# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

May 2014 , Volume 13 , Issue 3

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2014, 13(3): 977-990 doi: 10.3934/cpaa.2014.13.977 +[Abstract](1328) +[PDF](408.4KB)
Abstract:
In this paper, we study the positive solutions for the following integral system: \begin{eqnarray} u(x)=\int_{R^n_+}(\frac{1}{|x-y|^{n-\alpha}}-\frac{1}{|x^*-y|^{n-\alpha}})u^{\beta_1}(y)v^{\gamma_1}(y)dy ,\\ v(x)=\int_{R^n_+}(\frac{1}{|x-y|^{n-\alpha}}-\frac{1}{|x^*-y|^{n-\alpha}})u^{\beta_2}(y)v^{\gamma_2}(y)dy, \end{eqnarray} where $0 < \alpha < n$ and $x^*=(x_1,\cdots,x_{n-1},-x_n)$ is the reflection of the point $x$ about the plane $R^{n-1}$, and $\beta_1, \gamma_1, \beta_2, \gamma_2$ satisfy the condition$(f_1)$: \begin{eqnarray} 1 \leq \beta_1,\gamma_1,\beta_2,\gamma_2 \leq \frac{n+\alpha}{n-\alpha}\ \mbox{with}\ \beta_1+\gamma_1= \frac{n+\alpha}{n-\alpha}=\beta_2+\gamma_2, \beta_1\neq \beta_2, \gamma_1 \neq \gamma_2. \end{eqnarray}

This integral system is closely related to the PDE system with Navier boundary conditions, when $\alpha$ is a even number between $0$ and $n$, \begin{eqnarray} (- \Delta)^{\frac{\alpha}{2}}u(x)=u^{\beta_1}(x)v^{\gamma_1}(x), \mbox{in}\ R^n_+,\\ (- \Delta)^{\frac{\alpha}{2}}v(x)=u^{\beta_2}(x)v^{\gamma_2}(x), \mbox{in}\ R^n_+,\\ u(x)=-\Delta u(x)=\cdots =(-\Delta)^{\frac{\alpha}{2}-1} u(x)=0,\mbox{on}\ \partial{R^n_+},\\ v(x)=-\Delta v(x)=\cdots =(-\Delta)^{\frac{\alpha}{2}-1} v(x)=0,\mbox{on}\ \partial{R^n_+}. \end{eqnarray}

