Quasiconvergence in parabolic equations in one space dimension. (April 2019) - Li, Fang
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.