Perturbation theory of evolution inclusions on real Hilbert spaces with quasi-variational structures for inner products

Abstract

We consider an abstract Cauchy problem of an evolution inclusion with a singlevalued perturbation on a real Hilbert space. The evolution inclusion contains subdifferentials of time-dependent, proper, lower semicontinuous, convex functions which depends on a solution itself of the Cauchy problem. Moreover, the subdifferentials are taken with respect to inner products, which also depend on a solution of the Cauchy problem. Such structures are sometimes called quasi-variational structures for convex functions and inner products. The main purposes of this paper are to show the existence of strong solutions to the Cauchy problem of an evolution inclusion with a perturbation and to apply this result to a mass-conservative tumor invasion model with a degenerate cross diffusion.

Keywords

• an evolution inclusion
• a single-valued perturbation
• quasi-variational structures
• inner products
• a tumor invasion model
• degenerate cross diffusion