Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R / Peter Poláčik.

Format:

Book

Author/Creator:

Poláčik, P. (Peter), 1959- author.

Series:

Memoirs of the American Mathematical Society ; number 1278.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1278

Language:

English

Subjects (All):

Reaction-diffusion equations.

Differential equations, Partial.

Differential equations, Parabolic.

R (Computer program language).

Physical Description:

1 online resource (100 pages).

Edition:

1st ed.

Place of Publication:

Providence, RI : American Mathematical Society, [2020]

Summary:

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.

Contents:

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.

Notes:

Description based on print version record.

"March 2020, volume 264, number 1278 (first of 6 numbers)."

Includes bibliographical references.

ISBN:

1-4704-5806-3