Blowup in reaction-diffusion systems with dissipation of mass

We prove blowup in finite time of the solutions to some reaction-diffusion systems that preserve nonnegativity and for which the total mass of the components is uniformly bounded. (These are natural properties in applications.) This is done by presenting explicit counterexamples constructed with the help of formal computation software. Several partial results of global existence had been obtained previously in the literature. Our counterexamples explain a posteriori why extra conditions are needed. Negative results are also provided as a by-product for linear parabolic equations in nondivergence form and with discontinuous coefficients and for nonlinear Hamilton-Jacobi evolution equations.