  In this paper, by using scalar nonlinear parabolic equations, we construct
supersolutions for a class of nonlinear parabolic systems including $$
\left\{\begin{array}{ll} \partial_t u=\Delta u+v^p,\qquad &
x\in\Omega,\,\,\,t>0,\\ \partial_t v=\Delta v+u^q, & x\in\Omega,\,\,\,t>0,\\
u=v=0, & x\in\partial\Omega,\,\,\,t>0,\\ (u(x,0), v(x,0))=(u_0(x),v_0(x)), &
x\in\Omega, \end{array} \right. $$ where $p\ge 0$, $q\ge 0$, $\Omega$ is a
(possibly unbounded) smooth domain in ${\bf R}^N$ and both $u_0$ and $v_0$ are
nonnegative and locally integrable functions in $\Omega$. The supersolutions
enable us to obtain optimal sufficient conditions for the existence of the
solutions and optimal lower estimates of blow-up rate of the solutions.

Analysis of PDEs