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Communications on Pure & Applied Analysis

2012 , Volume 11 , Issue 2

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Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows
Stefano Bosia
2012, 11(2): 407-441 doi: 10.3934/cpaa.2012.11.407 +[Abstract](227) +[PDF](577.4KB)
We analyze a simplified Ericksen-Leslie model for nematic liquid crystal flows firstly introduced in [18] with non-autonomous forcing bulk term and boundary conditions on the order parameter field. We obtain existence of weak solutions in the two- and three-dimensional cases. We prove uniqueness, continuous dependence on initial conditions, forcing and boundary terms and also existence of strong solutions in the 2D case. Focusing on the 2D case, we then study the long term behavior of solutions by obtaining existence of global attractors for normal forcing terms (according to [21]). Finally, we prove the existence of exponential attractors for quasi-periodic forcing terms in the 2D model.
Regularity criterion of the Newton-Boussinesq equations in $R^3$
Zhengguang Guo and Sadek Gala
2012, 11(2): 443-451 doi: 10.3934/cpaa.2012.11.443 +[Abstract](171) +[PDF](329.9KB)
In this paper, we consider the regularity problem under the critical condition to the Newton-Boussinesq equations. The Serrin type regularity criteria are established in terms of the critical Morrey-Campanato spaces and Besov spaces.
Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains
Jingbo Dou and Qianqiao Guo
2012, 11(2): 453-464 doi: 10.3934/cpaa.2012.11.453 +[Abstract](339) +[PDF](419.1KB)
Consider the polyharmonic elliptic problem \begin{eqnarray*} (-\Delta)^m u=\lambda u+Q(x)|u|^{2^*-2}u & in \Omega, \\ (\frac{\partial}{\partial\nu })^j u |_{\partial\Omega}=0, j=0, 1, 2, \cdots, m-1,& \end{eqnarray*} where $\Omega$ is an open bounded domain with smooth boundary in $R^N$, $m\geq 1,N>2m, 2^*=\frac{2N}{N-2m}$ is the critical Sobolev exponent, $Q(x)$ is positive and continuous in $\overline{\Omega}$. We prove the existence of nontrivial and sign-changing solutions when $\Omega$, $Q(x)$ are invariant under a group of orthogonal transformations and $0 < \lambda < \lambda_1$, where $\lambda_1$ is the first Dirichlet eigenvalue of $(-\Delta)^m$ in $\Omega$.
Lyapunov-type inequalities for even order differential equations
Xiaofei He and X. H. Tang
2012, 11(2): 465-473 doi: 10.3934/cpaa.2012.11.465 +[Abstract](203) +[PDF](307.1KB)
In this paper, we establish several new Lyapunov-type inequalities for the $2n-$order differential equation

$x^{(2n)}(t)+(-1)^{n-1}q(t)x(t)=0, $

which are sharper than all related existing ones.

