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Communications on Pure & Applied Analysis

2010 , Volume 9 , Issue 6

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Global in time solution and time-periodicity for a smectic-A liquid crystal model
Blanca Climent-Ezquerra and Francisco Guillén-González
2010, 9(6): 1473-1493 doi: 10.3934/cpaa.2010.9.1473 +[Abstract](104) +[PDF](255.1KB)
In this paper some results are obtained for a smectic-A liquid crystal model with time-dependent boundary Dirichlet data for the so-called layer variable $\varphi$ (the level sets of $\varphi$ describe the layer structure of the smectic-A liquid crystal). First, the initial-boundary problem for arbitrary initial data is considered, obtaining the existence of weak solutions which are bounded up to infinity time. Second, the existence of time-periodic weak solutions is proved. Afterwards, the problem of the global in time regularity is attacked, obtaining the existence and uniqueness of regular solutions (up to infinity time) for both problems, i.e. the initial-valued problem and the time-periodic one, but assuming a dominant viscosity coefficient in the linear part of the diffusion tensor.
Elastic Herglotz functions in the plane
J. A. Barceló, M. Folch-Gabayet, S. Pérez-Esteva, A. Ruiz and M. C. Vilela
2010, 9(6): 1495-1505 doi: 10.3934/cpaa.2010.9.1495 +[Abstract](143) +[PDF](193.6KB)
We study spaces of solutions of the spectral Navier equation in the plane. We characterize the elastic Herglotz wave functions, namely the entire solutions $\mathbf{u}$ of the Navier equation with $L^2$ far-field-patterns. The characterization is in terms of a weighted $L^2$ norm involving $\mathbf{u}$ and its angular derivative $\partial_\theta \mathbf{u.}$ With respect to this norm, the space of elastic Herglotz wave functions is decomposed into the topological product of the compressional and shear elastic Herglotz wave functions. We also study the solutions of the Navier equation whose Lamé potentials are the Fourier transform of distributions in the circle. We prove that these are the entire solutions of the Navier equation with polynomial growth. This extends a result by Agmon for the Helmholtz equation.
Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case
Michael E. Filippakis, Donal O'Regan and Nikolaos S. Papageorgiou
2010, 9(6): 1507-1527 doi: 10.3934/cpaa.2010.9.1507 +[Abstract](102) +[PDF](270.2KB)
We consider a nonlinear elliptic equation of logistic type, driven by the $p$-Laplacian differential operator with a general superdiffusive reaction. We show that the equation exhibits a bifurcation phenomenon. Namely there is a critical value $\lambda_*$ of the parameter $\lambda>0$, such that, if $\lambda>\lambda_*$, the equation has two nontrivial positive smooth solutions, if $\lambda=\lambda_*,$ then there is one positive solution and finally if $\lambda\in (0,\lambda_*),$ then there is no positive solution.
Multiplicity results for periodic solutions to a class of second order delay differential equations
Zhiming Guo and Xiaomin Zhang
2010, 9(6): 1529-1542 doi: 10.3934/cpaa.2010.9.1529 +[Abstract](129) +[PDF](183.8KB)
In this paper, we study the following second order delay differential equation

$x''(t)=-f(x(t), x(t-\tau)).$

When $f$ possesses a symmetric property and grows asymptotically linear both at zero and at infinity, some new results for the existence and multiplicity of periodic solutions are obtained by using the critical point theory and $S^1$ geometrical index theory.

