Counting Ancient Solutions on A Strip with Exponential Growth

TL;DR

This paper investigates ancient solutions to parabolic equations on an infinite strip, demonstrating that polynomial growth solutions are constant and that solutions with sub-exponential growth form a finite-dimensional space.

Contribution

It establishes the constancy of polynomial growth ancient solutions and characterizes the finite-dimensionality of solutions with sub-exponential growth.

Findings

Polynomial growth ancient solutions are constant.

Solutions with growth slower than exponential form a finite-dimensional space.

Provides new insights into the structure of ancient solutions on strips.

Abstract

We study the ancient solutions of parabolic equations on an infinite strip. We show that any polynomial growth ancient solution for a class of parabolic equations must be constant. Furthermore, we show that the vector space of ancient solutions that grow slower than a fixed exponential order is of finite dimension.