Best asymptotic profile for the system of compressible adiabatic flow through porous media on quadrant
Shifeng Geng and Zhen Wang
2012, 11(2): 475-500 doi: 10.3934/cpaa.2012.11.475 +[Abstract](194) +[PDF](434.9KB)
We realize the best asymptotic profile for the solutions to the nonisentropic $p$-system with damping on quadrant is a particular solution of the IBVP for the corresponding nonlinear parabolic equation with special initial data, and we further show the convergence rates to this particular asymptotic profile. This rates are same to that for the isentropic case obtained by H. Ma and M. Mei (J. Differential Equations 249 (2010), 446--484).
The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$
Wen-ming He and Jun-zhi Cui
2012, 11(2): 501-516 doi: 10.3934/cpaa.2012.11.501 +[Abstract](173) +[PDF](399.4KB)
In this paper, for the second order elliptic problems with small periodic coefficients of the form $\frac{\partial}{\partial x_{i}} (a^{i j}(\frac{x}{\varepsilon})\frac{\partial u^{\varepsilon}(x)}{\partial x_{j}})=f(x)$, we shall discuss the multi-scale homogenization theory for Green's function $G_{y}^{\varepsilon}$ at point $y\in\Omega$ on Sobolev space $W^{1,q}(\Omega)$. Assume that $B(y,d)=\{x\in\Omega|dist(x,y)\leq d\},$ ${G}_{y}$ and $\theta_{G,y}^{\varepsilon}$ are the 1-order approximation and the boundary corrector of $G_{y}^{\varepsilon}$, respectively. We present an estimate for $\left\|G_{y}^{\varepsilon}-{G}_{y}-\theta_{G,y}^{\varepsilon}\right\|_{W^{1,q}(\Omega\ B(y,d))}$.
Nonradial positive solutions for a biharmonic critical growth problem
Zhongliang Wang
2012, 11(2): 517-545 doi: 10.3934/cpaa.2012.11.517 +[Abstract](179) +[PDF](536.8KB)
We investigate the existence of nonradial positive solutions for a critical semilinear biharmonic problem defined on a unit ball. The solution is obtained as a minimizer of the quotient functional associated to the problem restricted to appropriate subspaces of $H^2\cap H^1_0$ invariant for the action of a subgroup of $O(N)$. By making use of more careful estimates and some new arguments, we extend Serra's result in [41] to the biharmonic case.
Global solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary conditions
Jason Metcalfe and Jacob Perry
2012, 11(2): 547-556 doi: 10.3934/cpaa.2012.11.547 +[Abstract](151) +[PDF](349.0KB)
This article focuses on proving global existence for quasilinear wave equations with small initial data in homogeneous waveguides with infinite base of dimensions $n\geq 4$. The key estimate is a localized energy estimate for a perturbed wave equation.
On the constants in a Kato inequality for the Euler and Navier-Stokes equations
Carlo Morosi and Livio Pizzocchero
2012, 11(2): 557-586 doi: 10.3934/cpaa.2012.11.557 +[Abstract](154) +[PDF](690.4KB)
We continue an analysis, started in [10], of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a $d$-dimensional torus $T^d$. More specifically, we consider the quadratic term in these equations; this arises from the bilinear map $(v, w) \mapsto v \cdot \partial w$, where $v, w : T^d \to R^d$ are two velocity fields. We derive upper and lower bounds for the constants in some inequalities related to the above bilinear map; these bounds hold, in particular, for the sharp constants $G_{n d} \equiv G_n$ in the Kato inequality $| < v \cdot \partial w | w >_n | \leq G_n || v ||_n || w ||^2_n$, where $n \in (d/2 + 1, + \infty)$ and $v, w$ are in the Sobolev spaces $H^n_{\Sigma_0}, H^{n+1}_{\Sigma_0}$ of zero mean, divergence free vector fields of orders $n$ and $n+1$, respectively. As examples, the numerical values of our upper and lower bounds are reported for $d=3$ and some values of $n$. When combined with the results of [10] on another inequality, the results of the present paper can be employed to set up fully quantitative error estimates for the approximate solutions of the Euler/NS equations, or to derive quantitative bounds on the time of existence of the exact solutions with specified initial data; a sketch of this program is given.
The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy
Robert Magnus and Olivier Moschetta
2012, 11(2): 587-626 doi: 10.3934/cpaa.2012.11.587 +[Abstract](162) +[PDF](563.3KB)
The equation $-\varepsilon^2 \Delta u+F(V(x),u)=0$ is considered in $R^n$. It is assumed that $V$ possesses a set of critical points $B$ for which the values of $V$ and $D^2V$ satisfy certain compactness and uniformity properties. Under appropriate conditions on $F$ the problem is shown to possess for each $b\in B$ and small $\varepsilon>0$ a solution that concentrates at $b$ and has detailed uniformity and decay properties. This enables the construction of solutions that concentrate at arbitrary subsets of $B$ as $\varepsilon\to 0$. Examples are given in which $B$ is infinite and $V$ non-periodic.
On mappings of higher order and their applications to nonlinear equations
Dariusz Bugajewski and Piotr Kasprzak
2012, 11(2): 627-647 doi: 10.3934/cpaa.2012.11.627 +[Abstract](186) +[PDF](420.9KB)
One of the main goals of this paper is to investigate mappings of higher order which possess ``good'' properties, in particular, when we treat them as perturbations of nonlinear differential as well as integral equations. We draw a particular attention to nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan or in the sense of Young. We provide sufficient conditions which guarantee that nonlinear Hammerstein operators are of higher order in such spaces. We also prove a few extensions of Lovelady's fixed point theorem in Archimedean as well as non-Archimedean setting. Finally, we apply our results to prove the existence and uniqueness results to some commonly known nonlinear equations with perturbations.
On the critical exponents for porous medium equation with a localized reaction in high dimensions
Zhilei Liang
2012, 11(2): 649-658 doi: 10.3934/cpaa.2012.11.649 +[Abstract](184) +[PDF](368.5KB)
This paper is concerned with the critical exponents for the porous medium equation

