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

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.