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Communications on Pure & Applied Analysis

2008 , Volume 7 , Issue 6

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The inhomogeneous PME in several space dimensions. Existence and uniqueness of finite energy solutions
Guillermo Reyes and Juan-Luis Vázquez
2008, 7(6): 1275-1294 doi: 10.3934/cpaa.2008.7.1275 +[Abstract](329) +[PDF](283.6KB)
Abstract:
We study the questions of existence and uniqueness of non-negative solutions to the Cauchy problem

$\rho(x)\partial_t u= \Delta u^m\qquad$ in $Q$:$=\mathbb R^n\times\mathbb R_+$

$u(x, 0)=u_0$

in dimensions $n\ge 3$. We deal with a class of solutions having finite energy

$E(t)=\int_{\mathbb R^n} \rho(x)u(x,t) dx$

for all $t\ge 0$. We assume that $m> 1$ (slow diffusion) and the density $\rho(x)$ is positive, bounded and smooth. We prove existence of weak solutions starting from data $u_0\ge 0$ with finite energy. We show that uniqueness takes place if $\rho$ has a moderate decay as $|x|\to\infty$ that essentially amounts to the condition $\rho\notin L^1(\mathbb R^n)$. We also identify conditions on the density that guarantee finite speed of propagation and energy conservation, $E(t)=$const. Our results are based on a new a priori estimate of the solutions.

The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains
Dorina Mitrea, Marius Mitrea and Sylvie Monniaux
2008, 7(6): 1295-1333 doi: 10.3934/cpaa.2008.7.1295 +[Abstract](452) +[PDF](482.7KB)
Abstract:
We formulate and solve the Poisson problem for the exterior derivative operator with Dirichlet boundary condition in Lipschitz domains, of arbitrary topology, for data in Besov and Triebel-Lizorkin spaces.
Convexity of level curves for solutions to semilinear elliptic equations
David L. Finn
2008, 7(6): 1335-1343 doi: 10.3934/cpaa.2008.7.1335 +[Abstract](329) +[PDF](159.3KB)
Abstract:
Let $\Omega$ be a bounded strictly convex planar domain, and $f$ be a smooth function satisfying $f(0) < 0$ and $f'(t) \geq 0$. In this paper, we provide a simple proof using just the maximum principle that the level curves of the unique positive solution to $\Delta u = f(u)$ in $\Omega$ satisfying $u = 0$ on $\partial\Omega$ are convex and there is a unique critical point. We also provide generalization of this result to cover certain cases with $f'(t) < 0$.
Numerical mountain pass solutions of Ginzburg-Landau type equations
Dmitry Glotov and P. J. McKenna
2008, 7(6): 1345-1359 doi: 10.3934/cpaa.2008.7.1345 +[Abstract](323) +[PDF](560.8KB)
Abstract:
We study the numerical solutions of a system of Ginzburg-Landau type equations arising in the thin film model of superconductivity. These solutions are obtained by the Mountain Pass algorithm that was originally developed for semilinear elliptic equations. We prove a key hypothesis of the Mountain Pass theorem and investigate the physical features of the solutions such as the presence, the number, and the location of vortices and the numerical properties such as stability.
Nonuniformity of deformation preceding shear band formation in a two-dimensional model for Granular flow
Scott Gordon
2008, 7(6): 1361-1374 doi: 10.3934/cpaa.2008.7.1361 +[Abstract](240) +[PDF](240.8KB)
Abstract:
The onset of shear band formation in granular materials has been linked to the governing partial differential equations becoming ill-posed which has in turn been linked to nonassociativity of the flow rule. If uniform material properties and uniform deformation are assumed, ill-posedness occurs simultaneously at all points in the sample. This work derives a one-dimensional model from a two-dimensional model for granular flow with a nonassociative flow rule and shows that, shortly before the onset of ill-posedness, deformation can become highly non-uniform at a point where the material is slightly weakened.
Stabilisation of differential inclusions and PDEs without uniqueness by noise
Tomás Caraballo, José A. Langa and José Valero
2008, 7(6): 1375-1392 doi: 10.3934/cpaa.2008.7.1375 +[Abstract](341) +[PDF](243.2KB)
Abstract:
We prove that the asymptotic behaviour of partial differential inclusions and partial differential equations without uniqueness of solutions can be stabilised by adding some suitable Itô noise as an external perturbation. We show how the theory previously developed for the single-valued cases can be successfully applied to handle these set-valued cases. The theory of random dynamical systems is used as an appropriate tool to solve the problem.
Asymptotic behavior of a hyperbolic system arising in ferroelectricity
Sergio Frigeri
2008, 7(6): 1393-1414 doi: 10.3934/cpaa.2008.7.1393 +[Abstract](251) +[PDF](292.8KB)
Abstract:
We consider a coupled hyperbolic system which describes the evolution of the electromagnetic field inside a ferroelectric cylindrical material in the framework of the Greenberg-MacCamy-Coffman model. In this paper we analyze the asymptotic behavior of the solutions from the viewpoint of infinite-dimensional dissipative dynamical systems. We first prove the existence of an absorbing set and of a compact global attractor in the energy phase-space. A sufficient condition for the decay of the solutions is also obtained. The main difficulty arises in connection with the study of the regularity property of the attractor. Indeed, the physically reasonable boundary conditions prevent the use of a technique based on multiplication by fractional operators and bootstrap arguments. We obtain the desired regularity through a decomposition technique introduced by Pata and Zelik for the damped semilinear wave equation. Finally we provide the existence of an exponential attractor.
Linearization of smooth planar vector fields around singular points via commuting flows
Isaac A. García, Jaume Giné and Susanna Maza
2008, 7(6): 1415-1428 doi: 10.3934/cpaa.2008.7.1415 +[Abstract](340) +[PDF](203.0KB)
Abstract:
In this paper we propose a constructive procedure to get the change of variables that linearizes a smooth planar vector field on $\mathbb C^2$ around an elementary singular point (i.e., a singular point with associated eigenvalues $\lambda, \mu \in \mathbb C$ satisfying $\mu$≠$0$) or a nilpotent singular point from a given commutator. Moreover, it is proved that the near--identity change of variables that linearizes the vector field $\mathcal X = (x+\cdots) \partial_x + (y+\cdots) \partial_y$ is unique and linearizes simultaneously all the centralizers of $\mathcal X$. The method is used in order to obtain the linearization of some extracted examples of the existent literature.
On a time discretization of a weakly damped forced nonlinear Schrödinger equation
Olivier Goubet and Ezzeddine Zahrouni
2008, 7(6): 1429-1442 doi: 10.3934/cpaa.2008.7.1429 +[Abstract](312) +[PDF](205.9KB)
Abstract:
We consider a semi-discrete in time Crank-Nicolson scheme to discretize a damped forced nonlinear Schrödinger equation. This provides us with a discrete infinite-dimensional dynamical system. We prove the existence of a finite dimensional global attractor for this dynamical system.
Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation
Kazuhiro Kurata and Kotaro Morimoto
2008, 7(6): 1443-1482 doi: 10.3934/cpaa.2008.7.1443 +[Abstract](289) +[PDF](432.1KB)
Abstract:
In this paper, we are concerned with stationary solutions to the following reaction diffusion system which is called the Gierer-Meinhardt system:

