Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 46, Issue 5, Pages 3241-3276Publisher
SIAM PUBLICATIONS
DOI: 10.1137/130918289
Keywords
reaction-diffusion equation; Fisher-KPP equation; propagation of level sets; nonlinear fractional diffusion
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Funding
- Spanish Project [MTM2011-24696]
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We study the propagation properties of nonnegative and bounded solutions of the class of reaction-diffusion equations with nonlinear fractional diffusion: u(t) + (-Delta)(s) (u(m)) = f(u). For all 0 < s < 1 and m > m(c) = (N - 2s)(+)/N, we consider the solution of the initial-value problem with initial data having fast decay at infinity and prove that its level sets propagate exponentially fast in time, in contrast to the traveling wave behavior of the standard KPP case, which corresponds to putting s = 1, m = 1, and f(u) = u(1 - u). The proof of this fact uses as an essential ingredient the recently established decay properties of the self-similar solutions of the purely diffusive equation, u(t) + (-Delta)(s) u(m) = 0.
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