  The initial-value problem for the perturbed gradient flow \[
  B(t,u(t)) \in \partial\Psi_{u(t)}(u'(t))+\partial \mathcal E_t(u(t)) \text{
for a.a. } t\in (0,T),\qquad u(0)=u_0 \] with a perturbation $B$ in a Banach
space $V$ is investigated, where the dissipation potential $\Psi_u:
V\rightarrow [0,+\infty)$ and the energy functional $\mathcal E_t:V\rightarrow
(-\infty,+\infty]$ are nonsmooth and supposed to be convex and nonconvex,
respectively. The perturbation $B:[0,T]\times V \rightarrow V^*, (t,v)\mapsto
B(t,v)$ is assumed to be continuous and satisfies a growth condition. Under
additional 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.
