Three-dimensional stochastic wave equation with non-Lipschitz coefficients

Abstract

We consider the three-dimensional stochastic wave equation (SWE) driven by a multiplicative Gaussian noise that is white in time and colored in space: \[ \frac{\partial^2 u}{\partial t^2} = \Delta u + b\bigl(u\bigr) + \sigma\bigl(u\bigr)\,\dot{W}, \] where the drift function $ b $ and diffusion coefficient $\sigma$ 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 \[ b(u) = u (\log_+ u)^{\theta_1} \quad \text{and} \quad \sigma(u) = u (\log_+ u)^{\theta_2}, \] with parameters $\theta_1 \in (0,2)$ and $\theta_2 \in \bigl(0, \tfrac{\bar{\nu}+1}{2}\bigr)$, and $\log_+(z)=\log(z\vee e)$, where $\bar{\nu}$ is determined by the assumptions on $ f $.