Large-time behavior of solutions of parabolic equations on the real line with convergent initial data

TL;DR

This paper proves that bounded solutions of a semilinear parabolic equation on the real line with initial data having distinct limits at infinity tend to steady states as time approaches infinity.

Contribution

It establishes quasiconvergence for solutions with convergent initial data and distinct limits at infinity, extending understanding of long-term behavior of such equations.

Findings

Solutions are quasiconvergent under given conditions

Limit profiles as time goes to infinity are steady states

Bounded solutions with specified initial data converge to steady states

Abstract

We consider the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ We prove that if a solution $u$ of this equation is bounded and its initial value $u(x,0)$ has distinct limits at $x=\pm\infty,$ then the solution is quasiconvergent, that is, all its limit profiles as $t\to\infty$ are steady states.