The Well-Posedness for the Distributed-Order Wave Equation on R

Yan Ling Zhou ^[1] ; Yong Zhou ^[2] ; Xuan-Xuan Xi ^[1]
1. [1] Xiangtan University
  
  Xiangtan University
  
  China
2. [2] Macau University of Science and Technology
  
  Macau University of Science and Technology
  
  RAE de Macao (China)
Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 2, 2024
Idioma: inglés
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Resumen
- 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 (μ) in α ∈ [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 distributedorder 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 RN and used the contraction mapping principle to prove the existence and uniqueness of mild solution.