Blow up, global existence and growth rate estimates in nonlinear parabolic systems

We prove Fujita-type global existence and nonexistence theorems for a system of m equations (m > 1) with different diffusion coefficients, i.e. ${\displaystyle u_{it}-d_i\Delta u_i=\prod _{k=1}^m{u}_k^{p_k^i},i=1,...,m,x\in {ℝ}^N,t>0,}$ with nonnegative, bounded, continuous initial values and ${\displaystyle p_k^i\ge 0}$, ${\displaystyle i,k=1,...,m}$, ${\displaystyle d_i>0}$, ${\displaystyle i=1,...,m}$. For solutions which blow up at ${\displaystyle t=T<\le \infty }$, we derive the following bounds on the blow up rate: ${\displaystyle u_i(x,t)\le C{(T-t)}^{-{\alpha }_i}}$ with C > 0 and ${\displaystyle {\alpha }_i}$ defined in terms of ${\displaystyle p_k^i}$.