On the uniqueness of the limit for an asymptotically autonomous semilinear equation on R^N

TL;DR

This paper proves that under specific conditions, solutions to a certain parabolic PDE on R^N that are globally defined and bounded will converge uniquely to a steady state, extending understanding of asymptotic behavior.

Contribution

It establishes the uniqueness of the limit for solutions of a class of asymptotically autonomous semilinear equations on R^N, under certain conditions.

Findings

Globally defined bounded solutions converge to a single steady state.

Conditions on f and h ensure convergence and uniqueness of the limit.

The results extend previous understanding of asymptotic behavior in parabolic equations.

Abstract

We consider a parabolic equation of the form u_t=\Delta u +f(u)+h(x,t) in R^N\times (0,\infty), where f in C^1(R) is such that f(0)=0 and f'(0)<0 and h is a suitable function on R^N\times (0,\infty). We show that under certain conditions, each globally defined and nonnegative bounded solution u converges to a single steady state.