Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary fluxes

Abstract

This paper deals with simultaneous and non-simultaneous blow-up for heat equations coupled via nonlinear boundary fluxes $$\frac{\partial u}{\partial\eta} = u^{m} + v^{p}, \frac{\partial v}{\partial\eta} = u^{q} + v^{n}$$ . It is proved that, if m q + 1 or n > p + 1. We find three regions: (i) q + 1 q+1 and n > p+1, where both simultaneous and non-simultaneous blow-up are possible. Four different simultaneous blow-up rates are obtained under different conditions. It is interesting that different initial data may lead to different simultaneous blow-up rates even for the same values of the exponent parameters.