On blow‐up rate for sign‐changing solutions in a convex domain
On blow‐up rate for sign‐changing solutions in a convex domain
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DOI：
10.1002/mma.562
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发表时间：
2003-07
影响因子：
2.9
通讯作者：
Y. Giga;Shin’ya Matsui;S. Sasayama
中科院分区：
文献类型：
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作者：
Y. Giga;Shin’ya Matsui;S. Sasayama
研究了半线性热方程ut − Δu − u p−1 u=0在零边界条件下在凸区域D上的解在时刻T爆破的增长率.对于次临界p ∈（1，（n+2）/（n−2）），建立了一个增长率估计C（T-t）−1/（p−1），x ∈ D，t ∈（0，T），其中C独立于t，只要D是一致C2.该估计适用于变号解。最近，当D= n时，作者也建立了同样的估计。证明是类似的，但由于边界的存在，我们需要建立时间依赖域的Lh-Lk估计。版权所有© 2004年约翰威利父子有限公司。
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.