Well-posedness results for a class of semi-linear super-diffusive equations. (April 2019) - Alvarez, Edgardo; Gal, Ciprian G.; Keyantuo, Valentin; Warma, Mahamadi
Abstract: In this paper we investigate the following fractional order in time Cauchy problem D t α u ( t ) + A u ( t ) = f ( u ( t ) ), 1 < α < 2, u ( 0 ) = u 0, u ′ ( 0 ) = u 1 . 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 ∈ C 1 ( R ) that satisfies suitable growth conditions. Further theorems on the existence of strong solutions are also given in this general context.