On viscoelastic wave equation with nonlinear boundary damping and source term
Tae Gab Ha
2010, 9(6): 1543-1576 doi: 10.3934/cpaa.2010.9.1543 +[Abstract](161) +[PDF](335.1KB)
In this paper, we consider the viscoelastic wave equation with nonlinear boundary damping and source term. This work is devoted to prove the existence of solutions and uniform decay rates of the energy without imposing any restrictive growth assumption on the damping term and weakening the usual assumptions on the relaxation function.
Quasi-neutral limit of the two-fluid Euler-Poisson system
Qiangchang Ju, Hailiang Li, Yong Li and Song Jiang
2010, 9(6): 1577-1590 doi: 10.3934/cpaa.2010.9.1577 +[Abstract](143) +[PDF](182.9KB)
Quasi-neutral limit of the multidimensional isentropic two-fluid Euler-Poisson system is rigorously justified. For well-prepared initial data, as the Debye length goes to zero, the convergence of the bipolar Euler-Poisson system to the compressible Euler equations is proved in the time interval where a smooth solution of the limit problem exists.
Global wellposedness and blowup of solutions to a nonlocal evolution problem with singular kernels
Dong Li and Xiaoyi Zhang
2010, 9(6): 1591-1606 doi: 10.3934/cpaa.2010.9.1591 +[Abstract](80) +[PDF](223.4KB)
We consider a nonlocal evolution equation in $R^2$: $\partial_t u + \nabla \cdot (u K*u )= 0$, where $K(x) = \mu \frac x {|x|^\alpha}$, $\mu=\pm 1$ and $1 < \alpha < 2 $. We study wellposedness, continuation/blowup criteria and smoothness of solutions in Sobolev spaces. In the repulsive case ($\mu=1$), by using the sharp blowup criteria, we prove global wellposedness for any positive large initial data. In the attractive case ($\mu=-1$), by using a novel free energy inequality together with a mass localization technique, we construct finite time blowups for a large class of smooth initial data.
Structure of the set of bounded solutions for a class of nonautonomous second order differential equations
Yuan Guo, Xiaofei Gao and Desheng Li
2010, 9(6): 1607-1616 doi: 10.3934/cpaa.2010.9.1607 +[Abstract](119) +[PDF](167.0KB)
Some precise descriptions of the structure and behavior of bounded solutions for a class of nonautonomous second order differential equations $x''+g(x)x'+f(\theta_t h,x)=0$ are investigated. The results contain many previous ones in the literature as particular cases.
Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion
Andrea L. Bertozzi and Dejan Slepcev
2010, 9(6): 1617-1637 doi: 10.3934/cpaa.2010.9.1617 +[Abstract](182) +[PDF](276.2KB)
We present an energy-methods-based proof of the existence and uniqueness of solutions of a nonlocal aggregation equation with degenerate diffusion. The equation we study is relevant to models of biological aggregation.
Global dynamics of the periodic un-stirred chemostat with a toxin-producing competitor
Yifu Wang and Jingxue Yin
2010, 9(6): 1639-1651 doi: 10.3934/cpaa.2010.9.1639 +[Abstract](95) +[PDF](183.6KB)
This paper is concerned with the un-stirred chemostat with a toxin-producing competitor. The novelties of the modified model are the periodicity appearing in the boundary conditions, the different diffusive coefficients of the nutrient and the microorganisms, and some kinds of death rates. Both uniform persistence and global extinction of the microorganisms are established under suitable conditions in terms of principal eigenvalues of scalar periodic parabolic eigenvalue problems. Our result implies that the toxin inhibits the sensitive microorganism indeed. The techniques includes the theories of asymptotic periodic semi-flows, uniform persistence and the perturbation of global attractor.
On the upper semicontinuity of pullback attractors with applications to plate equations
Yonghai Wang
2010, 9(6): 1653-1673 doi: 10.3934/cpaa.2010.9.1653 +[Abstract](172) +[PDF](274.4KB)
In this paper, we give a technic method for verifying the upper semicontinuity of pullback attractors of dynamical systems under small nonautonomous perturbations. The method we give is suitable, in some sense, for weakly dissipative dynamical systems. And, we apply our result to plate equations.
Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition
Tsung-Fang Wu
2010, 9(6): 1675-1696 doi: 10.3934/cpaa.2010.9.1675 +[Abstract](120) +[PDF](251.6KB)
In this paper, we study a class of semilinear elliptic equations in $R_+^N$ with nonlinear boundary condition and sign-changing weight function. By means of the Lusternik-Schnirelman category, multiple positive solutions are obtained.
Regularity criteria of strong solutions to a problem of magneto-elastic interactions
Yong Zhou and Jishan Fan
2010, 9(6): 1697-1704 doi: 10.3934/cpaa.2010.9.1697 +[Abstract](127) +[PDF](140.7KB)
In this paper, various regularity criteria for the strong solutions to a problem arising in the study of magneto-elastic interactions are established. In particular, these regularity criteria are also true for the Landau-Lifshitz equation and give extensions of previous results.
Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain
M. Ben Ayed and Abdelbaki Selmi
2010, 9(6): 1705-1722 doi: 10.3934/cpaa.2010.9.1705 +[Abstract](94) +[PDF](238.0KB)
In this paper, we consider the problem $(Q_\varepsilon)$ : $\Delta ^2 u= u^9 +\varepsilon f(x)$ in $\Omega$, $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a bounded and smooth domain in $R^5$, $\varepsilon$ is a small positive parameter, and $f$ is a smooth function. Our main purpose is to characterize the solutions with some assumptions on the energy. We prove that these solutions blow up at a critical point of a function depending on $f$ and the regular part of the Green's function. Moreover, we construct families of solutions of $(Q_\varepsilon)$ which satisfy the conclusions of the first part.
A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials
Yongsheng Jiang and Huan-Song Zhou
2010, 9(6): 1723-1730 doi: 10.3934/cpaa.2010.9.1723 +[Abstract](102) +[PDF](196.8KB)
The aim of this paper is to establish a sharp decay estimate for radially symmetric solutions of the following type of nonlinear Schrödinger equations:

