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

TL;DR

This paper establishes well-posedness, including existence and regularity of solutions, for a class of fractional in time super-diffusive equations with nonlinear terms, using operator theory and fractional calculus.

Contribution

It provides new results on the existence and regularity of solutions for fractional super-diffusive equations with nonlinearities, extending previous work to more general operators and conditions.

Findings

Existence of local and global weak solutions.

Regularity results for solutions.

Conditions for strong solutions.

Abstract

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.