Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on $/mathbb {R}$ (eBook)

Peter Polacik (Autor)

eBook Download: PDF

87 Seiten
American Mathematical Society (Verlag)
978-1-4704-5806-5 (ISBN)

Infos

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.

Produkt-Details

• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• ISBN-10: 1-4704-5806-3 / 1470458063
• ISBN-13: 978-1-4704-5806-5 / 9781470458065