A nonlinear diffusion system with convection

This paper deals with a coupled nonlinear diffusion system with convection. A characteristic matrix equation covering all the six nonlinear exponents of the system is introduced to get a simple description for the blow-up criteria of the system. In addition, the influence of the four coefficients of the diffusion and the convection as well as the geometry of Ω on global boundedness or not of the solutions is shown clearly, especially for the situation with the critical sign of the characteristic parameters.