On the behavior of blow-up solutions to a parabolic problem with critical exponent

The blow-up behavior of the solution to a semilinear equation with critical exponent { u t = Δ u + | u | p − 1 u in  Ω × ( 0 , T ) , u = 0 on  ∂ Ω × ( 0 , T ) , u ( ⋅ , 0 ) = u 0 on  Ω is established. We obtain that if Ω is star-shaped about a ∈ Ω and n ⩾ 3 , p = n + 2 n − 2 , then lim t → T ( T − t ) β u ( a + y T − t , t ) exists and equals 0 or ± κ in L loc 2 ( R n ) , where β = 1 p − 1 , κ = β β .