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

Abstract

In this paper we investigate the following fractional order in time Cauchy problem Dtαu(t)+Au(t)=f(u(t)),1<α<2,u(0)=u0,u′(0)=u1.The fractional in time derivative is taken in the classical Caputo sense. In the scientific literature such equations are sometimes dubbed 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∈C1(R) that satisfies suitable growth conditions. Further theorems on the existence of strong solutions are also given in this general context.