Global Nonexistence for the Cauchy Problem of Some Nonlinear Reaction–Diffusion Systems

We, first, consider the parabolic equation ut = − − α/2um + a x ∇uq + f t x u ‘p + w t x t > 0 x ∈ N where − α/2 is the α/2−fractional power of the Laplacian − which for 0 < α ≤ 2 stands for impurities and f t x and w t x are given nonnegative functions, and find its critical exponent. Then, we consider the criticality for five systems of strongly coupled parabolic equations, two of them with nonlinear convective terms. Our results answer positively some open problems raised recently by Bandle, Deng, Levine, and Zhang.