Determining nodes for semilinear parabolic equations

TL;DR

This paper establishes that the asymptotic behavior of strong solutions to general semilinear parabolic equations can be uniquely determined throughout a domain from their behavior on a finite set, using energy methods.

Contribution

It introduces a novel approach to determine the entire asymptotic behavior of solutions from finite set data in semilinear parabolic equations.

Findings

Asymptotic behavior on finite set determines global behavior

Energy method effectively proves uniqueness

Applicable to general semilinear parabolic equations

Abstract

We are concerned with the uniqueness of the asymptotic behavior of strong solutions of the initial-boundary value problem for general semilinear parabolic equations by the asymptotic behavior of these strong solutions on a finite set of an entire domain. More precisely, if the asymptotic behavior of a strong solution is known on an appropriate finite set, then the asymptotic behavior of a strong solution itself is entirely determined in a domain. We prove the above property by the energy method.