Continuous reaction-diffusion mass balance on boxes, tori, and piecewise C1 domains

Since $u_t$ is continuous on the compact cylinder, the time difference quotient of $c^{T}u$ is the time average of $c^T{u}_t$ by [F4], and uniform continuity makes it converge uniformly to $c^T{u}_t$. Each stated domain has finite volume, so integrating this uniform convergence proves $\frac{d}{dt}{\int }_{\mathrm{\Omega }}{c}^{T}u={\int }_{\mathrm{\Omega }}{c}^T{u}_t$.

In the box branch, substitute [F1] into step 1.1. The diffusion contribution is ${\sum }_i{c}_i{d}_i{\int }_{\mathrm{\Omega }}\mathrm{\Delta }{u}_{i}dx$. For each coordinate $j$, [F3] permits integration in $x_j$ first, and [F4] gives ${\int }_0^{L_j}{\mathrm{\partial }}_{jj}{u}_{i}d{x}_j={\mathrm{\partial }}_j{u}_i{\mathrm{\mid }}_{x_j=L_j}-{\mathrm{\partial }}_j{u}_i{\mathrm{\mid }}_{x_j=0}$. This difference is zero for matching periodic traces and is zero for homogeneous Neumann data. Summing over components and coordinates makes the total diffusion contribution vanish.

In the specified piecewise $C^1$ branch, each vector field $\mathrm{\nabla }{u}_i$ is $C^1$ on the closure. Apply [F5] using the domain presentation [F6] and the axiom assumption [F7]: ${\int }_{\mathrm{\Omega }}\mathrm{\Delta }{u}_{i}dx={\sum }_j{\int }_{S_j}\mathrm{\nabla }{u}_i\cdot {\nu }_{j}dS$. By [A1], the integrand is zero on every face outside its surface-null edge set; hence every face integral vanishes. Thus the diffusion contribution to step 1.1 is also zero in this branch.

In either branch, after the diffusion term vanishes, the remaining derivative is ${\int }_{\mathrm{\Omega }}{c}^{T}F(u(x,t))dx$. By [F2] its integrand is identically zero in the conservation case and nonpositive in the dissipation case. Integrating proves the claimed mass derivative and monotonicity.