Doubly-nonlinear evolution equations of rate-independent type with irreversibility and energy balance law

Abstract

In this paper, the global-in-time $L^2$-solvability of the initial–boundary value problem for differential inclusions of doubly-nonlinear type is proved. This problem arises from fracture mechanics, and it is not covered by general existence theories due to the degeneracy and singularity of a dissipation potential along with the nonlinearity of elliptic terms. The existence of solutions is proved based on a minimizing movement scheme, which also plays a crucial role for deriving qualitative properties and asymptotic behaviors of strong solutions. Moreover, the solutions to the initial–boundary value problem comply with three properties intrinsic to brittle fracture: complete irreversibility, unilateral equilibrium of an energy and an energy balance law, which cannot generally be realized in dissipative systems. Furthermore, long-time dynamics of strong solutions are revealed, i.e., each stationary limit of the global-in-time solutions is characterized as a solution to the stationary problem.