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Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on $\mathbb R$
About this Title
Peter Poláčik, School of Mathematics, University of Minnesota, Minneapolis, MN 55455
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 264, Number 1278
ISBNs: 978-1-4704-4112-8 (print); 978-1-4704-5806-5 (online)
DOI: https://doi.org/10.1090/memo/1278
Published electronically: March 18, 2020
Keywords: Parabolic equations on $\mathbb R$,
minimal propagating terraces,
minimal systems of waves,
global attractivity,
limit sets,
quasiconvergence,
convergence,
spatial trajectories,
zero number
MSC: Primary 35K15, 35B40, 35B35, 35B05
Table of Contents
Chapters
- 1. Introduction
- 2. Main results
- 3. Phase plane analysis
- 4. Proofs of Propositions 2.8, 2.12
- 5. Preliminaries on the limit sets and zero number
- 6. Proofs of the main theorems
Abstract
We consider semilinear parabolic equations of the form \begin{equation*} u_t=u_{xx}+f(u),\quad x\in \mathbb {R},t>0, \end{equation*} where $f$ a $C^1$ function. Assuming that $0$ and $\gamma >0$ are constant steady states, we investigate 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, our 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). We prove 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 our theorems, we show that all front-like solutions are quasiconvergent: their $\omega$-limit sets with respect to the locally uniform convergence consist of steady states. In our proofs we employ 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.- Sigurd Angenent, The zero set of a solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79–96. MR 953678, DOI 10.1515/crll.1988.390.79
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