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

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  • 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 $s=1$.
    Mathematics Subject Classification: Primary: 35K58; Secondary: 35B50.


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