The asymptotic behaviour of the heat equation in a twisted Dirichlet-Neumann waveguide

TL;DR

This paper investigates how changing boundary conditions in a twisted waveguide improves the decay rate of the heat equation, revealing asymptotic boundary behavior and spectral effects using similarity variables and weighted Sobolev spaces.

Contribution

It demonstrates that switching boundary conditions in a twisted waveguide enhances heat decay rates and uncovers asymptotic boundary phenomena affecting spectral properties.

Findings

Switching boundary conditions improves decay rate by t^{-1/2}.

Asymptotic analysis reveals emergent Dirichlet boundary conditions.

Spectral lower bounds increase, enhancing heat dissipation.

Abstract

We consider the heat equation in a straight strip, subject to a combination of Dirichlet and Neumann boundary conditions. We show that a switch of the respective boundary conditions leads to an improvement of the decay rate of the heat semigroup of the order of $t^{-1/2}$. The proof employs similarity variables that lead to a non-autonomous parabolic equation in a thin strip contracting to the real line, that can be analyzed on weighted Sobolev spaces in which the operators under consideration have discrete spectra. A careful analysis of its asymptotic behaviour shows that an added Dirichlet boundary condition emerges asymptotically at the switching point, breaking the real line in two half-lines, which leads asymptotically to the 1/2 gain on the spectral lower bound, and the $t^{-1/2}$ gain on the decay rate in the original physical variables. This result is an adaptation to the case of strips with twisted boundary conditions of previous results by the authors on geometrically twisted Dirichlet tubes.