Sharp large time behaviour in $N$-dimensional reaction-diffusion equations of bistable type

TL;DR

This paper analyzes the long-term behavior of bistable reaction-diffusion equations in N dimensions, showing solutions converge to a traveling wave with a specific logarithmic correction and angular dependence.

Contribution

It extends previous results by precisely characterizing the asymptotic shape and location of solutions, including angular variations, in higher dimensions.

Findings

Solutions converge uniformly to a traveling wave profile.

The asymptotic shape includes a logarithmic correction term.

Angular dependence of the solution is described by a Lipschitz function.

Abstract

We study the large time behaviour of the reaction-diffsuion equation $\partial_t u=\Delta u +f(u)$ in spatial dimension $N$, when the nonlinear term is bistable and the initial datum is compactly supported. We prove the existence of a Lipschitz function $s^\infty$ of the unit sphere, such that $u(t,x)$ converges uniformly in $\mathbb{R}^N$, as $t$ goes to infinity, to $U_{c_*}\bigg(|x|-c_*t + \frac{N-1}{c_*} \mathrm{ln}t + s^\infty\Big(\frac{x}{|x|}\Big)\bigg)$, where $U_{c*}$ is the unique 1D travelling profile. This extends earlier results that identified the locations of the level sets of the solutions with $o_{t\to+\infty}(t)$ precision, or identified precisely the level sets locations for almost radial initial data.