AI 中文总结
本文研究具有非利普希茨系数的三维随机波动方程，在噪声空间协方差的适当假设下，证明了其整体温和解的存在与唯一性，并给出了适用的参数范围。
AI 中文摘要
我们研究由乘性高斯噪声驱动的三维随机波动方程（SWE），该噪声在时间上为白噪声、在空间上为有色噪声，方程形式为：∂²u/∂t² = Δu + b(u) + σ(u)Ẇ，其中漂移函数b和扩散系数σ被假设为局部利普希茨函数，且在无穷远处呈现对数超线性增长。我们在噪声Ẇ(t,x)的空间协方差函数f的适当假设下，证明了在任意固定时间区间[0,T]上整体温和解的存在性与唯一性。我们的结果适用于如下情形：b(u)=u(log₊u)^θ₁，σ(u)=u(log₊u)^θ₂，其中参数θ₁∈(0,2)，θ₂∈(0,(ν̄+1)/2)，且log₊(z)=log(z∨e)，ν̄由f的假设确定。
英文摘要
We consider the three-dimensional stochastic wave equation (SWE) driven by a multiplicative Gaussian noise that is white in time and colored in space: \[ \frac{\partial^2 u}{\partial t^2} = Δu + b\bigl(u\bigr) + σ\bigl(u\bigr)\,\dot{W}, \] where the drift function $ b $ and diffusion coefficient $σ$ are assumed to be locally Lipschitz and exhibit logarithmic superlinear growth at infinity. We establish the existence and uniqueness of a global mild solution on any fixed time interval $[0,T]$ under suitable assumptions on the spatial covariance function $ f $ of the noise $\dot W(t,x)$. Our results apply, for example, to the case \[ b(u) = u (\log_+ u)^{θ_1} \quad \text{and} \quad σ(u) = u (\log_+ u)^{θ_2}, \] with parameters $θ_1 \in (0,2)$ and $θ_2 \in \bigl(0, \tfrac{\barν+1}{2}\bigr)$, and $\log_+(z)=\log(z\vee e)$, where $\barν$ is determined by the assumptions on $ f $.
