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

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We study the nonlinear fractional reaction diffusion equation ∂ₜu + (-Δ)ˢ u= f(t,x,u), s∈(0,1) in a bounded domain Ω together with Dirichlet boundary conditions on 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 x₁=0, and convex in the x₁-direction, and that the nonlinearity f is even in x₁ and nonincreasing in |x₁|. 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 x₁ and strictly decreasing in |x₁|. 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.