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 $$\{\begin{array}{ll}{\partial }_{t}u=\Delta u+v^p,& x\in \Omega ,t>0,\\ {\partial }_{t}v=\Delta v+u^q,& x\in \Omega ,t>0,\\ u=v=0,& x\in \partial \Omega ,t>0,\\ (u(x,0),v(x,0))=(u_0(x),v_0(x)),& x\in \Omega ,\end{array}$$ where $p\ge 0$, $q\ge 0$, $\Omega$ is a (possibly unbounded) smooth domain in ${\mathbf{R}}^N$ 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.

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