Non-local evolution equations with Lévy diffusion: Well-posedness and limiting behavior

In this note we focus our attention on a class of nonlocal-in-time evolution equations with L\'{e}vy diffusion, they arise as models of unidirectional viscoelastic fluid flow and physical phenomena with memory effect.We 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 L\'{e}vy diffusion gets concentrated near two, we prove that the solution of nonlocal-in-time problem with L\'{e}vy diffusion uniformly converges to that of the generalized Rayleigh-Stokes equation and reveal the convergence rate.Finally, 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.