ISSN:

1534-0392

eISSN:

1553-5258

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## Communications on Pure & Applied Analysis

September 2015 , Volume 14 , Issue 5

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2015, 14(5): 1603-1621
doi: 10.3934/cpaa.2015.14.1603

*+*[Abstract](836)*+*[PDF](464.3KB)**Abstract:**

In this paper we strengthen some results on the existence and properties of pullback attractors for a 2D Navier-Stokes model with finite delay formulated in [Caraballo and Real, J. Differential Equations 205 (2004), 271--297]. Actually, we prove that under suitable assumptions, pullback attractors not only of fixed bounded sets but also of a set of tempered universes do exist. Moreover, thanks to regularity results, the attraction from different phase spaces also happens in $C([-h,0];V)$. Finally, from comparison results of attractors, and under an additional hypothesis, we establish that all these families of attractors are in fact the same object.

2015, 14(5): 1623-1639
doi: 10.3934/cpaa.2015.14.1623

*+*[Abstract](783)*+*[PDF](408.3KB)**Abstract:**

The shape optimization problem for the profile in compressible liquid crystals is considered in this paper. We prove that the optimal shape with minimal volume is attainable in an appropriate class of admissible profiles which subjects to a constraint on the thickness of the boundary. Such consequence is mainly obtained from the well-known weak sequential compactness method (see [25]).

2015, 14(5): 1641-1670
doi: 10.3934/cpaa.2015.14.1641

*+*[Abstract](778)*+*[PDF](573.1KB)**Abstract:**

In this paper, we consider the Cauchy problem for the generalized Davey-Stewartson system \begin{eqnarray} &i\partial_t u + \Delta u =-a|u|^{p-1}u+b_1uv_{x_1}, (t,x)\in R \times R^3,\\ &-\Delta v=b_2(|u|^2)_{x_1}, \end{eqnarray} where $a>0,b_1b_2>0$, $\frac{4}{3}+1< p< 5$. We first use a variational approach to give a dichotomy of blow-up and scattering for the solution of mass supercritical equation with the initial data satisfying $J(u_0)

2015, 14(5): 1671-1683
doi: 10.3934/cpaa.2015.14.1671

*+*[Abstract](705)*+*[PDF](370.1KB)**Abstract:**

In this paper, we investigate the initial boundary value problem for one-dimensional thermoelasticity with second sound in the half line. By using delicate energy estimates, together with a special form of Helmholtz free energy, we are able to show the global solutions exist under the Dirichlet boundary condition if the initial data are sufficient small.

2015, 14(5): 1685-1704
doi: 10.3934/cpaa.2015.14.1685

*+*[Abstract](820)*+*[PDF](522.3KB)**Abstract:**

We deal with a class of parabolic nonlinear evolution equations with state-dependent delay. This class covers several important PDE models arising in biology. We first prove well-posedness in a certain space of functions which are Lipschitz in time. This allows us to show that the model considered generates an evolution operator semigroup $S_t$ on a certain space of Lipschitz type functions over delay time interval. The operators $S_t$ are closed for all $t\ge 0$ and continuous for $t$ large enough. Our main result shows that the semigroup $S_t$ possesses compact global and exponential attractors of finite fractal dimension. Our argument is based on the recently developed method of quasi-stability estimates and involves some extension of the theory of global attractors for the case of closed evolutions.

2015, 14(5): 1705-1741
doi: 10.3934/cpaa.2015.14.1705

*+*[Abstract](807)*+*[PDF](596.9KB)**Abstract:**

We prove that the critical Wave Maps equation with target $S^2$ and origin $R^{2+1}$ admits energy class blow up solutions of the form \begin{eqnarray} u(t, r) = Q(\lambda(t)r) + \varepsilon(t, r) \end{eqnarray} where $Q:R^2\rightarrow S^2$ is the ground state harmonic map and $\lambda(t) = t^{-1-\nu}$ for any $\nu>0$. This extends the work [14], where such solutions were constructed under the assumption $\nu>\frac{1}{2}$. In light of a result of Struwe [23], our result is optimal for polynomial blow up rates.

