  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.
