Co-existence of Type II blow-ups with multiple blow-up rates for five-dimensional heat equation with critical nonlinear boundary conditions

We consider the following five-dimensional heat equation with critical boundary condition {equation*} ∂_t u=Δ u \ in \ R_+^5× (0,T) , -∂x_5u =|u|^2/3u \ on \ R^5_+ × (0,T) . {equation*} Given o distinct boundary points q[i] ∈ ∂ R₊⁵, and o integers lᵢ∈ N (possibly duplicated), i=1,2,, o, for $T>0$ sufficiently small, we construct a finite-time blow-up solution u with a type II blow-up rate (T-t)-3lᵢ -3 for x near q[i]. This seems to be the first result of the co-existence of type II blowups with different blow-up rates. To accommodate highly unstable blowups with different blowup rates, we first develop a unified linear theory for the inner problem with more time decay in the blow-up scheme through restriction on the spatial growth of the right-hand side, and then use vanishing adjustment functions for deriving multiple rates at distinct points. This paper is inspired by [25, 52, 60].