Quasiconvergence in parabolic equations in one space dimension. (April 2019)

• Record Type: Journal Article
• Authors: Li, Fang; Lou, Bendong; Lu, Junfan
• Is Part Of: Nonlinear analysis. Volume 46(2019)
• Page Start: 298
• Page End: 312
• Publication Date: 2019-04
• Subjects: Quasilinear parabolic equation -- Initial–boundary value problem -- Cauchy problem -- Asymptotic behavior -- Convergence and quasiconvergence; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2018.09.007
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: In this paper we first prove a quasiconvergence result (that is, the ω -limit set of a solution consists entirely of steady states) for bounded solutions of general quasilinear parabolic equations in unbounded or variable domains in one space dimension. Then, we use this result to study a one dimensional heterogeneous reaction–diffusion equation with combustion type of nonlinearity. We prove a trichotomy result on the asymptotic behavior of solutions of the Cauchy problem: any nonnegative solution converges as t → ∞ to either a positive steady state, or the ignition point, or the trivial solution 0.