Supersolutions for a class of nonlinear parabolic systems

Abstract

In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including $$ \left\{\begin{array}{ll} \partial_t u=Δu+v^p,\qquad & x\inΩ,\,\,\,t>0,\\ \partial_t v=Δv+u^q, & x\inΩ,\,\,\,t>0,\\ u=v=0, & x\in\partialΩ,\,\,\,t>0,\\ (u(x,0), v(x,0))=(u_0(x),v_0(x)), & x\inΩ, \end{array} \right. $$ where $p\ge 0$, $q\ge 0$, $Ω$ is a (possibly unbounded) smooth domain in ${\bf R}^N$ and both $u_0$ and $v_0$ are nonnegative and locally integrable functions in $Ω$. 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.