On Extinction Phenomena for Parabolic Systems

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 3x3 systems appearing as models in chemical engineering.