2015, 14(5): 1743-1757
doi: 10.3934/cpaa.2015.14.1743

*+*[Abstract](768)*+*[PDF](400.3KB)**Abstract:**

We investigate the uniqueness of nonnegative solutions to the following differential inequality \begin{eqnarray} div(A(x)|\nabla u|^{m-2}\nabla u)+V(x)u^{\sigma_1}|\nabla u|^{\sigma_2}\leq0, \tag{1} \end{eqnarray} on a noncompact complete Riemannian manifold, where $A, V$ are positive measurable functions, $m>1$, and $\sigma_1$, $\sigma_2\geq0$ are parameters such that $\sigma_1+\sigma_2>m-1$.

Our purpose is to establish the uniqueness of nonnegative solution to (1) via very natural geometric assumption on volume growth.

2015, 14(5): 1759-1780
doi: 10.3934/cpaa.2015.14.1759

*+*[Abstract](875)*+*[PDF](436.5KB)**Abstract:**

We consider the Cauchy problem in $ R^n$ for a strongly damped plate equation with a lower oder perturbation. We derive asymptotic profiles of solutions with weighted $L^{1,\gamma}(R^n)$ initial velocity by using a new method introduced in [7].

2015, 14(5): 1781-1801
doi: 10.3934/cpaa.2015.14.1781

*+*[Abstract](816)*+*[PDF](505.0KB)**Abstract:**

We study the long-time behavior of the solutions to a nonlinear damped driven Schrödinger type equation with quadratic potential on a strip. We prove that this behavior is described by a regular compact global attractor with finite fractal dimension.

2015, 14(5): 1803-1816
doi: 10.3934/cpaa.2015.14.1803

*+*[Abstract](991)*+*[PDF](410.2KB)**Abstract:**

In this paper, we study the following quasilinear Schrödinger equations of the form \begin{eqnarray} -\Delta u+V(x)u-[\Delta(1+u^2)^{\alpha/2}]\frac{\alpha u}{2(1+u^2)^{(2-\alpha)/2}}=\mathrm{g}(x,u)， \end{eqnarray} where $1 \le \alpha \le 2$, $N \ge 3$, $V\in C(R^N, R)$ and $\mathrm{g}\in C(R^N\times R, R)$. By using a change of variables, we get new equations, whose respective associated functionals are well defined in $H^1(R^N)$ and satisfy the geometric hypotheses of the mountain pass theorem. Using the special techniques, the existence of positive solutions is studied.

2015, 14(5): 1817-1840
doi: 10.3934/cpaa.2015.14.1817

*+*[Abstract](935)*+*[PDF](461.5KB)**Abstract:**

In this article, we are concerned with the initial value problem of fractional stochastic evolution equations in real separable Hilbert spaces. The existence of saturated mild solutions and global mild solutions is obtained under the situation that the nonlinear term satisfies some appropriate growth conditions by using $\alpha$-order fractional resolvent operator theory, the Schauder fixed point theorem and piecewise extension method. Furthermore, the continuous dependence of mild solutions on initial values and orders as well as the asymptotical stability in $p$-th moment of mild solutions for the studied problem have also been discussed. The results obtained in this paper improve and extend some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract results.

2015, 14(5): 1841-1863
doi: 10.3934/cpaa.2015.14.1841

*+*[Abstract](863)*+*[PDF](455.6KB)**Abstract:**

In this work, we propose and analyse approximation schemes for fully non-linear second order partial differential equations defined on the Heisenberg group. We prove that a consistent, stable and monotone scheme converges to a viscosity solution of a second order PDE on the Heisenberg group provided that comparison principles exists for the limiting equation. We also provide examples where this technique is applied.

