Convergence to equilibrium for linear parabolic systems coupled by matrix-valued potentials

We consider systems of parabolic linear equations, subject to Neumann boundary conditions on bounded domains in ${\mathbb{R}}^d$, that are coupled by a matrix-valued potential $V$, and investigate under which conditions each solution to such a system converges to an equilibrium as $t\to \mathrm{\infty }$. While this is clearly a fundamental question about systems of parabolic equations, it has been studied, up to now, only under certain positivity assumptions on the potential $V$. Without positivity, Perron-Frobenius theory cannot be applied and the problem is seemingly wide open. In the present article, we address this problem for all potentials that are ${\mathrm{\ell }}^p$-dissipative for some $p\in [1,\mathrm{\infty }]$. While the case $p=2$ can be treated by classical Hilbert space methods, the matter becomes more delicate for $p≠2$. We solve this problem by employing recent spectral theoretic results that are closely tied to the geometric structure of $L^p$-spaces.