Propagating Terraces and the Dynamics of Front-like Solutions of Reaction-diffusion Equations on Mathbb R - Couverture souple

Polacik, Peter

Couverture souple

ISBN 10 : 1470441128 ISBN 13 : 9781470441128

Editeur : American Mathematical Society, 2020

Synopsis

The author considers semilinear parabolic equations of the form $u_t=u_xx f(u),\quad x\in \mathbb R,t>0,$ where $f$ a $C^1$ function. Assuming that $0$ and $\gamma >0$ are constant steady states, the author investigates the large-time behavior of the front-like solutions, that is, solutions $u$ whose initial values $u(x,0)$ are near $\gamma $ for $x\approx -\infty $ and near $0$ for $x\approx \infty $. If the steady states $0$ and $\gamma $ are both stable, the main theorem shows that at large times, the graph of $u(\cdot ,t)$ is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). The author proves this result without requiring monotonicity of $u(\cdot ,0)$ or the nondegeneracy of zeros of $f$.

The case when one or both of the steady states $0$, $\gamma $ is unstable is considered as well. As a corollary to the author's theorems, he shows that all front-like solutions are quasiconvergent: their $\omega $-limit sets with respect to the locally uniform convergence consist of steady states. In the author's proofs he employs phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories $\{(u(x,t),u_x(x,t)):x\in \mathbb R\}$, $t>0$, of the solutions in question.

A propos du livre

À propos de l?auteur

Peter Polacik, University of Minnesota, Minneapolis, MN.

Détails de l'édition

• Éditeur: American Mathematical Society
• Date d'édition: 2020
• Langue: anglais
• ISBN 10: 1470441128
• ISBN 13: 9781470441128
• Reliure: Broché
• Nombre de pages: 87
• Coordonnées du fabricant: non disponible
• Personne responsable: non disponible