1203.4623 (C. B. Muratov et al.)

Threshold phenomena for symmetric decreasing solutions of
reaction-diffusion equations    [PDF]

C. B. Muratov, X. Zhong

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.

View original: http://arxiv.org/abs/1203.4623