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On blow‐up rate for sign‐changing solutions in a convex domain
Author(s) -
Giga Yoshikazu,
Matsui Shin'ya,
Sasayama Satoshi
Publication year - 2004
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.562
Subject(s) - mathematics , sign (mathematics) , domain (mathematical analysis) , regular polygon , boundary (topology) , zero (linguistics) , convex domain , heat equation , mathematical analysis , combinatorics , mathematical physics , geometry , philosophy , linguistics
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.