Propagation phenomena for time heterogeneous KPP reaction-diffusion equations

2011/04/19 by Grégoire Nadin, Luca Rossi, Nadin, Grégoire +1 · 1 citation
Computer Science · Mathematics · Medicine · #35B40 #35K57 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Differential Equations Analysis

paper · doi:10.48550/arxiv.1104.3686

openalex publication_date 2011/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate in this paper propagation phenomena for the heterogeneous reaction-diffusion equation ∂_t u -Δu = f(t,u), x∈ R^N, t∈\R, where f=f(t,u) is a KPP monostable nonlinearity which depends in a general way on t. A typical f which satisfies our hypotheses is f(t,u)=m(t) u(1-u), with m bounded and having positive infimum. We first prove the existence of generalized transition waves (recently defined by Berestycki and Hamel, Shen) for a given class of speeds. As an application of this result, we obtain the existence of random transition waves when f is a random stationary ergodic function with respect to t. Lastly, we prove some spreading properties for the solution of the Cauchy problem.

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