PDE Seminar Abstracts

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

$$\frac{d}{dt}\left(\begin{array}{c}\hfill {u}_{1}\hfill \\ \hfill \vdots \hfill \\ \hfill {u}_{N}\hfill \end{array}\right)=\left(\begin{array}{c}\hfill \Delta {u}_{1}\hfill \\ \hfill \vdots \hfill \\ \hfill \Delta {u}_{N}\hfill \end{array}\right)+V\left(\begin{array}{c}\hfill {u}_{1}\hfill \\ \hfill \vdots \hfill \\ \hfill {u}_{N}\hfill \end{array}\right),$$

subject to Neumann boundary conditions, where $V:\Omega \to {\mathbb{R}}^{N\times N}$ 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 ${\mathbb{R}}^{n}$ 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)

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