Regularity and Rothe method error estimates for parabolic hemivariational inequality

Abstract

This paper deals with regularity of solutions to the abstract operator version of parabolic partial differential inclusion of the form u ′ ( t ) + A u ( t ) + ι ⁎ ∂ J ( ι u ( t ) ) ∋ f ( t ) with the multivalued term given in the form of Clarke subdifferential of a locally Lipschitz functional J. Using the Rothe method, it is shown that under appropriate assumptions on the data, the solution has the increased regularity. Three regularity theorems are given. The first one concerns the regularity of solution in the Besov space B 2 ∞ 1 / 2 ( 0 , T ; H ) . The second theorem provides assumption under which the solution lies in the space H 1 ( 0 , T ; H ) ∩ L ∞ ( 0 , T ; V ) ∩ C ( [ 0 , T ] ; V weak ) . The last one shows that if the operator A is strongly monotone and the multivalued term satisfies the relaxed monotonicity assumption then the unique solution also belongs to H 1 ( 0 , T ; V ) ∩ W 1 , ∞ ( 0 , T ; H ) . In the last case the error estimates on the Rothe method are also proved.