More precisely, any solution of (1) multiplied by a constant satisfies (2). We use method of moving planes in integral forms introduced by Chen-Li-Ou to derive rotational symmetry, monotonicity, and non-existence of the positive solutions of (1) on the half space $R^n_+$.
2014, 13(3): 991-1015 doi: 10.3934/cpaa.2014.13.991 +[Abstract](1334) +[PDF](227.5KB)
Abstract:
In this paper, the almost sure global well-posedness of the cubic non linear wave equation on the sphere is studied when the initial datum is a random variable with values in low regularity spaces. The result is first proved on the 3D sphere, thanks to the existence of a uniformly bounded in $L^p$ basis of $L^2(S^3)$ and then it is extended to $R^3$ thanks to the Penrose transform.
2014, 13(3): 1017-1044 doi: 10.3934/cpaa.2014.13.1017 +[Abstract](1187) +[PDF](202.8KB)
Abstract:
We develop a general homogenisation procedure for Friedrichs systems. Under reasonable assumptions, the concepts of $G$ and $H$-convergence are introduced. As Friedrichs systems can be used to represent various boundary or initial-boundary value problems for partial differential equations, some additional assumptions are needed for compactness results. These assumptions are particularly examined for the stationary diffusion equation, the heat equation and a model example of a first order equation leading to memory effects. In the first two cases, the equivalence with the original notion of $H$-convergence is proved.
2014, 13(3): 1045-1060 doi: 10.3934/cpaa.2014.13.1045 +[Abstract](1159) +[PDF](423.1KB)
Abstract:
We derive a point-wise estimate for a map $u: \Omega \subset R^n \rightarrow R^m$ that minimizes $J_A(v): \int_A \frac{1}{2}|\nabla v|^2+U(v)$ subjected to the Dirichlet condition $v=u$ on $\partial\Omega$ for every open smooth and bounded set $A \subset \Omega$. We discuss some consequences of this basic estimate.
2014, 13(3): 1061-1074 doi: 10.3934/cpaa.2014.13.1061 +[Abstract](1414) +[PDF](379.4KB)
Abstract:
In this paper, we investigate the multi-valued solutions of a class of parabolic Monge-Ampère equation $-u_{t}\det(D^{2}u)=f$. Using the Perron method, we obtain the existence of finitely valued and infinitely valued solutions to the parabolic Monge-Ampère equations. We generalize the results of elliptic Monge-Ampère equations and Hessian equations.
2014, 13(3): 1075-1086 doi: 10.3934/cpaa.2014.13.1075 +[Abstract](1308) +[PDF](346.3KB)
Abstract:
We study a parametric nonlinear periodic problem driven by the scalar $p$-Laplacian. We show that if $\hat \lambda_1 >0$ is the first eigenvalue of the periodic scalar $p$-Laplacian and $\lambda> \hat \lambda_1$, then the problem has at least three nontrivial solutions one positive, one negative and the third nodal. Our approach is variational together with suitable truncation, perturbation and comparison techniques.
2014, 13(3): 1087-1104 doi: 10.3934/cpaa.2014.13.1087 +[Abstract](1221) +[PDF](364.3KB)
Abstract:
In this paper, we study the time periodic solution of a sixth order nonlinear parabolic equation, which arises in oil-water-surfactant mixtures. Based on Leray-Schauder's fixed point theorem and Campanato spaces, we prove the existence of time-periodic solutions in two space dimensions.
2014, 13(3): 1105-1117 doi: 10.3934/cpaa.2014.13.1105 +[Abstract](1635) +[PDF](405.3KB)
Abstract:
We study the asymptotic behaviour of the solutions of a class of linear neutral delay differential equations with discrete delay where the coefficients of the non neutral part are periodic functions which are rational multiples of all time delays. We show that this technique is applicable to a broader class where the coefficients of the neutral part are periodic functions as well.
2014, 13(3): 1119-1140 doi: 10.3934/cpaa.2014.13.1119 +[Abstract](1207) +[PDF](509.7KB)
Abstract:
In this article, we consider a non-autonomous multi-layer quasi-geostrophic equations of the ocean with a singularly oscillating external force $g^{\epsilon}= g_0(t) + \epsilon^{-\rho} g_1(t/\epsilon)$ depending on a small parameter $\epsilon > 0$ and $\rho \in [0, 1)$ together with the averaged system with the external force $g_0(t),$ formally corresponding to the case $\epsilon = 0.$ Under suitable assumptions on the external force, we prove as in [10] the boundness of the uniform global attractor $\mathcal{A}^{\epsilon}$ as well as the upper semi-continuity of the attractors $\mathcal{A}^{\epsilon}$ of the singular systems to the attractor $\mathcal{A}^0$ of the averaged system as $\epsilon \rightarrow 0^+.$ When the external force is small enough and the viscosity is large enough, the convergence rate is controlled by $K \epsilon^{(1 -\rho)}.$ Let us mention that the non-homogenous boundary conditions (and the non-local constraint) present in the multi-layer quasi-geostrophic model makes the estimates more complicated, [3]. These difficulties are overcome using the new formulation presented in [25].
2014, 13(3): 1141-1165 doi: 10.3934/cpaa.2014.13.1141 +[Abstract](1408) +[PDF](498.2KB)
Abstract:
This article is a continuation of our previous work [5], where we formulated general existence theorems for pullback exponential attractors for asymptotically compact evolution processes in Banach spaces and discussed its implications in the autonomous case. We now study properties of the attractors and use our theoretical results to prove the existence of pullback exponential attractors in two examples, where previous results do not apply.
2014, 13(3): 1167-1186 doi: 10.3934/cpaa.2014.13.1167 +[Abstract](1485) +[PDF](471.3KB)
Abstract:
We study the Cauchy problem for systems of quasilinear wave equations with multiple propagation speeds in four space dimensions. The nonlinear term in this problem may explicitly depends on the unknown function itself. By some new $L^{\infty}_{t}L^2_{x}$ estimates of the unknown function, combining with some Klainerman--Sideris type weighted estimates, we get the sharp lower bound of lifespan $T_{\varepsilon}\geq \exp{(\frac{c}{\varepsilon^2})}$ for the quasilinear system.
2014, 13(3): 1187-1202 doi: 10.3934/cpaa.2014.13.1187 +[Abstract](1382) +[PDF](393.2KB)
Abstract:
