Journal
NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
Volume 20, Issue 4, Pages 1519-1552Publisher
SPRINGER BASEL AG
DOI: 10.1007/s00030-013-0220-7
Keywords
Sharp transition; traveling waves; gradient flow
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Funding
- NSF [DMS-0718027, DMS-0908279, DMS-1119724]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1119724, 0908279] Funding Source: National Science Foundation
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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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