LAUSR

In this paper we study a class of nonlinear quasi-linear diffusion equations involving the fractional $$p(\cdot )$$p(·)-Laplacian with variable exponents, which is a fractional version of the nonhomogeneous $$p(\cdot )$$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 $$({\mathscr {P}}_{1})$$(P1) in a bounded domain $$\varOmega $$Ω of $${\mathbb {R}}^N$$RN 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 $$p(\cdot )$$p(·)-Laplacian operator generates a (nonlinear) submarkovian semigroup on $$L^{2}(\varOmega ).$$L2(Ω). In the second part of the paper we establish the existence of global attractors for problem $$({\mathscr {P}}_{2})$$(P2) under certain conditions in the potential $${\mathbb {V}}.$$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.