Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on $\mathbb {R}$

Peter Polacik

ISBN 10: 1470441128 / ISBN 13: 9781470441128

Editore: MP-AMM American Mathematical, 2020

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Codice articolo FW-9781470441128

Dati bibliografici

• Titolo: Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on $\mathbb {R}$
• Editore: MP-AMM American Mathematical
• Data di pubblicazione: 2020
• Lingua: Inglese
• ISBN 10: 1470441128
• ISBN 13: 9781470441128
• Rilegatura: PAP
• Condizione: New
• Peso: 290 grammi

Descrizione articolo

Riassunto

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.