Supersolutions for a class of nonlinear parabolic systems

TL;DR

This paper develops supersolutions for a class of nonlinear parabolic systems using scalar equations, providing optimal conditions for solution existence and blow-up rate estimates.

Contribution

It introduces a method to construct supersolutions for nonlinear parabolic systems, leading to precise existence criteria and blow-up rate bounds.

Findings

Established optimal existence conditions for solutions.

Derived lower bounds for blow-up rates.

Applied scalar equations to system analysis.

Abstract

In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including $$ \left\{\begin{array}{ll} \partial_t u=\Delta u+v^p,\qquad & x\in\Omega,\,\,\,t>0,\\ \partial_t v=\Delta v+u^q, & x\in\Omega,\,\,\,t>0,\\ u=v=0, & x\in\partial\Omega,\,\,\,t>0,\\ (u(x,0), v(x,0))=(u_0(x),v_0(x)), & x\in\Omega, \end{array} \right. $$ 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.