Blow-up phenomena of semilinear wave equations and their weakly coupled systems
Blow-up phenomena of semilinear wave equations and their weakly coupled systems
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DOI：
10.1016/j.jde.2019.05.029
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发表时间：
2018-07
影响因子：
2.4
通讯作者：
M. Ikeda;M. Sobajima;Kyouhei Wakasa
中科院分区：
文献类型：
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作者：
M. Ikeda;M. Sobajima;Kyouhei Wakasa
本文考虑含有未知函数的时间导数的具有幂类型非线性的波动方程及其弱耦合系统。我们提出了一个检验函数方法的框架，并简单地证明了非线性波动方程及其方程组的解的寿命的精确上界的推导。我们指出，对于各自的临界情况，我们使用一族包含Gauss超几何函数的线性波动方程的自相似解，这是由周[59]首次引入的。我们强调，即使在高维情形N≥4时，我们的框架也不要求初始数据的逐点正性。此外，我们发现了系统∂t2u−Δu=|v|q，∂t2v−Δv=|∂tu|p的一条新的(p，q)曲线，并在一个新的区域内估计了小解的寿命。
In this paper we consider the wave equations with power type nonlinearities including time-derivatives of unknown functions and their weakly coupled systems. We propose a framework of test function methods and give a simple proof of the derivation of sharp upper bounds for lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical cases, we use a family of self-similar solutions to the linear wave equation including Gauss's hypergeometric functions, which are originally introduced by Zhou [59]. We emphasize that our framework does not require the pointwise positivity of the initial data even in the high dimensional case N≥ 4. Moreover, we find a new (p, q)-curve for the system∂ t 2 u− Δ u=| v| q,∂ t 2 v− Δ v=|∂ t u| p with lifespan estimates for small solutions in a new region.