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

Abstract

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 $t\to \infty$. 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 ${\ell }^p$-dissipative for some $p\in [1,\infty ]$. While the case $p=2$ can be treated by classical Hilbert space methods, the matter becomes more delicate for $p\ne 2$. We solve this problem by employing recent spectral theoretic results that are closely tied to the geometric structure of $L^p$-spaces.

arXiv comment18 pages. This article originated from Section 2 in version 5 of arXiv:2001.00523, which was removed there in version 6. The contents have been thoroughly rewritten and extended, which resulted in the current article. This is version 2. Compared to version 1, the symmetry assumption on the coefficients in Subsections 3.2 and 3.5 has been removed