Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems. (January 2016)

• Record Type: Journal Article
• Authors: Ley, Olivier; Nguyen, Vinh Duc
• Is Part Of: Nonlinear analysis. Volume 130(2015)
• Page Start: 76
• Page End: 101
• Publication Date: 2016-01
• Subjects: primary 35J70 -- secondary 35K40 35B40 35B45 35B50 35F21 49L25; Nonlinear degenerate parabolic equations -- Nonlinear degenerate elliptic equations -- Hamilton–Jacobi equations -- Monotone systems -- Gradient bounds -- Oscillation -- Strong maximum principle -- Ergodic problem -- Asymptotic behavior -- Viscosity 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.2015.09.012
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We obtain new oscillation and gradient bounds for the viscosity solutions of fully nonlinear degenerate elliptic equations where the Hamiltonian is a sum of a sublinear and a superlinear part in the sense of Barles and Souganidis (2001). We use these bounds to study the asymptotic behavior of weakly coupled systems of fully nonlinear parabolic equations. Our results apply to some "asymmetric systems" where some equations contain a sublinear Hamiltonian whereas the others contain a superlinear one. Moreover, we can deal with some particular case of systems containing some degenerate equations using a generalization of the strong maximum principle for systems.