Existence, uniqueness, and longtime behavior for a nonlinear Volterra integrodifferential equation

We consider an initial and boundary value problem for a nonlinear Volterra integrodifferential equation. This equation governs the evolution of a pair of state variables, u and ϑ, which are mutually related by a maximal monotone graph γ in R× R. The model can be viewed, for instance, as a generalized Stefan problem within the theory of heat conduction in materials with memory. Besides, it can be used for describing some diffusion processes in fractured media. The relation defined by γ is properly interpreted and generalized in terms of a subdifferential operator associated with γ and acting from H¹(Ω) to its dual space. Then, the generalized problem is formulated as an abstract Cauchy problem for a perturbation of a nonlinear semigroup, and existence and uniqueness of a solution (u,ϑ) can be proved via a fixed-point argument whatever the maximal monotone graph γ is. Moreover, the meaning of γ as a pointwise relationship is recovered almost everywhere, in the case when γ is bounded on bounded subsets of R. Finally, under some other restrictions on γ, the longtime behavior of the solution is investigated, in a more specific context related to the generalized Stefan problem.