TY - JOUR

T1 - Propagating terraces and the dynamics of front-like solutions of reaction-diffusion equations on R

AU - Poláčik, Peter

N1 - Publisher Copyright:
© 2020 American Mathematical Society.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - We consider semilinear parabolic equations of the form ut = uxx + f(u), x ∈ R, t > 0, where f a C1 function. Assuming that 0 and γ > 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 γ for x ≈ −∞ and near 0 for x ≈ ∞. If the steady states 0 and γ are both stable, our main theorem shows that at large times, the graph of u(·, t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). We prove this result without requiring monotonicity of u(·, 0) or the nondegeneracy of zeros of f. The case when one or both of the steady states 0, γ is unstable is considered as well. As a corollary to our theorems, we show that all front-like solutions are quasiconvergent: their ω-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), ux(x, t)): x ∈ R}, t > 0, of the solutions in question.

AB - We consider semilinear parabolic equations of the form ut = uxx + f(u), x ∈ R, t > 0, where f a C1 function. Assuming that 0 and γ > 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 γ for x ≈ −∞ and near 0 for x ≈ ∞. If the steady states 0 and γ are both stable, our main theorem shows that at large times, the graph of u(·, t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). We prove this result without requiring monotonicity of u(·, 0) or the nondegeneracy of zeros of f. The case when one or both of the steady states 0, γ is unstable is considered as well. As a corollary to our theorems, we show that all front-like solutions are quasiconvergent: their ω-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), ux(x, t)): x ∈ R}, t > 0, of the solutions in question.

KW - Convergence

KW - Global attractivity

KW - Limit sets

KW - Minimal propagating terraces

KW - Minimal systems of waves

KW - Parabolic equations on R

KW - Quasiconvergence

KW - Spatial trajectories

KW - Zero number

UR - http://www.scopus.com/inward/record.url?scp=85085647742&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85085647742&partnerID=8YFLogxK

U2 - 10.1090/memo/1278

DO - 10.1090/memo/1278

M3 - Article

AN - SCOPUS:85085647742

VL - 264

SP - 1

EP - 100

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

SN - 0065-9266

IS - 1278

ER -