The existence of weak solutions for parabolic variational inequalities with (p(x,t), q(x,t))-growth

Abstract

In this paper, we introduce a kind of variable exponent Sobolev spaces and obtain some new properties of these spaces. Under the framework of these spaces, we study a class of parabolic variational inequalities with ()-growth, and obtain the existence of weak solutions by means of penalty method and Galerkin’s approximation method.

Keywords:

• parabolic variational inequality
• variable exponent space
• penalty method

AMS Subject Classifications:

• 35D30
• 35K86
• 35K99

Acknowledgments

The authors would like to thank the referee for his/her many helpful comments and suggestions.