we determine the limiting behavior of the solutions when the dispersive and dissipative parameters goes to zero. In addition, we will discuss the asymptotic behavior (as $|x|\to \infty$) of the solutions by solving the equation in weighted Sobolev spaces.
2016, 15(6): 2203-2219 doi: 10.3934/cpaa.2016034 +[Abstract](1568) +[PDF](429.0KB)
Abstract:
The Maxwell-Klein-Gordon equations in 2+1 dimensions in temporal gauge are locally well-posed for low regularity data even below energy level. The corresponding (3+1)-dimensional case was considered by Yuan. Fundamental for the proof is a partial null structure in the nonlinearity which allows to rely on bilinear estimates in wave-Sobolev spaces by d'Ancona, Foschi and Selberg, on an $(L^p_x L^q_t)$ - estimate for the solution of the wave equation, and on the proof of a related result for the Yang-Mills equations by Tao.
2016, 15(6): 2221-2245 doi: 10.3934/cpaa.2016035 +[Abstract](1858) +[PDF](501.6KB)
Abstract:
In this paper, we consider the existence of random attractors in a weighted space $l_\rho ^2$ for first-order non-autonomous stochastic lattice system with random coupled coefficients and multiplicative/additive white noise, and establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.
2016, 15(6): 2247-2280 doi: 10.3934/cpaa.2016036 +[Abstract](1542) +[PDF](576.2KB)
Abstract:
We study the Cauchy problem of the Klein-Gordon-Zakharov system in spatial dimension $d \ge 4$ with radial or non-radial initial datum $(u, \partial_t u, n,$ $\partial_t n)|_{t=0} \in H^{s+1}(R^d) \times H^s(R^d) \times\dot{H}^s(R^d) \times \dot{H}^{s-1}(R^d)$. The critical value of $s$ is $s_c=d/2-2$. By the radial Strichartz estimates and $U^2, V^2$ type spaces, we prove that the small data global well-posedness and scattering hold at $s=s_c$ in $d \ge 4$ for radial initial datum. For non-radial initial datum, we prove that the local well-posedness hold at $s=1/4$ when $d=4$ and $s=s_c+1/(d+1)$ when $d \ge 5$.
2016, 15(6): 2281-2300 doi: 10.3934/cpaa.2016037 +[Abstract](1651) +[PDF](170.6KB)
Abstract:
In the case of oscillatory potentials, we present Lyapunov type inequalities for $n$th order forced differential equations of the form \begin{eqnarray} x^{(n)}(t)+\sum_{j=1}^{m}q_j(t)|x(t)|^{\alpha_j-1}x(t)=f(t) \end{eqnarray} satisfying the boundary conditions \begin{eqnarray} x(a_i)=x'(a_i)=x''(a_i)=\cdots=x^{(k_i)}(a_i)=0;\qquad i=1,2,\ldots,r, \end{eqnarray} where $a_1 < a_2 < \cdots < a_r$, $0\leq k_i$ and \begin{eqnarray} \sum_{j=1}^{r}k_j+r=n;\qquad r\geq 2. \end{eqnarray} No sign restriction is imposed on the forcing term and the nonlinearities satisfy \begin{eqnarray} 0 < \alpha_1 < \cdots < \alpha_j < 1 < \alpha_{j+1} < \cdots < \alpha_m < 2. \end{eqnarray} The obtained inequalities generalize and compliment the existing results in the literature.
2016, 15(6): 2301-2328 doi: 10.3934/cpaa.2016038 +[Abstract](1618) +[PDF](216.0KB)
Abstract:
We consider a (nonlinear) Berger plate in the absence of rotational inertia acted upon by nonlinear boundary dissipation. We take the boundary to have two disjoint components: a clamped (inactive) portion and a controlled portion where the feedback is active via a hinged-type condition. We emphasize the damping acts only in one boundary condition on a portion of the boundary. In [24] this type of boundary damping was considered for a Berger plate on the whole boundary and shown to yield the existence of a compact global attractor. In this work we address the issues arising from damping active only on a portion of the boundary, including deriving a necessary trace estimate for $(\Delta u)\big|_{\Gamma_0}$ and eliminating a geometric condition in [24] which was utilized on the damped portion of the boundary.
Additionally, we use recent techniques in the asymptotic behavior of hyperbolic-like dynamical systems [11, 18] involving a stabilizability" estimate to show that the compact global attractor has finite fractal dimension and exhibits additional regularity beyond that of the state space (for finite energy solutions).
2016, 15(6): 2329-2355 doi: 10.3934/cpaa.2016039 +[Abstract](1707) +[PDF](515.2KB)
Abstract:
We study two models of differential equations for the stationary flow of an incompressible viscous magnetic fluid subjected to an external magnetic field. The first model, called Rosensweig's model, consists of the incompressible Navier-Stokes equations, the angular momentum equation, the magnetization equation of Bloch-Torrey type, and the magnetostatic equations. The second one, called Shliomis model, is obtained by assuming that the angular momentum is given in terms of the magnetic field, the magnetization field and the vorticity. It consists of the incompressible Navier-Stokes equation, the magnetization equation and the magnetostatic equations. We prove, for each of the differential systems posed in a bounded domain of $\mathbb{R}^3$ and equipped with boundary conditions, existence of weak solutions by using regularization techniques, linearization and the Schauder fixed point theorem.
2016, 15(6): 2357-2372 doi: 10.3934/cpaa.2016040 +[Abstract](1622) +[PDF](400.6KB)
Abstract:
In this paper we give sufficient conditions on $\alpha \ge 0$ and $c\in R$ ensuring that the space of test functions $C_c^\infty(R^N)$ is a core for the operator \begin{eqnarray} L_0u=(1+|x|^\alpha )\Delta u+\frac{c}{|x|^2}u=:Lu+\frac{c}{|x|^2}u, \end{eqnarray} and $L_0$ with a suitable domain generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p(R^N), 1 < p < \infty$. The proofs are based on some $L^p$-weighted Hardy's inequality and perturbation techniques.