$u_{t}=\triangle u^m+a(x)u^p, (x,t)\in R^N\times (0,T), $

where $m>1, p>0,$ and the function $a(x)\geq 0$ has a compact support. Suppose the space dimension $N\geq 2$, we prove that the global exponent $p_0$ and the Fujita type exponent $p_c$ are both $m$: if $0 < p < m$ every solution is global in time, if $ p = m $ all the solutions blow up and if $p > m$ both the blowing up solutions and the global solutions exist. While for the one-dimensional case, it is proved $p_0=\frac{m+1}{2} < m+1 = p_c$ by [E. Ferreira, A. Pablo, J. Vazquez, Classification of blow-up with nonlinear diffusion and localized reaction, J. Diff. Eqns., 231(2006) 195-211].

Long-time dynamics in plate models with strong nonlinear damping
Igor Chueshov and Stanislav Kolbasin
2012, 11(2): 659-674 doi: 10.3934/cpaa.2012.11.659 +[Abstract](213) +[PDF](460.8KB)
We study long-time dynamics of a class of abstract second order in time evolution equations in a Hilbert space with the damping term depending both on displacement and velocity. This damping represents the nonlinear strong dissipation phenomenon perturbed with relatively compact terms. Our main result states the existence of a compact finite dimensional attractor. We study properties of this attractor. We also establish the existence of a fractal exponential attractor and give the conditions that guarantee the existence of a finite number of determining functionals. In the case when the set of equilibria is finite and hyperbolic we show that every trajectory is attracted by some equilibrium with exponential rate. Our arguments involve a recently developed method based on the "compensated" compactness and quasi-stability estimates. As an application we consider the nonlinear Kirchhoff, Karman and Berger plate models with different types of boundary conditions and strong damping terms. Our results can be also applied to the nonlinear wave equations.
On a heated incompressible magnetic fluid model
Youcef Amirat and Kamel Hamdache
2012, 11(2): 675-696 doi: 10.3934/cpaa.2012.11.675 +[Abstract](136) +[PDF](467.0KB)
In this paper we study the equations describing the dynamics of heat transfer in an incompressible magnetic fluid under the action of an applied magnetic field. The system consists of the Navier-Stokes equations, the magnetostatic equations and the temperature equation. We prove global-in-time existence of weak solutions to the system posed in a bounded domain of $R^3$ and equipped with initial and boundary conditions. The main difficulty comes from the singularity of the term representing the Kelvin force due to magnetization.
On the blow-up boundary solutions of the Monge -Ampére equation with singular weights
Haitao Yang and Yibin Chang
2012, 11(2): 697-708 doi: 10.3934/cpaa.2012.11.697 +[Abstract](199) +[PDF](144.3KB)
We consider the Monge-Ampére equations det$D^2 u = K(x) f(u)$ in $\Omega$, with $u|_{\partial\Omega}=+\infty$, where $\Omega$ is a bounded and strictly convex smooth domain in $R^N$. When $f(u) = e^u$ or $f(u)= u^p$, $p>N$, and the weight $K(x)\in C^\infty (\Omega )$ grows like a negative power of $d(x)=dist(x, \partial \Omega)$ near $\partial \Omega$, we show some results on the uniqueness, nonexistence and exact boundary blow-up rate of strictly convex solutions for this problem. Existence of such solutions will be also studied in a more general case.
Rough solutions for the periodic Korteweg--de~Vries equation
Massimiliano Gubinelli
2012, 11(2): 709-733 doi: 10.3934/cpaa.2012.11.709 +[Abstract](145) +[PDF](530.7KB)
We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg--de Vries (KdV) equation on a periodic domain and with initial condition in $F L^{\alpha,p}$ spaces. We discuss convergence of Galerkin approximations, a modified Euler scheme and the presence of a random force of white-noise type in time.
The moving boundary problem in a chemotaxis model
Hua Chen and Shaohua Wu
2012, 11(2): 735-746 doi: 10.3934/cpaa.2012.11.735 +[Abstract](174) +[PDF](328.6KB)
In this paper, we prove the local existence and uniqueness of a moving boundary problem modeling chemotactic phenomena. We also get the explicit representative for the moving boundary and show the finite-time blow-up and chemotactic collapse for the solution of the problem.
On the regularity of solutions to the Navier-Stokes equations
Vittorino Pata
