Wellposedness and regularity of a nonlinear variable-order fractional wave equation

Abstract

We prove the wellposedness of a nonlinear variable-order fractional wave differential equation that describes the vibration of a single particle attached to a viscoelastic material. We then prove that the high-order regularity of its solution depends on the behavior of the variable orders α(t) and β(t) of the fractional damping terms at t=0. In particular, we prove that the solution to the model recovers full regularity when the fractional damping terms exhibit (the physically relevant) elastic restoration force and integer-order damping at the initial time t=0, which, thus, eliminates the incompatibility between the nonlocal differential operators and the locality of classical initial condition.