Non-local evolution equations with L\'{e}vy diffusion: Well-posedness and limiting behavior

In this note we focus our attention on a class of nonlocal-in-time evolution equations with Lévy diffusion, they arise as models of unidirectional viscoelastic fluid flow and physical phenomena with memory this http URL first consider the existence of the classical solution to a nonlocal linear evolution problem under conditions on the involved memory kernels which allows complete positivity.

Then we investigate the limit of this model to a generalized Rayleigh-Stokes equation, as the index of Lévy diffusion gets concentrated near two, we prove that the solution of nonlocal-in-time problem with Lévy diffusion uniformly converges to that of the generalized Rayleigh-Stokes equation and reveal the convergence this http URL, the existence and limiting behavior of the mild solution to a nonlocal evolution problem with nonlinearity are established.

The proofs are based on subordination principle and relaxation function theory.