Li-Yau type and Harnack estimates for systems of reaction-diffusion equations via hybrid curvature-dimension condition

Abstract

We prove Li-Yau and Harnack inequalities for systems of linear reaction-diffusion equations. By introducing an additional discrete spatial variable, the system is rewritten as a scalar diffusion equation with an operator sum. For such operators in a mixed continuous and discrete setting, we introduce the hybrid curvature-dimension condition $C{D}_{hyb}(\kappa ,d)$, which is a combination of the Bakry-Émery condition $CD(\kappa ,d)$ and one of its discrete analogues, the condition $C{D}_{{\rm Y}}(\kappa ,d)$. We establish a hybrid tensorisation principle and prove that under $C{D}_{hyb}(0,d)$ with $d<\infty$ a differential Harnack estimate of Li-Yau type holds, from which a Harnack inequality can be deduced by an integration argument.