Weak solutions for nonlocal evolution variational inequalities involving gradient constraints and variable exponent

In this article, we study a class of nonlocal quasilinear parabolic variational inequality involving p(x)-Laplacian operator and gradient constraint on a bounded domain. Choosing a special penalty functional according to the gradient constraint, we transform the variational inequality to a parabolic equation. By means of Galerkin's approximation method, we obtain the existence of weak solutions for this equation, and then through a priori estimates, we obtain the weak solutions of variational inequality.