A variational principle for nonpotential perturbations of gradient flows of nonconvex energies

We investigate a variational approach to nonpotential perturbations of gradient flows of nonconvex energies in Hilbert spaces. We prove existence of solutions to elliptic-in-time regularizations of gradient flows by combining the minimization of a parameter-dependent functional over entire trajectories and a fixed-point argument. These regularized solutions converge up to subsequences to solutions of the gradient flow as the regularization parameter goes to zero. Applications of the abstract theory to nonlinear reaction-diffusion systems are presented.