On extinction phenomena for parabolic system

We investigate the extinction phenomena for some linear combinations of components of the vector--valued solutions to classes of semilinear parabolic systems. The crucial assumption on simultaneous splitting of the matrix-valued elliptic operators and the nonlinear source term allow us to uncouple the systems into a linear part and a scalar nonlinear equation depending on the solutions of the linear part. We propose necessary conditions and sufficient conditions on the existence of the extinction time for the solutions . We recapture as particular case previous results and apply our abstract theorem to a class of $3\times 3$ systems appearing as models in Chemical Engineering