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
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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 -