Virtual Linearity for KPP Reaction-Diffusion Equations

2022/02/15 by Andrej Zlatoš, Zlatos, Andrej
Computer Science · Mathematics · Medicine · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models

paper · pdf · doi:10.48550/arxiv.2202.07743

openalex publication_date 2022/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that long time solution dynamic for general reaction-advection-diffusion equations with KPP reactions is virtually linear in the following sense. Its leading order depends on the non-linear reaction only through its linearization at u=0, and it can also be recovered for general initial data by instead solving the PDE for restrictions of the initial condition to unit cubes on \bbR^d (the latter means that non-linear interaction of these restricted solutions has only lower order effects on the overall solution dynamic). The result holds under a uniform bound on the advection coefficient, which we show to be sharp. We also extend it to models with non-local diffusion and KPP reactions.

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