Approximation of solutions to integro-differential time fractional wave equations in $ L^{p}- $space

Yongqiang Zhao ¹,
Yanbin Tang ^{1,2
,
,}

1.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China
2.
Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China

Received: 07 January 2023 Revised: 09 March 2023 Accepted: 13 March 2023 Published: 24 March 2023

In this paper, we investigate the abstract integro-differential time-fractional wave equation with a small positive parameter $ \varepsilon $. The $ L^{p}-L^{q} $ estimates for the resolvent operator family are obtained using the Laplace transform, the Mittag-Leffler operator family, and the $ C_{0}- $semigroup. These estimates serve as the foundation for some fixed point theorems that demonstrate the local-in-time existence of the solution in weighted function space. We first demonstrate that, for acceptable indices $ p\in[1, +\infty) $ and $ s\in(1, +\infty) $, the mild solution of the approximation problem converges to the solution of the associated limit problem in $ L^{p}((0, T), L^{s}({\bf R}^{n})) $ as $ \varepsilon\rightarrow 0^{+} $. The resolvent operator family and a set of kernel $ k(t) $ assumptions form the foundation of the proof's primary methodology for evaluating norms. Moreover, we consider the asymptotic behavior of solutions as $ \alpha\rightarrow 2^{-} $.
- time fractional derivative,
- integro-differential equation,
- time fractional wave equation,
- heterogeneous parameter,
- homogeneous limit problem,
- analytic semigroup,
- $ C_{0}- $semigroup
Citation: Yongqiang Zhao, Yanbin Tang. Approximation of solutions to integro-differential time fractional wave equations in $ L^{p}- $space[J]. Networks and Heterogeneous Media, 2023, 18(3): 1024-1058. doi: 10.3934/nhm.2023045

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Abstract

In this paper, we investigate the abstract integro-differential time-fractional wave equation with a small positive parameter $ \varepsilon $. The $ L^{p}-L^{q} $ estimates for the resolvent operator family are obtained using the Laplace transform, the Mittag-Leffler operator family, and the $ C_{0}- $semigroup. These estimates serve as the foundation for some fixed point theorems that demonstrate the local-in-time existence of the solution in weighted function space. We first demonstrate that, for acceptable indices $ p\in[1, +\infty) $ and $ s\in(1, +\infty) $, the mild solution of the approximation problem converges to the solution of the associated limit problem in $ L^{p}((0, T), L^{s}({\bf R}^{n})) $ as $ \varepsilon\rightarrow 0^{+} $. The resolvent operator family and a set of kernel $ k(t) $ assumptions form the foundation of the proof's primary methodology for evaluating norms. Moreover, we consider the asymptotic behavior of solutions as $ \alpha\rightarrow 2^{-} $.

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