Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian

We obtain the one-dimensional symmetry and monotonicity of the entire positive solutions to some reaction-diffusion equations involving fractional p-Laplacian by virtue of the sliding method. More precisely, we consider the following problem {partial derivative u/partial derivative t (x, t) + (-Delta)(p)(s)u(x ,t) = f(T, u(x, t)), (x, t) is an element of Omega x R, u(x, t) > 0, (x, t) is an element of Omega x R, u(x ,t) = 0, (x, t) is an element of Omega(c) x R, where s is an element of(0,1), p >= 2, (-Delta)(p)(s) is the fractional p-Laplacian, f(t,u)is some continuous function, the domain Omega subset of R-n is unbounded and Omega (c) = R-n\Omega. Firstly, we establish a maximum principle involving the parabolic p-Laplacian operator. Then, under certain conditions off, we prove the asymptotic behavior of solutions faraway from the boundary uniformly in t is an element of R. Finally, the sliding method is implemented to derive the mono-tonicity and uniqueness of the bounded positive entire solutions. To our best knowledge, there has not been any results on the symmetry and monotonicity properties of solutions to the parabolic fractional p-Laplacian equations before.