Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments

DOI10.1016/J.MATPUR.2022.09.001zbMATH Open1507.35040arXiv2101.06698MaRDI QIDQ6358455

King-Yeung Lam, Xiao Yu

Publication date: 17 January 2021

Abstract: We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in heterogeneous shifting habitats, with any number of shifting speeds, by further developing the method based on the theory of viscosity solutions of Hamilton-Jacobi equations. Our framework addresses both reaction-diffusion equation and integro-differential equations with a distributed time-delay. The latter leads to a class of limiting equations of Hamilton-Jacobi-type depending on the variable $x/t$ and in which the time and space derivatives are coupled together. We will first establish uniqueness results for these Hamilton-Jacobi equations using elementary arguments, and then characterize the spreading speed in terms of a reduced equation on a one-dimensional domain in the variable $s=x/t$. In terms of the standard Fisher-KPP equation, our results leads to a new class of "asymptotically homogeneous" environments which share the same spreading speed with the corresponding homogeneous environments.

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) Reaction-diffusion equations (35K57) Partial functional-differential equations (35R10) Traveling wave solutions (35C07) Viscosity solutions to PDEs (35D40)

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