Double grow-up and global convergence of solutions for a parabolic prescribed mean curvature problem. (September 2015)

• Record Type: Journal Article
• Authors: Pan, Hongjing; Xing, Ruixiang
• Is Part Of: Nonlinear analysis. Volume 125(2015)
• Page Start: 201
• Page End: 226
• Publication Date: 2015-09
• Subjects: 35J93 -- 35K93 -- 53C44 -- 53A10; Infinite time gradient blow-up -- Prescribed mean curvature equation -- Mean curvature flow -- Double grow-up -- Quasilinear parabolic problem -- Global convergence; 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.2015.05.031
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The paper is concerned with a parabolic mean curvature type problem with a varying parameter λ . We study large time behavior of global solutions for different ranges of λ and initial data and present some new asymptotic results about global convergence and infinite time blow-up. In particular, it is shown that for suitable ranges of parameter λ and initial data, there exists a double grow-up phenomenon: the solution itself blows up at every interior point and its gradient blows up at the boundary of the domain as t → + ∞ . We also establish an interesting connection between global convergence and the non-classical solution of the associated stationary problem: if initial data are smaller than the non-classical solution, then the solutions must decay to zero in C 1 norm as t → ∞ .