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2024 | OriginalPaper | Buchkapitel

4. Well-Posedness and Regularity of Fractional Wave Equations

verfasst von : Yong Zhou

Erschienen in: Fractional Diffusion and Wave Equations

Verlag: Springer Nature Switzerland

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Abstract

In this chapter, we first study the well-posedness and regularity of mild solutions for a class of time fractional damped wave equations. A concept of mild solutions is introduced to prove the existence for the linear problem, as well as the regularity of the solutions. We also establish a well-posedness result for nonlinear problem. As an application, we discuss a case of time fractional telegraph equations. Section 4.2 studies the semilinear time fractional wave equation on a whole Euclidean space, also known as the superdiffusive equations. Based on the initial data taken in the fractional Sobolev spaces and some known Sobolev embeddings, we prove the local/global well-posedness results of \(L^2\)-solutions for the linear and semilinear problems. In Sect. 4.3, we concern with an exponential nonlinearity for a fractional wave equation in the whole space, and we establish the local existence of solutions in a dense subspace of the Orlicz classification. Moreover, we obtain the global existence of solutions for small initial data in lower dimension \(1\leq d\leq 3\). Our proofs base on the analyticity of the Mittag-Leffler functions, the framework of prior estimates, and the type of exponential nonlinearity. The material in Sect. 4.1 is due to Zhou and He (Monatsh Math 194(2):425–458, 2021) . The results in Sect. 4.2 are taken from Zhou, He, Alsaedi, and Ahmad Zhou et al. (Elec Res Arch 30(8):2981–3003, 2022). The results in Sect. 4.3 are adopted from He and Zhou (Bull Sci Math 189:103357, 2023).

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Titel: Well-Posedness and Regularity of Fractional Wave Equations
verfasst von: Yong Zhou
Verlag: Springer Nature Switzerland
Buch: Fractional Diffusion and Wave Equations
Print ISBN: 978-3-031-74030-5

Electronic ISBN: 978-3-031-74031-2

Copyright-Jahr: 2024
DOI: https://doi.org/10.1007/978-3-031-74031-2_4

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