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arXiv · 2607.13163

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

Xi Huang·Li Peng·Yong Zhou
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Abstract

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.

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Xi Huang, Li Peng, Yong Zhou. 2026-07-14. Non-local evolution equations with L\'{e}vy diffusion: Well-posedness and limiting behavior. https://arxiv.org/abs/2607.13163

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