2015, 14(5): 1865-1884
doi: 10.3934/cpaa.2015.14.1865

*+*[Abstract](841)*+*[PDF](472.4KB)**Abstract:**

In this paper, we consider the global well-posedness of the incompressible magnetohydrodynamic equations with initial data $(u_0,b_0)$ in the critical Besov space $\dot{B}_{2,1}^{1/2}(\mathbb{R}^3)\times \dot{B}_{2,1}^{1/2}(\mathbb{R}^3)$. Compared with [30], making full use of the algebraical structure of the equations, we relax the smallness condition in the third component of the initial velocity field and magnetic field. More precisely, we prove that there exist two positive constants $\varepsilon_0$ and $C_0$ such that if \begin{eqnarray} (\|u_0^h\|_{\dot{B}_{2,1}^{1/2}} +\|b_0^h\|_{\dot{B}_{2,1}^{1/2}}) \exp\{C_0(\frac{1}{\mu}+\frac{1}{\nu})^3 (\|u_0^3\|_{\dot{B}_{2,1}^{1/2}} +\|b_0^3\|_{\dot{B}_{2,1}^{1/2}})^2\} \le \varepsilon_0\mu\nu, \end{eqnarray} then the 3-D incompressible magnetohydrodynamic system has a unique global solution $(u,b)\in C([0,+\infty);\dot{B}_{2,1}^{1/2})\cap L^1((0,+\infty);\dot{B}_{2,1}^{5/2})\times C([0,+\infty);\dot{B}_{2,1}^{1/2})\cap L^1((0,+\infty);\dot{B}_{2,1}^{5/2}).$ Finally, we analyze the long behavior of the solution and get some decay estimates which imply that for any $t>0$ the solution $(u(t),b(t))\in C^{\infty}(\mathbb{R}^3)\times C^{\infty}(\mathbb{R}^3)$.

2015, 14(5): 1885-1902
doi: 10.3934/cpaa.2015.14.1885

*+*[Abstract](769)*+*[PDF](409.8KB)**Abstract:**

Let $P$ be a self-adjoint positive elliptic (-pseudo) differential operator on a smooth compact manifold $M$ without boundary. In this paper, we obtain a refined $L^p$ bound of the maximal function of the multiplier operators associated to $P$ satisfying the Hörmander-Mikhlin condition.

2015, 14(5): 1903-1913
doi: 10.3934/cpaa.2015.14.1903

*+*[Abstract](678)*+*[PDF](381.5KB)**Abstract:**

We address the problem of local and global well-posedness of Gross-Neveu (GN) equations for low regularity initial data. Combined with the standard machinery of $X_R$, $Y_R$ and $X^{s,b}$ spaces, we obtain local-wellposedness of (GN) for initial data $u, v \in H^s$ with $s\geq 0$. To prove the existence of global solution for the critical space $L^2$, we show non concentration of $L^2$ norm.

2015, 14(5): 1915-1927
doi: 10.3934/cpaa.2015.14.1915

*+*[Abstract](837)*+*[PDF](394.2KB)**Abstract:**

In this paper, we establish some Liouville type theorems for positive solutions of fractional Hénon equation and system in $\mathbb{R}^n$. First, under some regularity conditions, we show that the above equation and system are equivalent to the some integral equation and system, respectively. Then, we prove Liouville type theorems via the method of moving planes in integral forms.

2015, 14(5): 1929-1940
doi: 10.3934/cpaa.2015.14.1929

*+*[Abstract](741)*+*[PDF](372.7KB)**Abstract:**

Based on a generalized linking theorem for the strongly indefinite functionals, we study the existence of homoclinic orbits of the second order self-adjoint discrete Hamiltonian system \begin{eqnarray} \triangle [p(n)\triangle u(n-1)]-L(n)u(n)+\nabla W(n, u(n))=0, \end{eqnarray} where $p(n), L(n)$ and $W(n, x)$ are $N$-periodic on $n$, and $0$ lies in a gap of the spectrum $\sigma(\mathcal{A})$ of the operator $\mathcal{A}$, which is bounded self-adjoint in $l^2(\mathbb{Z}, \mathbb{R}^{\mathcal{N}})$ defined by $(\mathcal{A}u)(n)=\triangle [p(n)\triangle u(n-1)]-L(n)u(n)$. Under weak superquadratic conditions, we establish the existence of homoclinic orbits.

2015, 14(5): 1941-1960
doi: 10.3934/cpaa.2015.14.1941

*+*[Abstract](758)*+*[PDF](450.6KB)**Abstract:**

In this paper, we investigate the dynamics of a boson gas with three-body interactions in $T^2$. We prove that when the particle number $N$ tends to infinity, the BBGKY hierarchy of $k$-particle marginals converges to a infinite Gross-Pitaevskii(GP) hierarchy for which we prove uniqueness of solutions, and for the asymptotically factorized $N$-body initial datum, we show that this $N\rightarrow\infty$ limit corresponds to the quintic nonlinear Schrödinger equation. Thus, the Bose-Einstein condensation is preserved in time.

