Non-local Diffusion Equations Involving the Fractional p(·) -Laplacian

In this paper we study a class of nonlinear quasi-linear diffusion equations involving the fractional p(·) -Laplacian with variable exponents, which is a fractional version of the nonhomogeneous p(·) -Laplace operator. The paper is divided into two parts. In the first part, under suitable conditions on the nonlinearity f, we analyze the problem (P 1 ) in a bounded domain Ω of R N and we establish the well-posedness of solutions by using techniques of monotone operators. We also study the large-time behaviour and extinction of solutions and we prove that the fractional p(·) -Laplacian operator generates a (nonlinear) submarkovian semigroup on L 2 (Ω). In the second part of the paper we establish the existence of global attractors for problem (P 2 ) under certain conditions in the potential V. Our results are new in the literature, both for the case of variable exponents and for the fractional p-laplacian case with constant exponent.