Remark on upper bound for lifespan of solutions to semilinear evolution equations in a two-dimensional exterior domain
Remark on upper bound for lifespan of solutions to semilinear evolution equations in a two-dimensional exterior domain
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DOI：
10.1016/j.jmaa.2018.10.004
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发表时间：
2017-11
影响因子：
1.3
通讯作者：
M. Ikeda;M. Sobajima
中科院分区：
文献类型：
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作者：
M. Ikeda;M. Sobajima
在本文中,我们考虑下面的初边值问题的非线性| | u型p与1 < p≤2在一个二维表面域(0.1){τ∂t 2 u (x, t)−Δu (x, t) + e我ζt u (x, t) =λ∂u (x, t) | | p (x, t)∈Ω×(0,t), u (x, t) = 0, (x, t)∈∂Ω×(0,t), u (x, 0) =εf (x), x∈Ω,∂g t u (x, 0) =ε(x) x∈Ω,Ω由Ω= {x∈R 2;| x|> 1}, ζ∈[−π 2, π 2], λ∈C, τ∈{0,1}转换抛物线性，色散性和双曲性。注意，2= 1+ 2/N被称为藤田指数。如果p> 2，那么对于一些足够小的初始数据（见Ikehata[11]）存在一个小的全局实时解（0.1），如果p< 2，那么对于任何正的初始数据（见Ogawa-Takeda[22]和Lai-Yin[14]）不存在全局实时解。结果表明，对于给定的初始数据(f, τ g)∈H 0 1 （Ω）× l2 （Ω）满足（f+ τ g） log (|) x|∈l1 （Ω）并满足一定的条件，解在有限时间爆炸，并且在p= 2时给出了（0.1）解的寿命上界：寿命(u)≤exp (C ε−1)[exp (C ε−1)]。关键思想是使用近似于满足Dirichlet边界条件的调和函数log (|) x（|）的测试函数和由[9]改进而来的技术。
In this paper we consider the following initial-boundary value problem with the power type nonlinearity| u| p with 1< p≤ 2 in a two-dimensional exterior domain (0.1){τ∂ t 2 u (x, t)− Δ u (x, t)+ e i ζ∂ t u (x, t)= λ| u (x, t)| p,(x, t)∈ Ω×(0, T), u (x, t)= 0,(x, t)∈∂ Ω×(0, T), u (x, 0)= ε f (x), x∈ Ω,∂ t u (x, 0)= ε g (x), x∈ Ω, where Ω is given by Ω={x∈ R 2;| x|> 1}, ζ∈[− π 2, π 2], λ∈ C and τ∈{0, 1} switches the parabolicity, dispersivity and hyperbolicity. Remark that 2= 1+ 2/N is well-known as the Fujita exponent. If p> 2, then there exists a small global-in-time solution of (0.1) for some initial data small enough (see Ikehata [11]), and if p< 2, then global-in-time solutions cannot exist for any positive initial data (see Ogawa–Takeda [22] and Lai–Yin [14]). The result is that for given initial data (f, τ g)∈ H 0 1 (Ω)× L 2 (Ω) satisfying (f+ τ g) log| x|∈ L 1 (Ω) with some requirement, the solution blows up at finite time, and moreover, the upper bound for lifespan of solutions to (0.1) is given as the following double exponential type when p= 2: LifeSpan (u)≤ exp[exp(C ε− 1)]. The crucial idea is to use test functions which approximates the harmonic function log| x| satisfying Dirichlet boundary condition and the technique modified from [9].