The asymptotic behaviour of the heat equation in a sheared unbounded strip

TL;DR

This paper investigates how shearing deformation in an unbounded strip improves the decay rate of the heat semigroup, using Hardy inequalities and self-similar variables to analyze the heat equation.

Contribution

It introduces a novel analysis of the heat equation in sheared geometries, demonstrating enhanced decay rates due to geometric deformation.

Findings

Shearing deformation improves heat semigroup decay rates.

Hardy inequality is key to the analysis.

Self-similar variables facilitate the study of asymptotic behaviour.

Abstract

We show that the geometric deformation of shearing yields an improved decay rate for the heat semigroup associated with the Dirichlet Laplacian in an unbounded strip. The proof is based on the Hardy inequality due to the shearing established in [2] and the method of self-similar variables and weighted Sobolev spaces for the heat equation.