Non-simultaneous blow-up of n components for nonlinear parabolic systems

This paper deals with non-simultaneous and simultaneous blow-up for radially symmetric solution ( u 1 , u 2 , … , u n ) to heat equations coupled via nonlinear boundary ∂ u i ∂ η = u i p i u i + 1 q i + 1 ( i = 1 , 2 , … , n ) . It is proved that there exist suitable initial data such that u i ( i ∈ { 1 , 2 , … , n } ) blows up alone if and only if q i + 1 < p i . All of the classifications on the existence of only two components blowing up simultaneously are obtained. We find that different positions (different values of k, i, n) of u i − k and u i leads to quite different blow-up rates. It is interesting that different initial data lead to different blow-up phenomena even with the same requirements on exponent parameters. We also propose that u i − k , u i − k + 1 , … , u i ( i ∈ { 1 , 2 , … , n } , k ∈ { 0 , 1 , 2 , … , n − 1 } ) blow up simultaneously while the other ones remain bounded in different exponent regions. Moreover, the blow-up rates and blow-up sets are obtained.