An existence result and evolutionary -convergence for perturbed gradient systems

The initial-value problem for the perturbed gradient flow with a perturbation B in a Banach space V is investigated, where the dissipation potential u:V[0,+) and the energy functional Et:V(-,+] are non-smooth and supposed to be convex and nonconvex, respectively. The perturbation B:[0,T]xVV,(t,v)?B(t,v) is assumed to be continuous and satisfies a growth condition. Under suitable assumptions on the dissipation potential and the energy functional, existence of strong solutions is shown by proving convergence of a semi-implicit discretization scheme with a variational approximation technique. Moreover, for perturbed gradient systems (V,E epsilon,epsilon,B epsilon) depending on a small parameter epsilon>0, we develop a theory of evolutionary -convergence in terms of the suitable convergences of E epsilon, epsilon, and B epsilon to the limit system (V,epsilon(0),Psi(0),B-0).