Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R
著者
書誌事項
Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R
（Memoirs of the American Mathematical Society, no. 1278）
American Mathematical Society, c2020
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注記
"March 2020, volume 264, number 1278 (first of 6 numbers)"
Includes bibliographical reference (p. 86-87)
内容説明・目次
内容説明
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.
目次
Introduction
Main results
Phase plane analysis
Proofs of Propositions 2.8, 2.12
Preliminaries on the limit sets and zero number
Proofs of the main theorems
Bibliography.
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