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= u \ in \ R_+⁵ (0, T), -ₗ䃕u =|u|²3u \ on \ R⁵_+ (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.