2015, 14(5): 1961-1986
doi: 10.3934/cpaa.2015.14.1961

*+*[Abstract](667)*+*[PDF](549.9KB)**Abstract:**

Pointwise estimate for the solutions of elliptic equations in periodic perforated domains is concerned. Let $\epsilon$ denote the size ratio of the period of a periodic perforated domain to the whole domain. It is known that even if the given functions of the elliptic equations are bounded uniformly in $\epsilon$, the $C^{1,\alpha}$ norm and the $W^{2,p}$ norm of the elliptic solutions may not be bounded uniformly in $\epsilon$. It is also known that when $\epsilon$ closes to $0$, the elliptic solutions in the periodic perforated domains approach a solution of some homogenized elliptic equation. In this work, the Hölder uniform bound in $\epsilon$ and the Lipschitz uniform bound in $\epsilon$ for the elliptic solutions in perforated domains are proved. The $L^\infty$ and the Lipschitz convergence estimates for the difference between the elliptic solutions in the perforated domains and the solution of the homogenized elliptic equation are derived.

2015, 14(5): 1987-2007
doi: 10.3934/cpaa.2015.14.1987

*+*[Abstract](697)*+*[PDF](464.1KB)**Abstract:**

In 1993, V. Šverák proved that if a sequence of uniformly bounded domains $\Omega_n\subset R^2$ such that $\Omega_n\to \Omega$ in the sense of the Hausdorff complementary topology, verify that the number of connected components of its complements are bounded, then the solutions of the Dirichlet problem for the Laplacian with source $f\in L^2(R^2)$ converges to the solution of the limit domain with same source. In this paper, we extend Šverák result to variable exponent spaces.

2015, 14(5): 2009-2020
doi: 10.3934/cpaa.2015.14.2009

*+*[Abstract](894)*+*[PDF](379.3KB)**Abstract:**

In this paper, by using the (variant) Fountain Theorem, we obtain that there are infinitely many solutions for a Kirchhoff type equation that involves a nonlocal operator.

2015, 14(5): 2021-2042
doi: 10.3934/cpaa.2015.14.2021

*+*[Abstract](872)*+*[PDF](495.7KB)**Abstract:**

In this paper we study an asymptotic behavior of a solution to the initial boundary value problem for a viscous liquid-gas two-phase flow model in a half line $R_+:=(0,\infty).$ Our idea mainly comes from [23] and [29] which describe an isothermal Navier-Stokes equation in a half line. We obtain the convergence rate of the time global solution towards corresponding stationary solution in Eulerian coordinates. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. These theorems are proved by a weighted energy method.

2015, 14(5): 2043-2067
doi: 10.3934/cpaa.2015.14.2043

*+*[Abstract](1201)*+*[PDF](481.1KB)**Abstract:**

Let $\Omega\subset R^N$ be a bounded open set with Lipschitz continuous boundary $\partial \Omega$. We define a fractional Dirichlet-to-Neumann operator and prove that it generates a strongly continuous analytic and compact semigroup on $L^2(\partial \Omega)$ which can also be ultracontractive. We also use the fractional Dirichlet-to-Neumann operator to compare the eigenvalues of a realization in $L^2(\Omega)$ of the fractional Laplace operator with Dirichlet boundary condition and the regional fractional Laplacian with the fractional Neumann boundary conditions.

2015, 14(5): 2069-2094
doi: 10.3934/cpaa.2015.14.2069

*+*[Abstract](917)*+*[PDF](504.0KB)**Abstract:**

The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using a proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.

2015, 14(5): 2095-2115
doi: 10.3934/cpaa.2015.14.2095

*+*[Abstract](974)*+*[PDF](458.1KB)**Abstract:**

In this paper, we study the global dynamics of a population model with age structure. The model is given by a nonlocal reaction-diffusion equation carrying a maturation time delay, together with the homogeneous Dirichlet boundary condition. The non-locality arises from spatial movements of the immature individuals. We are mainly concerned with the case when the birth rate decays as the mature population size becomes large. The analysis is rather subtle and it is inadequate to apply the powerful theory of monotone dynamical systems. By using the method of super-sub solutions, combined with the careful analysis of the kernel function in the nonlocal term, we prove nonexistence, existence and uniqueness of the positive steady states of the model. By establishing an appropriate comparison principle and applying the theory of dissipative systems, we obtain some sufficient conditions for the global asymptotic stability of the trivial solution and the unique positive steady state.

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