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The Well-Posedness for the Distributed-Order Wave Equation on \(\mathbb {R}^N\)

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Abstract

Distributed-order calculus can summarize the intrinsic multiscale effects of integer and fractional order operators, and construct a more complex physical model. The paper is devoted to study the time distributed-order wave equation. First, we give the definition of distributed-order integral operators \(I ^{(\mu )}\) in \(\alpha \in [1,2]\), and from the definition of the integral operator, we found that the operator has similar properties to the fractional integral operators. Next, according to the properties of the distributed-order integral operator and Laplace transform, we obtain the expression of the solution of the distributed-order wave equation. Then we use the resolvent operator to estimate the solution operators. At last, we further studied the liner or semilinear wave problem with the distributed-order derivative on \(\mathbb {R}^N\) and used the contraction mapping principle to prove the existence and uniqueness of mild solution.

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Funding

This work was supported by National Natural Science Foundation of China (12071396).

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Authors and Affiliations

  1. Faculty of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Yan Ling Zhou & Xuan-Xuan Xi

  2. School of Computer Science and Engineering, Macau University of Science and Technology, Macau, 999078, People’s Republic of China

    Yong Zhou

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  1. Yan Ling Zhou
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  2. Yong Zhou
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  3. Xuan-Xuan Xi
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YLZ and XXX wrote the main manuscript text after discussing main technique with YLZ. All authors reviewed the manuscript.

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Correspondence to Yong Zhou.

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Zhou, Y.L., Zhou, Y. & Xi, XX. The Well-Posedness for the Distributed-Order Wave Equation on \(\mathbb {R}^N\). Qual. Theory Dyn. Syst. 23, 58 (2024). https://doi.org/10.1007/s12346-023-00915-8

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