Thursday, March 22, 2012

1203.4623 (C. B. Muratov et al.)

Threshold phenomena for symmetric decreasing solutions of
reaction-diffusion equations

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.
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