A critical point theory for non-differentiable functionals defined on a closed convex subset of a Banach space is worked out. Special attention is paid to the notion of critical point and possible compactness conditions of Palais-Smale's type. Two Mountain-Pass like theorems are also established. Concepts and results are compared with those already existing in the literature.
2014, 13(3): 1203-1222 doi: 10.3934/cpaa.2014.13.1203 +[Abstract](1212) +[PDF](447.6KB)
Abstract:
In this paper, we study the Cauchy problem for the Boussinesq-type equation \begin{eqnarray} \partial^2_t u-\varepsilon \partial_t \Delta u=-\Delta ^2 u+\Delta u+\Delta g(u), \end{eqnarray} where $g(u)=O(u^\rho),$ $\rho \geq 2.$ By means of long wave-short wave decomposition, Green's function method and energy method, we show that the Cauchy Problem admits a global classical solution in multi dimension. We also show the pointwise estimate of the time asymptotic shape of the solutions in odd dimensional space.
2014, 13(3): 1223-1236 doi: 10.3934/cpaa.2014.13.1223 +[Abstract](1338) +[PDF](366.8KB)
Abstract:
In this paper, we make an exhaustive study of the third order linear operators $u''' +Mu$, $u'''+Mu'$ and $u'''+Mu''$ coupled with $(k, 3-k)$-conjugate boundary conditions, where $k=1,2$. We obtain the optimal intervals on which the Green's functions are of one sign. The main tool is the disconjugacy theory. As an application of our results, we develop a monotone iteration method to obtain positive solutions of the nonlinear problem $u'''+Mu''+f(t,u)=0$, $u(0)=u'(0)=u(1)=0$.
2014, 13(3): 1237-1265 doi: 10.3934/cpaa.2014.13.1237 +[Abstract](1022) +[PDF](552.1KB)
Abstract:
Under suitable conditions, we obtain some characterizations of supercontractivity, ultraboundedness and ultracontractivity of the evolution operator $G(t,s)$ associated to a class of nonautonomous second order parabolic equations with unbounded coefficients defined in $I\times R^d$, where $I$ is a right-halfline. For this purpose, we establish an Harnack type estimate for $G(t,s)$ and a family of logarithmic Sobolev inequalities with respect to the unique tight evolution system of measures $\{\mu_t: t \in I\}$ associated to $G(t,s)$. Sufficient conditions for the supercontractivity, ultraboundedness and ultracontractivity to hold are also provided.
2014, 13(3): 1267-1282 doi: 10.3934/cpaa.2014.13.1267 +[Abstract](1849) +[PDF](455.6KB)
Abstract:
We show the existence of ground state and orbital stability of standing waves of fractional Schrödinger equations with power type nonlinearity. For this purpose we establish the uniqueness of weak solutions.
2014, 13(3): 1283-1304 doi: 10.3934/cpaa.2014.13.1283 +[Abstract](1442) +[PDF](446.3KB)
Abstract:
In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly establish the local well-posedness result in Besov spaces, and then present a precise blow-up scenario for strong solutions. Furthermore, we show the existence of single peakon by the method of analysis.
2014, 13(3): 1305-1315 doi: 10.3934/cpaa.2014.13.1305 +[Abstract](1303) +[PDF](379.9KB)
Abstract:
We consider bilinear multipliers that appeared as a distinguished particular case in the classification of two-dimensional bilinear Hilbert transforms by Demeter and Thiele [9]. In this note we investigate their boundedness on Sobolev spaces. Furthermore, we study structurally similar operators with symbols that also depend on the spatial variables. The new results build on the existing $\mathrm{L}^p$ estimates for a paraproduct-like operator previously studied by the authors in [5] and [10]. Our primary intention is to emphasize the analogies with Coifman-Meyer multipliers and with bilinear pseudodifferential operators of order $0$.
2014, 13(3): 1317-1325 doi: 10.3934/cpaa.2014.13.1317 +[Abstract](1258) +[PDF](333.3KB)
Abstract:
In this paper, we study the exact boundary controllability for the cubic focusing semilinear wave equation on Schwarzschild black hole background in radially symmetrical case. When the initial data and the final data are in the so called potential well, we find that the sufficient condition for the global existence is also sufficient to ensure the exact boundary controllability of the problem. Moreover, under the assumption of radial symmetry, our problem is changed to one space dimension case, and then the control time can be that of the linear wave equation.
2014, 13(3): 1327-1336 doi: 10.3934/cpaa.2014.13.1327 +[Abstract](1231) +[PDF](364.9KB)
Abstract:
In this paper we study the initial problem for the Hall-magnetohydrodynamics system with partial viscosity in $R^n(n=2, 3)$. We obtain a Beale-Kato-Majda type blow up criterion of smooth solutions.
2014, 13(3): 1337-1345 doi: 10.3934/cpaa.2014.13.1337 +[Abstract](1344) +[PDF](343.0KB)
Abstract:
We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for the incompressible MHD equations in bounded smooth domains of $\mathbb R^3$ under some suitable smallness conditions. The initial density is allowed to have vacuum, in particular, it can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities $\|\sqrt\rho_0u_0\|_{L^2(\Omega)}^2+\|H_0\|_{L^2(\Omega)}^2$ and $\|\nabla u_0\|_{L^2(\Omega)}^2+\|\nabla H_0\|_{L^2(\Omega)}^2$ is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.
2014, 13(3): 1347-1359 doi: 10.3934/cpaa.2014.13.1347 +[Abstract](1220) +[PDF](351.1KB)
Abstract:
In this article we investigate a general class of Monge-Ampère equations in the plane, including the constant Gauss curvature equation. Our first aim is to prove some maximum and minimum principles for suitable $P$-functions, in the sense of L.E. Payne. Then, these new principles are employed to solve a general class of overdetermined Monge-Ampère problems and to investigate two boundary value problems for the constant Gauss curvature equation. More precisely, when the constant Gauss curvature equation is subject to the homogeneous Dirichlet boundary condition, we prove several isoperimetric inequalities, while when it is subject to the contact angle boundary condition, some necessary conditions of solvability, involving the curvature of the boundary of the underlying domain and the given contact angle, are derived.

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