Solvability Cauchy Problem for Time-Space Fractional Diffusion-Wave Equation with Variable Coefficient

This work investigates the Cauchy problem for the $$n$$ -dimensional time-space fractional diffusion wave equation with variable coefficient. In order to provide solution to the Cauchy problem, a fundamental solution of this equation is constructed and the properties of this solution are studied. The fundamental solutions of the considered equations, which can be expressed in the H-function, are constructed and checked using the asymptotic expansions of the H-function. By the Fourier method, this problem is reduced to equivalent integral equations, which contain Mittag-Leffler type functions in free terms and kernels. An integral equation equivalent to Cauchy’s problem is obtained. For this equation, we construct their fundamental solution, analyze asymptotic behaviors of solutions, and study gradient estimates and large time behaviors. It was verified that the solution of the Cauchy problem satisfies the equation in the sense of a classical solution. The existence and uniqueness of the solution of the integral equation was proved.