Local uniform convergence and eventual positivity of solutions to biharmonic heat equations

TL;DR

This paper investigates the long-term behavior and positivity of solutions to biharmonic heat equations on infinite cylinders and Euclidean space, using spectral and Fourier analysis techniques.

Contribution

It establishes local eventual positivity of solutions to biharmonic heat equations and their generalizations, advancing understanding of their asymptotic properties.

Findings

Solutions become locally positive after large times.

The analysis applies Fourier and spectral methods.

Results extend to biharmonic heat equations on Euclidean space.

Abstract

We study the evolution equation associated with the biharmonic operator on infinite cylinders with bounded smooth cross-section subject to Dirichlet boundary conditions. The focus is on the asymptotic behaviour and positivity properties of the solutions for large times. In particular, we derive the local eventual positivity of solutions. We furthermore prove the local eventual positivity of solutions to the biharmonic heat equation and its generalisations on Euclidean space. The main tools in our analysis are the Fourier transform and spectral methods.