Monotonicity for some reaction-diffusion systems with delay and Dirichlet boundary conditions (Q2738668)

The reaction-diffusion system NEWLINE\[NEWLINE\begin{aligned} {\partial u\over\partial t}(x, t) &= D\Delta u(x,t)+ f(x, u_t(x)),\;x\in \Omega,\;t> 0,\\ u(x,t) & =0,\;x\in\partial\Omega,\;t> 0,\\ u_0(x) &= \varphi(x),\;x\in\Omega\end{aligned}NEWLINE\]NEWLINE is considered and the fact that the system operator does not have dense domain is mitigated by the assumption that the Hille-Yosida conditions are satisfied. This gives rise to standard existence and uniqueness results. The main point of the paper is to prove monotonicity of solutions, i.e., if \(V(t)\varphi= u_t(\cdot,\varphi)\) denotes the semiflow, then \(\varphi_1\leq \varphi_2\) implies \(V(t)\varphi_1\leq V(t)\varphi_2\) for all \(t\) for which the solutions exist.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00044].