On blow‐up rate for sign‐changing solutions in a convex domain
Yoshikazu Giga, S. Matsui, Satoshi Sasayama · Mathematical Methods in the Applied Sciences · 2004
Abstract This paper studies a growth rate of a solution blowing up at time T of the semilinear heat equation ut − Δu − ∣u∣p−1 u=0 in a convex domain D in ℝn 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 C2. The estimate applies to sign‐changing solutions. The same estimate has been recently established when D=ℝn by authors. The proof is similar but we need to establish Lh – Lk estimate for a time‐dependent domain because of the presence of the boundary. Copyright © 2004 John Wiley & Sons, Ltd.