ASYMPTOTIC SYMMETRY FOR A CLASS OF NONLINEAR FRACTIONAL REACTION-DIFFUSION EQUATIONS

We study the nonlinear fractional reaction-diffusion equation partial derivative(t)u + (-Delta)(s)u = f(t, x, u), s is an element of (0, 1) in a bounded domain Omega together with Dirichlet boundary conditions on R-N \ 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 vertical bar x(1)vertical bar. 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 vertical bar x(1)vertical bar. 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.