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Regularity of solutions to space–time fractional wave equations: A PDE approach

Enrique Otárola EMAIL logo and Abner J. Salgado

Abstract

We consider an evolution equation involving the fractional powers, of order s ∈ (0, 1), of a symmetric and uniformly elliptic second order operator and Caputo fractional time derivative of order γ ∈ (1, 2]. Since it has been shown useful for the design of numerical techniques for related problems, we also consider a quasi–stationary elliptic problem that comes from the realization of the spatial fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi–infinite cylinder. We provide existence and uniqueness results together with energy estimates for both problems. In addition, we derive regularity estimates both in time and space; the time–regularity results show that the usual assumptions made in the numerical analysis literature are problematic.

Acknowledgements

EO is partially supported by CONICYT through FONDECYT Project 3160201. AJS is partially supported by NSF Grant DMS-1720213.

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Received: 2017-10-16
Published Online: 2019-01-13
Published in Print: 2018-10-25

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