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

Abstract

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.

Keywords

• Nonlocal evolution variational inequality
• variable exponent space
• Galerkin approximation
• penalty method