On blow‐up rate for sign‐changing solutions in a convex domain

This paper studies a growth rate of a solution blowing up at time T of the semilinear heat equation u t − Δ u − ∣ u ∣ p −1 u =0 in a convex domain D in ℝ n with zero‐boundary condition. For a subcritical p ∈ (1,( n +2)/( n −2)) a growth rate estimate ∣ u ( x , t )∣⩽ C ( T − t ) −1/( p −1) , x ∈ D , t ∈ (0, T ) is established with C independent of t provided that D is uniformly C 2 . The estimate applies to sign‐changing solutions. The same estimate has been recently established when D =ℝ n by authors. The proof is similar but we need to establish L h – L k estimate for a time‐dependent domain because of the presence of the boundary. Copyright © 2004 John Wiley & Sons, Ltd.