$A_t=\varepsilon^2 \Delta A-A+\frac{A^2}{H(1+kA^2)},\ A>0,\ $ in $\Omega\times (0,\infty), $

$\tau H_t=D\Delta H-H+A^2,\ H>0,\ $ in $\Omega \times (0,\infty),$

$\frac{\partial A}{\partial \nu}=\frac{\partial H}{\partial \nu}=0,\ $ on $\partial \Omega\times (0,\infty),$

where $\varepsilon>0$, $\tau \geq 0$, $k>0$. The unknowns $A=A(x,t)$, $H=H(x,t)$ represent the concentrations of the activator and the inhibitor at a point $x\in \Omega \subset R^N$ and at a time $t>0$. Here $\Delta$ := $\sum_{j=1}^N\frac{\partial^2}{\partial x^2_j}$ is the Laplace operator in $R^N$, $\Omega$ is a bounded smooth domain in $R^N$, and $\nu=\nu(x)$ is the outer unit normal at $x\in \partial \Omega$. When $\Omega$ is an $x_N$-axially symmetric domain and $2\leq N\leq 5$, for sufficiently small $\varepsilon>0$ and sufficiently large $D>0$ we construct a multi-peak stationary solution peaked at arbitrarily chosen intersections of $x^N$-axis and $\partial \Omega$, under the condition that $4k\varepsilon^{-2N}|\Omega|^2$ converges to some $k_0\in[0,\infty)$ as $\varepsilon\to 0$.

The threshold for persistence of parasites with multiple infections
M. P. Moschen and A. Pugliese
2008, 7(6): 1483-1496 doi: 10.3934/cpaa.2008.7.1483 +[Abstract](267) +[PDF](213.5KB)
Abstract:
We analyse a model for macro-parasites in an age-structured host population, with infections of hosts occurring in clumps of parasites. The resulting model is an infinite system of partial differential equations of the first order, with non-local boundary conditions. We establish a condition for the parasite--free equilibrium to be asymptotically stable, in terms of $R_0 < 1$, where $R_0$ is a quantity interpreted as the reproduction number of parasites. To show this, we prove that $s(B-A)<0$ [$>0$] if and only if $\rho(B(A)^{-1} )< 1$ [$>1$], where $B$ is a positive operator, and $A$ generates a positive semigroup of negative type. Finally, we discuss how $R_0$ depends on the parameters of the system, especially on the mean size of infecting clumps.
Existence results for nonlinear elliptic equations related to Gauss measure in a limit case
Giuseppina di Blasio, Filomena Feo and Maria Rosaria Posteraro
2008, 7(6): 1497-1506 doi: 10.3934/cpaa.2008.7.1497 +[Abstract](285) +[PDF](182.8KB)
Abstract:
The aim of this paper is to prove existence results for nonlinear elliptic equations whose the prototype is -div$(|\nabla u|^{p-2}\nabla u\varphi) =g\varphi $ in a open subset $ \Omega $ of $R^n,$ $u=0$ on $\partial \Omega $, where $p\geq 2$, the function $\varphi (x)=(2\pi)^{-\frac{n}{2}}$exp$( -|x|^2 /2) $ is the density of Gauss measure and $g\in L^1$ (log $L)^{\frac{1}{2}}( \varphi, \Omega)$. This condition on the function $g$ is sharp in the class of Zygmund spaces.

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