Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations

TL;DR

This paper proves that solutions to certain nonlinear fractional reaction-diffusion equations become symmetric over time under specific geometric and functional conditions, using new estimates for antisymmetric solutions.

Contribution

It establishes asymptotic symmetry for solutions of fractional reaction-diffusion equations with minimal technical assumptions, introducing new estimates for antisymmetric supersolutions.

Findings

Solutions become symmetric and decreasing in the spatial variable over time.

New maximum principles are developed for fractional operators.

Asymptotic symmetry holds under weak geometric and monotonicity conditions.

Abstract

We study the nonlinear fractional reaction diffusion equation $\partial_{t}u + (-\Delta)^{s} u= f(t,x,u)$, $s\in(0,1)$ in a bounded domain $\Omega$ together with Dirichlet boundary conditions on $\R^N \setminus \Omega$. We prove asymptotic symmetry of nonnegative globally bounded solutions in the case where the underlying data obeys some symmetry and monotonicity assumptions. More precisely, we assume that $\Omega$ is symmetric with respect to reflection at a hyperplane, say ${x_1=0}$, and convex in the $x_1$-direction, and that the nonlinearity $f$ is even in $x_1$ and nonincreasing in $|x_1|$. Under rather weak additional technical assumptions, we then show that any nonzero element in the $\omega$-limit set of nonnegative globally bounded solution is even in $x_1$ and strictly decreasing in $|x_1|$. This result, which is obtained via a series of new estimates for antisymmetric supersolutions of a corresponding family of linear equations, implies a strong maximum type principle which is not available in the non-fractional case $s=1$.