Non-simultaneous blow-up for [formula omitted]-componential parabolic systems

Abstract

This paper deals with the non-simultaneous and simultaneous blow-up for some parabolic systems ( u i ) t = Δ u i + u i p i coupled via nonlinear boundary flux ∂ u i ∂ η = u i + 1 q i + 1 ( i = 1 , 2 , … , n ) . For radially symmetric solutions, we obtain that one component can blow up by itself and may provide sufficient help to the blow-up of the other k ( ∈ { 0 , 1 , … , n − 2 } ) ones under suitable initial data. In particular, such phenomena happen for every initial data in some exponent regions. It is interesting that there exist initial data such that any two components blow up simultaneously, either of which blows up depending on itself and also can give sufficient help to the other components blowing up simultaneously. A necessary and sufficient condition is obtained on the simultaneous blow-up of at least two components for all initial data. Moreover, the non-simultaneous and simultaneous blow-up rates and sets are determined.