Well-posedness and regularity theory for the fractional diffusion-wave equation in Lebesgue spaces

Abstract: This paper is dedicated to the study of the semilinear fractional diffusion-wave equation. We provide estimates on the families of linear operators related to the problem in the fractional power scale associated with the Laplace operator. Furthermore, we analyze the existence and uniqueness of local mild solutions and their possible continuation to a maximal interval of existence. We also consider the issue of spatial regularity and continuous dependence with respect to initial data.