Well-posedness results for a class of semi-linear super-diffusive equations

In this paper we investigate the following fractional order in time Cauchy problem \begin{equation*} \begin{cases} \mathbb{D}_{t}^{\alpha }u(t)+Au(t)=f(u(t)), & 1<\alpha <2, u(0)=u_{0},\,\,\,u^{\prime }(0)=u_{1}. & \end{cases}% \end{equation*}% The fractional in time derivative is taken in the classical Caputo sense. In the scientific literature such equations are sometimes dubbed as fractional-in time wave equations or super-diffusive equations. We obtain results on existence and regularity of local and global weak solutions assuming that $A$ is a nonnegative self-adjoint operator with compact resolvent in a Hilbert space and with a nonlinearity $f\in C^{1}({\mathbb{R}}$ that satisfies suitable growth conditions. Further theorems on the existence of strong solutions are also given in this general context.