2016, 15(6): 2373-2400 doi: 10.3934/cpaa.2016041 +[Abstract](1715) +[PDF](473.9KB)
Abstract:
The Riemann solutions for the relativistic Euler equations for generalized Chaplygin gas are considered. It is rigorously proved that, as the pressure vanishes, they tend to the two kinds of Riemann solutions to the zero-pressure relativistic Euler equations, which include a delta shock formed by a weighted $\delta$-measure and a vacuum state.
2016, 15(6): 2401-2445 doi: 10.3934/cpaa.2016042 +[Abstract](1633) +[PDF](942.7KB)
Abstract:
We consider an evolutionary, non-degenerate, symmetric $p$-Laplacian. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes a breakdown of the Uhlenbeck structure. We derive interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient $p$-Laplacian case with lower-order dependencies. Finally, having our regularity result for time derivatives, we obtain respective regularity of the main part. The Appendix on Nikolskii-Bochner spaces, that includes theorems on their embeddings and interpolations, may be of independent interest.
2016, 15(6): 2447-2456 doi: 10.3934/cpaa.2016043 +[Abstract](1267) +[PDF](384.0KB)
Abstract:
We study the quenching behaviors for the generalized microelectromechanical system (MEMS) equation $u_{t}-\Delta u=\lambda\rho(x)f(u)$, $0 < u < A$ ($A=1$ or $+\infty$), in $\Omega\times(0,+\infty)$, $u(x,t)=0$ on $\partial\Omega\times(0,+\infty)$, $u(x,0)=u_{0}(x)\in[0,A)$ in $\Omega$, where $\lambda>0$, $\Omega\subset R^N$ is a bounded domain, $0\le \rho(x) \in C^{\alpha}(\overline{\Omega})$, $\rho\not\equiv0$, for some constant $0 < \alpha < 1$, $0 < f \in C^{2}((0,A))$ such that $f'(s)\ge0$, $f''(s)\ge0$ for any $s\in[0,A)$ and $u_{0}$ is smooth, $u_{0}=0$ on $\partial\Omega$. It is well known that quenching does occur and corresponds to a touchdown phenomenon. We establish an interesting quenching rate, and based on which we then prove that touchdown cannot occur at zero points of $\rho(x)$ or at the boundary of $\Omega$, without the assumption of compactness of the touchdown set.
2016, 15(6): 2457-2473 doi: 10.3934/cpaa.2016044 +[Abstract](1918) +[PDF](444.4KB)
Abstract:
In this paper, we study the existence, multiplicity and concentration of positive solutions for the following indefinite semilinear elliptic equations involving concave-convex nonlinearities: \begin{eqnarray} \left\{\begin{array}{l} -\Delta u+V_{\lambda }\left( x\right) u=f\left( x\right) \left\vert u\right\vert ^{q-2}u+g\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{in }\mathbb{R}^{N}, \\ u\geq 0, & \text{in }\mathbb{R}^{N}, \end{array}\right. \end{eqnarray} where$1 < q < 2 < p < 2^{\ast} (2^{\ast } = \frac{2N}{N-2}$ for $N \geq 3)$ the potential $V_{\lambda }(x)=\lambda V^{+}(x)-V^{-}(x)$ with $V^{\pm }=\max \left\{ \pm V,0\right\}$ and the parameter $\lambda >0.$ We assume that the functions $f,g$ and $V$ satisfy suitable conditions with the potential $V$ and the weight function $g$ without the assumptions of infinite limits.
2016, 15(6): 2475-2487 doi: 10.3934/cpaa.2016045 +[Abstract](1940) +[PDF](375.8KB)
Abstract:
We study nonsymmetric second order elliptic operators with Wentzell boundary conditions in general domains with sufficiently smooth boundary. The ambient space is a space of $L^p$- type, $1\le p\le \infty$. We prove the existence of analytic quasicontractive $(C_0)$-semigroups generated by the closures of such operators, for any $1< p< \infty$. Moreover, we extend a previous result concerning the continuous dependence of these semigroups on the coefficients of the boundary condition. We also specify precisely the domains of the generators explicitly in the case of bounded domains and $1 < p < \infty$, when all the ingredients of the problem, including the boundary of the domain, the coefficients, and the initial condition, are of class $C^{\infty}$.
2016, 15(6): 2489-2507 doi: 10.3934/cpaa.2016046 +[Abstract](1559) +[PDF](399.9KB)
Abstract:
We consider a semilinear Robin problem driven by the negative Laplacian plus an indefinite and unbounded potential and a superdiffusive lotistic-type reaction. We prove bifurcation results describing the dependence of the set of positive solutions on the parameter of the problem. We also establish the existence of extreme positive solutions and determine their properties.
2016, 15(6): 2509-2526 doi: 10.3934/cpaa.2016047 +[Abstract](2112) +[PDF](597.0KB)
Abstract:
In this paper, we investigate the existence and uniqueness of crossing limit cycle for a planar nonlinear Liénard system which is discontinuous along a straight line (called a discontinuity line). By using the Poincaré mapping method and some analysis techniques, a criterion for the existence, uniqueness and stability of a crossing limit cycle in the discontinuous differential system is established. An application to Schnakenberg model of an autocatalytic chemical reaction is given to illustrate the effectiveness of our result. We also consider a class of discontinuous piecewise linear differential systems and give a necessary condition of the existence of crossing limit cycle, which can be used to prove the non-existence of crossing limit cycle.

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