Cauchy problem for fractional diffusion-wave equations with variable coefficients

Abstract

We consider an evolution equation with the Caputo-Dzhrbashyan fractional derivative of order with respect to the time variable, and the second order uniformly elliptic operator with variable coefficients acting in spatial variables. This equation describes the propagation of stress pulses in a viscoelastic medium. Its properties are intermediate between those of parabolic and hyperbolic equations. In this paper, we construct and investigate a fundamental solution of the Cauchy problem, prove existence and uniqueness theorems for such equations.

Keywords:

• fractional diffusion-wave equation
• Caputo-Dzhrbashyan fractional derivative
• fundamental solution of the Cauchy problem

AMS Subject Classifications:

• Primary: 35R11
• Secondary: 35K70
• 35Q74