Abstract
In this paper, we study the well-posedness and regularity of mild solutions for a class of time fractional damped wave equations, which the fractional derivatives in time are taken in the sense of Caputo type. A concept of mild solutions is introduced to prove the existence for the linear problem, as well as the regularity of the solution. We also establish a well-posed result for nonlinear problem. By applying finite dimensional approximation method, a compact result of solution operators is presented, following this, an existence criterion shows that the Lipschitz condition or smoothness of nonlinear force functions in some literatures can be removed. As an application, we discuss a case of time fractional telegraph equations.