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Regularity of Solutions to Space–Time Fractional Wave Equations: A PDE Approach

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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.

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Authors and Affiliations

  1. Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile

    Enrique Otárola

  2. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA

    Abner J. Salgado

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  1. Enrique Otárola
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  2. Abner J. Salgado
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Correspondence to Enrique Otárola.

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Otárola, E., Salgado, A.J. Regularity of Solutions to Space–Time Fractional Wave Equations: A PDE Approach. FCAA 21, 1262–1293 (2018). https://doi.org/10.1515/fca-2018-0067

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