Blowup in reaction-diffusion systems with dissipation of mass

We prove possible blowup in finite time of the solutions to reaction-diffus ion systems which preserve nonnegativity and for which the total mass of th e components is nonincreasing in time (two natural proper ties in applicati ons). 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 counterexampl es explain a posteriori why extra conditions were needed. Negative results: of independent interest are also provided as a by-product for linear parab olic equations in nondivergence form and with discontinuous coefficients an d for nonlinear Hamilton-Jacobi evolution equations.