Nonuniqueness for a reversible cubic reaction--diffusion system

We construct two global renormalized solutions with the same nonnegative finite-entropy initial data for the reversible cubic reaction $A+2B\rightleftharpoons3B$ on a three-dimensional ball. The two diffusion coefficients are equal. One species starts from zero, while the other has an integrable $|x|^{-1}$ singularity. The zero-species solution coexists with a second solution obtained from seeds whose initial masses tend to zero. A compact self-similar subsolution keeps the second solution nonzero. We prove its exact initial trace and the square-root Sobolev bounds needed for the full renormalized formulation. Thus entropy dissipation and local Lipschitz continuity of the reaction do not imply uniqueness for all nonnegative finite-entropy initial data.