2012, 11(2): 747-761 doi: 10.3934/cpaa.2012.11.747 +[Abstract](220) +[PDF](437.7KB)
This article is concerned with the incompressible Navier-Stokes equations in a three-dimensional domain. A criterion of Prodi-Serrin type up to the boundary for global existence of strong solutions is established.
Global solutions to the incompressible magnetohydrodynamic equations
Xiaoli Li and Dehua Wang
2012, 11(2): 763-783 doi: 10.3934/cpaa.2012.11.763 +[Abstract](260) +[PDF](433.2KB)
An initial-boundary value problem of the three-dimensional incompressible magnetohydrodynamic (MHD) equations is considered in a bounded domain. The homogeneous Dirichlet boundary condition is prescribed on the velocity, and the perfectly conducting wall condition is prescribed on the magnetic field. The existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small initial data. Furthermore, the weak-strong uniqueness of solutions is also proved, which shows that the weak solution is equal to the strong solution with certain initial data.
Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings
Kaïs Ammari, Denis Mercier, Virginie Régnier and Julie Valein
2012, 11(2): 785-807 doi: 10.3934/cpaa.2012.11.785 +[Abstract](214) +[PDF](452.1KB)
We consider $N$ Euler-Bernoulli beams and $N$ strings alternatively connected to one another and forming a particular network which is a chain beginning with a string. We study two stabilization problems on the same network and the spectrum of the corresponding conservative system: the characteristic equation as well as its asymptotic behavior are given. We prove that the energy of the solution of the first dissipative system tends to zero when the time tends to infinity under some irrationality assumptions on the length of the strings and beams. On another hand we prove a polynomial decay result of the energy of the second system, independently of the length of the strings and beams, for all regular initial data. Our technique is based on a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.
On the structure of the global attractor for non-autonomous dynamical systems with weak convergence
Tomás Caraballo and David Cheban
2012, 11(2): 809-828 doi: 10.3934/cpaa.2012.11.809 +[Abstract](170) +[PDF](495.9KB)
The aim of this paper is to describe the structure of global attractors for non-autonomous dynamical systems with recurrent coefficients (with both continuous and discrete time). We consider a special class of this type of systems (the so--called weak convergent systems). It is shown that, for weak convergent systems, the answer to Seifert's question (Does an almost periodic dissipative equation possess an almost periodic solution?) is affirmative, although, in general, even for scalar equations, the response is negative. We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our paper to the study of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of different classes of differential equations.
Multi-bump solutions for a class of quasilinear equations on $R$
Claudianor O. Alves, Olímpio H. Miyagaki and Sérgio H. M. Soares
2012, 11(2): 829-844 doi: 10.3934/cpaa.2012.11.829 +[Abstract](138) +[PDF](408.5KB)
This paper is concerned with the existence of multi-bump solutions to a class of quasilinear Schrödinger equations in $R$. The proof relies on variational methods and combines some arguments given by del Pino and Felmer, Ding and Tanaka, and Séré.
The supercooled Stefan problem in one dimension
Lincoln Chayes and Inwon C. Kim
2012, 11(2): 845-859 doi: 10.3934/cpaa.2012.11.845 +[Abstract](170) +[PDF](402.8KB)
We study the 1D contracting Stefan problem with two moving boundaries that describes the freezing of a supercooled liquid. The problem is borderline ill--posed with a density in excess of unity indicative of the dividing line. We show that if the initial density, $\rho_0(x)$ does not exceed one and is not too close to one in the vicinity of the boundaries, then there is a unique solution for all times which is smooth for all positive times. Conversely if the initial density is too large, singularities may occur. Here the situation is more complex: the solution may suddenly freeze without any hope of continuation or may continue to evolve after a local instant freezing but, sometimes, with the loss of uniqueness.

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