The profile and boundary layer for parabolic system with critical simultaneous blow-up exponent

This paper deals with simultaneous blow-up solutions to a Dirichlet initial–boundary problem of the parabolic equations ut=div(a(x)∇u)+∫Ωumvsdx and vt=div(b(x)∇v)+∫Ωuqvpdx in Ω×[0,T)Ω×[0,T). We complete the previous known results by covering the whole range of possible exponents. Then uniform blow-up profile is obtained for all simultaneous blow-up solutions through proving new rules for some auxiliary systems. At last, boundary layer is studied.