GLOBAL EXISTENCE FOR COUPLED REACTION DIFFUSION EQUATIONS WITH A BALANCE LAW AND NONLINEARITIES WITH NON CONSTANT SIGN.

The paper proves uniform-in-time bounds for the Lyapunov functional by first smoothing the piecewise-constant weights near the sign-change set of c and carefully controlling the extra interface terms that arise upon integrating by parts; see the setup of the functional (13)–(18) and Theorem 1 with its truncation argument (19a)–(27) and the handling of Jn ≤ 0 and the Xn/Grönwall step, which together yield the claimed bounds . By contrast, the candidate solution integrates by parts on Ω− and Ω+ and discards the internal-boundary terms on the zero set of c on the grounds that it has measure zero; this is not valid (notably in 1D the internal boundary contribution is a point evaluation, not an integral that vanishes by measure-zero arguments). The paper’s smoothing step is precisely what is needed to control these terms. The candidate also asserts L′(t) ≤ 0 for the unsmoothed functional, which the paper neither proves nor requires. The PDE, hypotheses, and Lyapunov structure are exactly as in (1)–(4), (13)–(18) of the paper .