Existence and weak–strong uniqueness for damage systems in viscoelasticity

In this paper we investigate the existence of solutions and their weak–strong uniqueness property for a PDE system modelling damage in viscoelastic materials. In fact, we address two solution concepts, weak and strong solutions. For the former, we obtain a global-in-time existence result, but the highly nonlinear character of the system prevents us from proving their uniqueness. For the latter, we prove local-in-time existence. Then, we show that the strong solution, as long as it exists, is unique in the class of weak solutions. This weak–strong uniqueness statement is proved by means of a suitable relative energy inequality.