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

This paper deals with simultaneous and non-simultaneous blow-up for heat equations coupled via nonlinear boundary fluxes . It is proved that, if m < q + 1 and n < p + 1, then blow-up must be simultaneous, and that, for radially symmetric and nondecreasing in time solutions, non-simultaneous blow-up occurs for some initial data if and only if m > q + 1 or n > p + 1. We find three regions: (i) q + 1 < m < p/(p + 1 − n) and n < p+1, (ii) p + 1 < n < q/(q + 1 − m) and m < q+1, (iii) m > 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. Supported by the National Natural Science Foundation of China.