Superdiffusive fractional dynamics: Unveiling regularity results in systems with general positive self-adjoint operators

Abstract

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*}% where $\mathbb{D}_{t}^{\alpha }u(\cdot )$ is the Caputo time-fractional derivative of order $\alpha\in (1, 2)$ of the function $u$. Such problems are increasingly used in concrete models in applied sciences, notably phenomena with memory effects. We obtain results on existence and regularity of weak and strong energy solutions assuming that $A$ is any positive self-adjoint operator in a Hilbert space, when the nonlinearity $f\in C^{1}({\mathbb{R}}) $ satisfies suitable growth conditions. Our aim is to obtain regularity results {without} assuming that the operator $A$ has compact resolvent readily extending our recent results from our previous paper \cite{AGKW}. Examples of operators $A$ are considered, mainly differential operators such as Schr\"odinger operators, as well as various nonlocal operators.