Sharp lifespan estimates of blowup solutions to semi-linear wave equations with time-dependent effective damping
Sharp lifespan estimates of blowup solutions to semi-linear wave equations with time-dependent effective damping
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DOI：
10.1142/s0219891619500176
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发表时间：
2018-08
影响因子：
0.7
通讯作者：
M. Ikeda;M. Sobajima;Yuta Wakasugi
中科院分区：
文献类型：
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作者：
M. Ikeda;M. Sobajima;Yuta Wakasugi
考虑一类含时变有效阻尼项的半线性波动方程的初值问题。有趣的是，考虑到随着时间趋于无穷大，阻尼系数的渐近分布，解的寿命的行为。Ikeda and Wakasugi(2015)和Ikeda and Inui(2019)分别讨论了有效阻尼的简单情况和过阻尼意义下的阈值情况。我们在这里讨论了在所讨论的某种假设下的一般阻尼项。本文的结果是包括以前研究中处理的典型案例的井喷解的精确寿命估计。寿命上界的证明是对Ikeda-Sobajima(2019)的检验函数方法的修正，下界的证明是基于一维情形的Gallay和Rauel(1998)和高维情形的Wakasugi(2017)引入的尺度变量技术。
We consider the initial value problem for a semi-linear wave equation with a time-dependent effective damping term. The interest is the behavior of lifespan of solutions in view of the asymptotic profile of the coefficient of damping as time tends to infinity. A simple case of effective damping and a threshold case in the sense of overdamping were discussed in Ikeda and Wakasugi (2015) and Ikeda and Inui (2019), respectively. We discuss here general damping terms under a certain assumption that is discussed. The result of this paper is the sharp lifespan estimates of blowup solutions including the typical cases treated in previous studies. The proof of the upper bound of lifespan is a modification of the test-function method by Ikeda–Sobajima (2019) and the one of lower bound is based on the technique of scaling variables introduced by Gallay and Raugel (1998) for the one-dimensional case and Wakasugi (2017) for the higher-dimensional case.