PDE Seminar Abstracts

Coupled systems of heat equations and convergence to equilibrium

Jochen Glück
University of Passau, Germany
Mon 9th Mar 2020, 2-3pm, Carslaw Room 829 (AGR)


On a bounded domain in Ω d, consider the coupled heat equation

d dt u1 u N = Δu1 Δu N +V u1 u N ,

subject to Neumann boundary conditions, where V : Ω N×N is a matrix-valued potential. While the solution to a single heat equation is well-known to converge to an equilibrium as t , 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 . We shall see that well-behavedness of the potential V with respect to the p-unit ball in 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 p2: in the first case, standard Hilbert space techniques can be used, while the case p2 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)