On family of potential wells and applications to semilinear hyperbolic equations and parabolic equations. (September 2020)

• Record Type: Journal Article
• Authors: Chen, Tianlong; Chen, Yuxuan
• Is Part Of: Nonlinear analysis. Volume 198(2020)
• Publication Date: 2020-09
• Subjects: Semilinear hyperbolic equations -- Semilinear parabolic equation -- Potential wells -- Global existence -- Nonexistence -- Asymptotic behavior; 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.2020.111898
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study the initial boundary value problem for semilinear hyperbolic equations and parabolic equations with convex function nonlinear terms. By introducing a family of potential wells we not only obtain the invariant sets and vacuum isolation of solutions, but also give a threshold result of global existence and nonexistence of solutions. Moreover we discuss the global existence of solution for problem with critical initial condition E ( 0 ) = d (or J ( u 0 ) = d ), which fills some important gaps regarding this problem. Finally we prove the global nonexistence of solution and asymptotic behavior of solution for semilinear parabolic equations.