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: \[ \frac{\partial^2 u}{\partial t^2} = Δu + b\bigl(u\bigr) + σ\bigl(u\bigr)\,\dot{W}, \] 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 \[ b(u) = u (\log_+ u)^{θ_1} \quad \text{and} \quad σ(u) = u (\log_+ u)^{θ_2}, \] with parameters ${\theta }_1\in (0,2)$ and ${\theta }_2\in (0,\frac{\bar{\nu }+1}{2})$, and ${\log }_{+}(z)=\log (z\vee e)$, where $\bar{\nu }$ is determined by the assumptions on $f$.