Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations

Document Type

Article

Publication Date

2-1-2017

Abstract

We study the long time behavior of positive solutions of the Cauchy problem for nonlinear reaction-diffusion equations in ℝN with bistable, ignition or monostable nonlinearities that exhibit threshold behavior. For L2 initial data that are radial and non-increasing as a function of the distance to the origin, we characterize the ignition behavior in terms of the long time behavior of the energy associated with the solution. We then use this characterization to establish existence of a sharp threshold for monotone families of initial data in the considered class under various assumptions on the nonlinearities and spatial dimension. We also prove that for more general initial data that are sufficiently localized the solutions that exhibit ignition behavior propagate in all directions with the asymptotic speed equal to that of the unique one-dimensional variational traveling wave.

Identifier

85006508299 (Scopus)

Publication Title

Discrete and Continuous Dynamical Systems Series A

External Full Text Location

https://doi.org/10.3934/dcds.2017038

e-ISSN

15535231

ISSN

10780947

First Page

915

Last Page

944

Issue

2

Volume

37

Grant

1313687

Fund Ref

National Science Foundation

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