| Abstract: |
| This talk is concerned with the solvability of time-fractional
gradient flow equations for nonconvex energies in Hilbert spaces. Main
results consist of local and global (in time) existence of
(continuous) strong solutions to time-fractional evolution equations
governed by the difference of two subdifferential operators in Hilbert
spaces. In contrast with classical evolution equations (with standard
time-derivatives), there arise several new difficulties such as lack
of chain-rule identity and low regularity of solutions from the
subdiffusive nature of the problem. To prove the main results,
integral forms of chain-rule formulae for time-fractional derivatives,
a Lipschitz perturbation theory for time-fractional gradient flows for
convex energies and Gronwall-type lemmas for nonlinear Volterra
integral inequalities are developed. These abstract results are also
applied to the Cauchy-Dirichlet problem for some $p$-Laplace
subdiffusion equations with blow-up terms. |
|