TY - JOUR

T1 - Dynamics of nonnegative solutions of one-dimensional reaction–diffusion equations with localized initial data. Part I

T2 - A general quasiconvergence theorem and its consequences

AU - Matano, H.

AU - Poláčik, P.

N1 - Publisher Copyright:
© 2016, Copyright © Taylor & Francis Group, LLC.

PY - 2016/5/3

Y1 - 2016/5/3

N2 - We consider the Cauchy problem (Formula presented.) where f is a locally Lipschitz function on ℝ with f(0) = 0, and u0 is a nonnegative function in C0(ℝ), the space of continuous functions with limits at ± ∞ equal to 0. Assuming that the solution u is bounded, we study its large-time behavior from several points of view. One of our main results is a general quasiconvergence theorem saying that all limit profiles of u(·, t) in (Formula presented.) are steady states. We also prove convergence results under additional conditions on u0. In the bistable case, we characterize the solutions on the threshold between decay to zero and propagation to a positive steady state and show that the threshold is sharp for each increasing family of initial data in C0(ℝ).

AB - We consider the Cauchy problem (Formula presented.) where f is a locally Lipschitz function on ℝ with f(0) = 0, and u0 is a nonnegative function in C0(ℝ), the space of continuous functions with limits at ± ∞ equal to 0. Assuming that the solution u is bounded, we study its large-time behavior from several points of view. One of our main results is a general quasiconvergence theorem saying that all limit profiles of u(·, t) in (Formula presented.) are steady states. We also prove convergence results under additional conditions on u0. In the bistable case, we characterize the solutions on the threshold between decay to zero and propagation to a positive steady state and show that the threshold is sharp for each increasing family of initial data in C0(ℝ).

KW - Convergence

KW - generalized omega-limit set

KW - localized initial data

KW - parabolic equations on ℝ

KW - quasiconvergence

KW - threshold solutions

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U2 - 10.1080/03605302.2016.1156697

DO - 10.1080/03605302.2016.1156697

M3 - Article

AN - SCOPUS:84973874588

VL - 41

SP - 785

EP - 811

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 5

ER -