We consider a class of semilinear elliptic equations of the form$$-{\epsilon }^2\Delta u(x,y)+a\left(x\right)W^{\text{'}}\left(u(x,y)\right)=0,(x,y)\in {ℝ}^2$$where $\epsilon \>0$, $a:ℝ\to ℝ$ is a periodic, positive function and $W:ℝ\to ℝ$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={(s^2-1)}^2$. We look for solutions to (1) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty$ uniformly with respect to $y\in ℝ$. We show via variational methods that if $\epsilon$ is sufficiently small and $a$ is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

We consider a class of semilinear elliptic equations of the form 15.7cm -${\epsilon }^2\Delta u(x,y)+a\left(x\right)W^{\text{'}}\left(u(x,y)\right)=0,(x,y)\in {ℝ}^2$ where $\epsilon >0$, $a:ℝ\to ℝ$ is a periodic, positive function and $W:ℝ\to ℝ$ is modeled on the classical two well Ginzburg-Landau potential $W\left(s\right)={(s^2-1)}^2$. We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions $u(x,y)\to \pm 1$ as $x\to \pm \infty$ uniformly with respect to $y\in ℝ$. We show via variational methods that if ε is sufficiently small and a is not constant, then ([see full textsee full text]) admits infinitely many of such solutions, distinct...

We consider scalar reaction-diffusion equations on bounded and extended domains, both with the autonomous and time-periodic nonlinear term. We discuss the meaning and implications of the ergodic Poincaré-Bendixson theorem to dynamics. In particular, we show that in the extended autonomous case, the space-time topological entropy is zero. Furthermore, we characterize in the extended nonautonomous case the space-time topological and metric entropies as entropies of a pair of commuting planar homeomorphisms....

On étudie la position des pôles de diffusion du problème de Dirichlet pour l’équation des ondes amorties du type ${\partial }_t^2-\Delta +a\left(x\right){\partial }_t$ dans un domaine extérieur. Sous la condition du « contrôle géométrique extérieur », on déduit alors le comportement des solutions en grand temps. On calcule en particulier le meilleur taux de décroissance de l’énergie locale en dimension impaire d’espace.

In this paper we describe some existence and uniqueness theorems for radial ground states of a class of quasilinear elliptic equations. In particular, the mean curvature operator and the degenerate Laplace operator are considered.

In this paper we focus on melting and solidification processes described by phase-field models and obtain rigorous estimates for such processes. These estimates are derived in Section 2 and guarantee the convergence of solutions to non-constant equilibrium patterns. The most basic results conclude with the inequality (E2.31). The estimates in the remainder of Section 2 illustrate what obtains if the initial data is progressively more regular and may be omitted on first reading. We also present some...

In this paper we focus on melting and solidification processes described by phase-field models and obtain rigorous estimates for such processes. These estimates are derived in Section 2 and guarantee the convergence of solutions to non-constant equilibrium patterns. The most basic results conclude with the inequality (E2.31). The estimates in the remainder of Section 2 illustrate what obtains if the initial data is progressively more regular and may be omitted on first reading. We also present...

This is a survey of known results on estimating the principal Lyapunov exponent of a timedependent linear differential equation possessing some monotonicity properties. Equations considered are mainly strongly cooperative systems of ordinary differential equations and parabolic partial differential equations of second order. The estimates are given either in terms of the principal (dominant) eigenvalue of some derived time-independent equation or in terms of the parameters of the equation itself....