Necessary conditions for reaction-diffusion system with delay preserving positivity

Abstract

We consider the reaction--diffusion system with delay \begin{equation*} \left\{\begin{aligned} &\frac{\partial u}{\partial t}=A(t,x)\Delta u-\sum_{i=1}^{k}\gamma_{i}(t,x)\partial_{x_{i}}u +f(t,u_{t}) , &x\in \Omega; \\ &B(u)|_{\partial \Omega}=0.\\ \end{aligned} \right. \end{equation*} We show that this system with delay preserves positivity if and only if its diffusion matrix $A$ and convection matrix $\gamma_{i}$ are diagonal with non-negative elements and nonlinear delay term $f$ satisfies the normal tangential condition.

Keywords

• positivity
• monotonicity
• reaction-diffusion equation with delay