Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations

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

2012-03-20

arxiv.org/abs/1203.4623v1

Mathematics

Analysis of PDEs

Scientific paper

We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the long time behavior of the solution and the limit value of its energy for symmetric decreasing initial data in $L^2$ under minimal assumptions on the nonlinearities. The obtained relation allows to establish sharp threshold results between propagation and extinction for monotone families of initial data in the considered general setting.

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