Well-posedness results for a class of semi-linear super-diffusive equations. (April 2019)

• Record Type: Journal Article
• Authors: Alvarez, Edgardo; Gal, Ciprian G.; Keyantuo, Valentin; Warma, Mahamadi
• Is Part Of: Nonlinear analysis. Volume 181(2019)
• Page Start: 24
• Page End: 61
• Publication Date: 2019-04
• Subjects: 26A33 -- 35R11 -- 35G31 -- 34A12 -- 74G20 -- 74G25; Fractional semi-linear wave equations -- Polynomial growth condition -- Weak and strong solutions -- Existence and uniqueness of local and global solutions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2018.10.016
• Languages: English
• ISSNs: 0362-546X

Abstract

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.