$-\Delta u + V(|x|)u =Q(|x|)|u|^{p-2}u, x\in R^N, $

$u(x)\rightarrow 0$ as $|x|\rightarrow+\infty,$

where $N\geq 3$, $p\in(2,+\infty)$, $V(x)$ and $Q(x)$ are continuous functions which vanishes at infinity and may change sign. As a special case, our result shows that the solutions obtained by Su-Wang-Willem in [11, Theorem 3] must decay precisely like $|x|^{-(N-2)}$ as $|x|\rightarrow+\infty$ if $V(|x|)$ decays faster than $|x|^{-2}$ at infinity.

On the shape of the least-energy solutions to some singularly perturbed mixed problems
Luigi Montoro
2010, 9(6): 1731-1752 doi: 10.3934/cpaa.2010.9.1731 +[Abstract](108) +[PDF](1470.2KB)
In this paper we want to \emph{characterize} and \emph{visualize} the shape of some solutions to a singularly perturbed problem \eqref{eq:pe} with mixed Dirichlet and Neumann boundary conditions. Such type of problem arises in several situations as reaction-diffusion systems, nonlinear heat conduction and also as limit of reaction-diffusion systems coming from chemotaxis. In particular we are interested in showing the location and the shape of {\it least energy solutions} when the singular perturbation parameter goes to zero, analyzing the geometrical effect of the \emph{curved boundary} of the domain.
$\sum $-convergence and reiterated homogenization of nonlinear parabolic operators
Jean Louis Woukeng
2010, 9(6): 1753-1789 doi: 10.3934/cpaa.2010.9.1753 +[Abstract](133) +[PDF](453.7KB)
We study the reiterated homogenization of nonlinear parabolic differential equations associated with monotone operators. Contrary to what is usually done in the deterministic homogenization theory, we present a new approach based on a deterministic assumption on the coefficients of the operators, which allows us to consider the concrete homogenization problems from a true and natural perspective, taking into account the discontinuities in general. Based on this new approach we obtain very general homogenization results, and we solve several concrete homogenization problems. Our main tool is the theory of homogenization structures, and our homogenization approach falls within the scope of multiscale convergence method.

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