Well-Posedness and Regularity of Fractional Wave Equations

In this chapter, we first study the well-posednesswell-posedness and regularityregularity of mild solutionsmild solution(s) for a class of time fractional damped wave equationsdamped wave equation(s). A concept of mild solutionsmild solution(s) is introduced to prove the existenceexistence for the linear problem, as well as the regularityregularity 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 equationfractional wave equation(s) on a whole Euclidean space, also known as the superdiffusive equations. Based on the initial data taken in the fractional Sobolev spacesSobolev space(s) and some known Sobolev embeddings, we prove the local/global well-posednesswell-posedness results of L 2 $$L^2$$ -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 ≤ d ≤ 3 $$1\leq d\leq 3$$ . 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).