Elliptic-like regularization of a fully nonlinear evolution inclusion and applications

Abstract

Consider in a Hilbert space [Formula: see text] the Cauchy problem [Formula: see text]: [Formula: see text], and associate with it the second-order problem [Formula: see text]: [Formula: see text], where [Formula: see text] is a (possibly set-valued) maximal monotone operator, [Formula: see text] is a Lipschitz operator, and [Formula: see text] is a positive small parameter. Note that [Formula: see text] is an elliptic-like regularization of [Formula: see text] in the sense suggested by Lions in his book on singular perturbations. We prove that the solution [Formula: see text] of [Formula: see text] approximates the solution [Formula: see text] of [Formula: see text]: [Formula: see text]. Applications to the nonlinear heat equation as well as to the nonlinear telegraph system and the nonlinear wave equation are presented.