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Communications on Pure & Applied Analysis

2002 , Volume 1 , Issue 4

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Boundary spikes in the Gierer-Meinhardt system
Manuel del Pino, Patricio Felmer and Michal Kowalczyk
2002, 1(4): 437-456 doi: 10.3934/cpaa.2002.1.437 +[Abstract](604) +[PDF](233.2KB)
Abstract:
In this paper we consider the Gierer-Meinhardt system in dimension $N=2,3$. Assuming small diffusion of the activator $\varepsilon$ « 1 and large diffusion of the inhibitor $D$ » $1/\varepsilon^N$ we show that there exists a solution to the Gierer-Meinhardt system such that the activator is concentrated at the critical point of the curvature of the domain. To establish this result we use the topological degree argument.
Some existence and concentration results for nonlinear Schrödinger equations
Teresa D'Aprile
2002, 1(4): 457-474 doi: 10.3934/cpaa.2002.1.457 +[Abstract](523) +[PDF](230.5KB)
Abstract:
In this paper we are concerned with the existence of solutions with non-vanishing angular momentum for a class of nonlinear Schrödinger equations of the form

$ i \h$$ \frac{\partial\psi}{\partial t}=-$ $\frac{ \h^2}{2m}\Delta \psi+V(x)\psi-\gamma|\psi|^{p-2}\psi,$ $\gamma>0,$ $ x\in\mathbb R^{N}$

where $\h$$ >0$, $p>2$, $\psi:\mathbb R^{N}\rightarrow\mathbb C,$ and the potential $V$ satisfies some symmetric properties. In particular the cases $N=2$ with $V$ radially symmetric and $N=3$ with $V$ having a cylindrical symmetry are discussed. Our main purpose is to study the asymptotic behaviour of such solutions in the semiclassical limit (i.e. as $\hbar \rightarrow 0^+$) when a concentration phenomenon around a point of $\mathbb R^N$ appears.

Bifurcations of periodics from homoclinics in singular O.D.E.: applications to discretizations of travelling waves of P.D.E.
Michal Fečkan and Vassilis Rothos
2002, 1(4): 475-483 doi: 10.3934/cpaa.2002.1.475 +[Abstract](510) +[PDF](145.0KB)
Abstract:
Bifurcations of periodic solutions from homoclinic ones are investigated for certain singularly perturbed systems of autonomous ordinary differential equations in $\mathbb R^4$. Results are applied to discretization of travelling waves of certain p.d.e.
A note on the convergence in the limit of a long wave vortex density superconductivity model to the Bean model
V. Styles
2002, 1(4): 485-494 doi: 10.3934/cpaa.2002.1.485 +[Abstract](483) +[PDF](154.8KB)
Abstract:
We show that the unique solution of the Bean model for superconductivity is the $p\to \infty$ limit of the solution to a two-dimensional vortex density model with a p-Laplacian velocity law.
Phase-field systems with vectorial order parameters including diffusional hysteresis effects
Nobuyuki Kenmochi and Jürgen Sprekels
2002, 1(4): 495-511 doi: 10.3934/cpaa.2002.1.495 +[Abstract](503) +[PDF](216.9KB)
Abstract:
This paper is concerned with phase-field systems of Penrose-Fife type which model the dynamics of a phase transition with non-conserved vectorial order parameter. The main novelty of the model is that the evolution of the order parameter vector is governed by a system consisting of one partial differential equation and one partial differential inclusion, which in the simplest case may be viewed as a diffusive approximation of the so-called multi-dimensional stop operator, which is one of the fundamental hysteresis operators. Results concerning existence, uniqueness and continuous dependence on data are presented which can be viewed as generalizations of recent results by the authors to cases where a diffusive hysteresis occurs.
Existence result for a class of nonconservative and nonstrictly hyperbolic systems
Graziano Crasta and Philippe G. LeFloch
2002, 1(4): 513-530 doi: 10.3934/cpaa.2002.1.513 +[Abstract](441) +[PDF](219.8KB)
Abstract:
We consider the class of nonconservative hyperbolic systems

