**SMS scnews item created by Daniel Daners at Sat 7 Mar 2020 1137**

Type: Seminar

Distribution: World

Expiry: 9 Mar 2020

**Calendar1: 9 Mar 2020 1400-1500**

**CalLoc1: AGR Carslaw 829**

CalTitle1: PDE Seminar: Coupled systems of heat equations and convergence to equilibrium (Glueck)

Auth: daners@d27-99-2-95.bla1.nsw.optusnet.com.au (ddan2237) in SMS-SAML

### PDE Seminar

# Coupled systems of heat equations and convergence to equilibrium

### Glueck

Jochen Glück

University of Passau, Germany

Mon 9th Mar 2020, 2-3pm, Carslaw Room 829 (AGR)

## Abstract

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

$$\frac{d}{dt}\left(\begin{array}{}="array"\; columnalign="center">{u}_{1}="array">\end{array}="array"\; columnalign="center">\vdots ="array">="array"\; columnalign="center">{u}_{N}\right)$$

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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