在偏微分方程研究领域, 非线性偏微分方程是重要分支, 其中, 对数非线性问题日益引起学者的关注并被广泛应用于许多物理场景, 但目前大部分研究都是在位势阱框架下完成的，研究方法过于单一. 本文主要采用一种新的研究方法——能量法以及解的生命线估计，对含强阻尼和对数非线性源项的波动方程解的有限时间爆破性进行研究.

对于含强阻尼和对数非线性源项的确定性波动方程

u_tt-Δu-Δu_t=u|u|^p-2ln|u|,

本文首先通过Fuedo-Galerkin法、对数不等式以及压缩映射原理, 获得了局部解的存在唯一性; 其次通过能量法以及解的生命线估计, 获得e(0)>0,e(0)=0 , e(0)<0三种状态下解在有限时间T^*的爆破性, 并对爆破时间T^*的上界进行了估计.

对于含强阻尼和对数非线性源项的随机性波动方程

u_tt-Δu-Δu_t=u|u|^p-2ln|u|+εσ(x,t)W_t(x,t),

本文首先通过Galerkin法、B-D-G不等式以及对数不等式，获得了解的存在唯一性; 其次借助Itô公式以及一些微分不等式技巧, 通过能量法以及解的生命线估计, 在期望意义下获得了Ee(0)>0, Ee(0)=0以及Ee(0)<0三种情况下解在有限时间T^*的爆破性, 并给出了爆破时间T^*的上界估计.

In the research field of partial differential equations, logarithmic nonlinearity is an important branch. Among them, logarithmic nonlinear problems are increasingly attracting scholars' attention and are widely used in many physical scenarios. However, most of the research is done under the framework of potential wells, and the research methods are too simple. In this paper, we study the finite time blow up of solutions of wave equation with strong damping and logarithmic nonlinear source terms by using a new research method energy method and lifespan estimation of solutions

For deterministic wave equation with strongly damped and logarithmically nonlinear source terms

u_tt-Δu-Δu_t=u|u|^p-2ln|u|,

Firstly, the existence and uniqueness of local solutions were obtained through the Fuedo Galerkin method, logarithmic inequality, and contraction mapping principle; Secondly, by using the energy method and estimating the lifespan of the solution, we obtained the blow-up solution in finite time under three different states, and estimated the upper bound of the blowup time.

  For the Stochastic wave equation with Strongly Damped and Logarithmically Nonlinear Source Terms

u_tt-Δu-Δu_t=u|u|^p-2ln|u|+εσ(x,t)W_t(x,t),

Firstly, the existence and uniqueness of the solution are obtained through Galerkin's method, B-D-G inequality, and logarithmic inequality; Secondly, using the Itô formula and some differential inequality techniques, the energy method and the lifespan estimation of the solution were used to obtain the blowup of the solution in finite time in the expected sense, as well as in three cases, and an upper bound estimate of the blowup time was given.