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

We study the nonlinear fractional reaction diffusion equation $\partial_{t}u

(-\Delta)^{s} u= f(t,x,u) $,$ s\in(0,1) $i n a b o u n d e d d o m a i n$ \Omega$ together with Dirichlet boundary conditions on $R^{N} ∖ Ω$ . 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 $Ω$ 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 $ω$ -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 $s = 1$ .

We study the nonlinear fractional reaction diffusion equation $\partial_{t}u

(-\Delta)^{s} u= f(t,x,u) $,$ s\in(0,1) $i n a b o u n d e d d o m a i n$ \Omega$ together with Dirichlet boundary conditions on $R^{N} ∖ Ω$ . 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 $Ω$ 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 $ω$ -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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