Semilinear vector reaction-diffusion systems

Let $m\ge 1$, let $D=\mathrm{diag}(d_1,\dots ,d_m)$ with $d_i\ge 0$, and let $F:U\to {\mathbb{R}}^m$ be as in Reaction vector fields and quasi-positivity. A semilinear vector reaction-diffusion system is

$$u_t=D\mathrm{\Delta }u+F(u),u(x,0)=u_0(x),$$

on either a flat periodic torus ${\mathbb{T}}_L^d$ or a bounded connected polyhedral domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with homogeneous no-flux conditions $d_i{\mathrm{\partial }}_n{u}_i=0$ on each boundary face. Equivalently, ${\mathrm{\partial }}_n{u}_i=0$ is required for components with $d_i>0$; components with $d_i=0$ require no boundary condition. The diffusion matrix is diagonal and independent of $u$; this definition does not include cross-diffusion.

A classical solution is bounded in the rectangle $R$ on $[0,T]$ if $u(x,t)\in R$ at every point and time in its stated solution class, with $R$ as in Inward reaction rectangles. This item fixes the continuous PDE convention; the finite-volume and periodic discrete operators are defined separately.