Classification of connecting solutions of semilinear parabolic equations

TL;DR

This paper investigates the behavior of solutions to semilinear parabolic equations with polynomial nonlinearities, showing that some solutions not blowing up tend to equilibria under a finite energy constraint.

Contribution

It characterizes solutions that tend to equilibria based on a finite energy condition for a broad class of semilinear parabolic equations.

Findings

Solutions with finite energy tend to equilibria.

Not all solutions blow up; some stabilize over time.

The equations can be expressed as an $L^2$ gradient of a functional.

Abstract

For a given semilinear parabolic equation with polynomial nonlinearity, many solutions blow up in finite time. For a certain large class of these equations, we show that some of the solutions which do not blow up actually tend to equilibria. The characterizing property of such solutions is a finite energy constraint, which comes about from the fact that this class of equations can be written as the $L^2$ gradient of a certain functional.