$\partial_t u+A(u) \partial_x u =0,\quad\partial_t v + A(u) \partial_x v =0,$

where $u=u(x,t),\quad v=v(x,t)\in\mathbb R^N$ are the unknowns and $A(u)$ is a strictly hyperbolic matrix. Relying on the notion of weak solution proposed by Dal Maso, LeFloch, and Murat ("Definition and weak stability of nonconservative products", J. Math. Pures Appl. 74 (1995), 483--548), we establish the existence of weak solutions for the corresponding Cauchy problem, in the class of bounded functions with bounded variation. The main steps in our proof are as follows:
(i) We solve the Riemann problem based on a prescribed family of paths.
(ii) We derive a uniform bound on the total variation of corresponding wave-front tracking approximations $u^h$, $v^h$.
(iii) We justify rigorously the passage to the limit in the nonconservative product $A(u^h) \partial_x v^h$, based on the local uniform convergence properties of $u^h$, by extending an argument due to LeFloch and Liu ("Existence theory for nonlinear hyperbolic systems in nonconservative form", Forum Math. 5 (1993), 261--280). Our results provide a generalization to the existence theorem established earlier in the scalar case ($N=1$) by the second author ("An existence and uniqueness result for two nonstrictly hyperbolic systems", IMA Volumes in Math. and its Appl. 27, "Nonlinear evolution equations that change type", ed. B.L. Keyfitz and M. Shearer, Springer Verlag, 1990, pp. 126--138.)

Signed solution for a class of quasilinear elliptic problem with critical growth
Claudianor Oliveira Alves, Paulo Cesar Carrião and Olímpio Hiroshi Miyagaki
2002, 1(4): 531-545 doi: 10.3934/cpaa.2002.1.531 +[Abstract](423) +[PDF](205.3KB)
Abstract:
In this paper we will study the existence of signed solutions for problems of the type

$-L u=\lambda h(x)|x|^{\delta}(u_{+})^q-|x|^{\gamma}(u_{-})^p, \quad $ in $\Omega$,

$u_{\pm}$ ≠0, $\quad u\in E,$ $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ (P)

where $\Omega$ is either a whole space $\mathbb R^N$ or a bounded smooth domain, $Lu =:$ div$(|x|^{\alpha}|\nabla u|^{m-2}\nabla u), $ $\lambda >0, \quad0 < q < m-1 < p \leq m$*$-1,$ $\alpha, $ $\delta $ and $\gamma $ are real numbers, $ N> m-\alpha, $ $m$*$=\frac{(\gamma+N)m}{(\alpha+N-m)}$, $h:\Omega \rightarrow \mathbb R$ is a positive continuous function, $u_{\pm}=\max \{\pm u,0\}$ and $E$ is a Banach space that will be defined later on. We will show that (P) has a solution that changes sign in several situations. The proof of the main results are done by using variational methods applied to the energy functional associated to $(P)$.

Boundary layer on a high-conductivity domain
D. Sanchez
2002, 1(4): 547-564 doi: 10.3934/cpaa.2002.1.547 +[Abstract](421) +[PDF](211.8KB)
Abstract:
In this paper we study the diffraction of an electromagnetic wave on a high conductivity domain. We observe the formation of a boundary layer in the obstacle and we give a complete asymptotic expansion of this boundary layer.
On equality of relaxations for linear elastic strains
Kewei Zhang
2002, 1(4): 565-573 doi: 10.3934/cpaa.2002.1.565 +[Abstract](371) +[PDF](156.4KB)
Abstract:
We use the quadratic rank-one convex envelope $qr_e(f)$ for $f:M_s^{n} \to \mathbb R$ defined on the space of linear elastic strains with $n\geq 2$ to study conditions for equality of semiconvex envelopes. We also use the corresponding quadratic rank-one convex hull $qr_e(K)$ for compact sets $K\subset M_s^{n}$ to give a condition for equality of semiconvex hulls. We show that $L^e_c(K)=C(K)$ if and only if $qr_e(K)=C(K)$, where $L^e_c(K)$ is the closed lamination convex hull on linear strains. We also establish that for functions satisfying $f(A)\geq c|A|^2-C_1$ for $A\in M_s^{n}$, $R_e(f)=C(f)$ if and only if $qr_e(f)=C(f)$.

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