Large time monotonicity of solutions of reaction-diffusion equations in R^N

TL;DR

This paper proves that solutions to certain reaction-diffusion equations become monotonically increasing over time in large spatial domains, with specific results for one-dimensional cases and applications to medical imaging.

Contribution

It establishes large-time monotonicity of solutions for heterogeneous Fisher-KPP type equations, including cases with compact initial support and one-dimensional homogeneous outside intervals.

Findings

Solutions become time-increasing at large times for initial conditions.

In 1D, solutions are time-increasing if the equation is homogeneous outside a bounded interval.

Results are motivated by applications in medical imaging.

Abstract

In this paper, we consider nonnegative solutions of spatially heterogeneous Fisher-KPP type reaction-diffusion equations in the whole space. Under some assumptions on the initial conditions, including in particular the case of compactly supported initial conditions, we show that, above any arbitrary positive value, the solution is increasing in time at large times. Furthermore, in the one-dimensional case, we prove that, if the equation is homogeneous outside a bounded interval and the reaction is linear around the zero state, then the solution is time-increasing in the whole line at large times. The question of the monotonicity in time is motivated by a medical imagery issue.