The blow-up property for a system of heat equations with nonlinear boundary conditions

Abstract

This paper deals with the blow-up properties of solutions to the systems u t=Δu,vt=Δv in B RX(0,T) subject to nonlinear boundary conditions $$\frac{{\partial u}}{{\partial \eta }} = \upsilon ^p ,\frac{{\partial \upsilon }}{{\partial \eta }} = u^q $$ , in S RX(0,T). It is shown that under certain conditions the solution blows up at a finite time and the blow-up only occurs on the boundary. The self-similar solution for the one-dimensional case has been studied. Moreover, the exact blow-up rates are also derived.