Propagating Terraces and the Dynamics of Front-like Solutions of Reaction-diffusion Equations on Mathbb R (Memoirs of the American Mathematical Society) - Softcover

Polacik, Peter

Softcover

ISBN 10: 1470441128 ISBN 13: 9781470441128

Publisher: Amer Mathematical Society, 2020

Synopsis

The author considers semilinear parabolic equations of the form $ut=uxx f(u),quad xin mathbb R,t0,$ where $f$ a $C1$ 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 $xapprox -infty $ and near $0$ for $xapprox 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 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 $(u(x,t),ux(x,t)):xin mathbb R$, $t0$, of the solutions in question.

Edition details

• Publisher: Amer Mathematical Society
• Publication date: 2020
• Language: English
• ISBN 10: 1470441128
• ISBN 13: 9781470441128
• Binding: Paperback
• Number of pages: 87