  We establish a weak-strong uniqueness principle for solutions to
entropy-dissipating reaction-diffusion equations: As long as a strong solution
to the reaction-diffusion equation exists, any weak solution and even any
renormalized solution must coincide with this strong solution. Our assumptions
on the reaction rates are just the entropy condition and local Lipschitz
continuity; in particular, we do not impose any growth restrictions on the
reaction rates. Therefore, our result applies to any single reversible reaction
with mass-action kinetics as well as to systems of reversible reactions with
mass-action kinetics satisfying the detailed balance condition. Renormalized
solutions are known to exist globally in time for reaction-diffusion equations
with entropy-dissipating reaction rates; in contrast, the global-in-time
existence of weak solutions is in general still an open problem - even for
smooth data - , thereby motivating the study of renormalized solutions. The key
ingredient of our result is a careful adjustment of the usual relative entropy
functional, whose evolution cannot be controlled properly for weak solutions or
renormalized solutions.
