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

Sven Jarohs; Tobias Weth

doi:10.3934/dcds.2014.34.2581

Article Contents

2014, Volume 34, Issue 6: 2581-2615. Doi: 10.3934/dcds.2014.34.2581

This issue Previous Article Liouville type results and regularity of the extremal solutions of biharmonic equation with negative exponents Next Article Classification of radial solutions to Liouville systems with singularities

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

Sven Jarohs^1, and
Tobias Weth^1,

1.
Institut für Mathematik, Goethe-Universität, Frankfurt, Robert-Mayer-Straße 10, 60054 Frankfurt

Received: February 2013

Published: June 2014

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Abstract

We study the nonlinear fractional reaction-diffusion equation $∂_t u + (-\Delta)^s u = f(t,x,u)$, $s\in(0,1)$ in a bounded domain $\Omega$ together with Dirichlet boundary conditions on $\mathbb{R}^N \setminus \Omega$. 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 $s=1$.

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Mathematics Subject Classification: Primary: 35K58; Secondary: 35B50.

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