Coupled systems of heat equations and convergence to equilibrium

Abstract

On a bounded domain in $\Omega \subseteq {ℝ}^d$, consider the coupled heat equation

subject to Neumann boundary conditions, where $V:\Omega \to {ℝ}^{N\times N}$ is a matrix-valued potential. While the solution to a single heat equation is well-known to converge to an equilibrium as $t\to \infty$, the matrix potential $V$ can for instance introduce the existence of periodic solutions to the equation.

In this talk, we will discuss sufficient conditions for the solutions to the above equation to converge as $t\to \infty$. We shall see that well-behavedness of the potential $V$ with respect to the ${\ell }^p$-unit ball in ${ℝ}^n$ is a crucial property, here – more precisely speaking, we need that $V$ is $p$-dissipative.

What makes our analysis quite interesting is the fact that we need completely different methods for the cases $p=2$ and $p\ne 2$: in the first case, standard Hilbert space techniques can be used, while the case $p\ne 2$ requires more sophisticated methods from spectral geometry, the geometry of Banach spaces and semigroup theory.

This talk is based on joint work the Alexander Dobrick (Christian-Albrechts-Universität zu Kiel)