Super linear stochastic wave equations in dimension three

$$ \frac{\partial^2 u}{\partial t^2} (t,x) = \frac{\partial^2 u}{\partial x^2}(t,x) + b(u(t,x))+ \sigma(u(t,x))\dot{W}(t,x) \quad t\geq 0, \; x \in \mathbb{R}^3\,, $$ where \(\dot{W}(t,x)\) is a Gaussian noise white in time and colored in space with spatial covariance function \(f\). The drift \(b(u)\) and diffusion coefficient \(\sigma(u)\) are assumed to be locally Lipschitz satisfying the growth condition $$ b(u) = O (|u|(\log |u|)^{\theta_1}) \quad \text {and} \quad \sigma(u) = O (|u|(\log |u|)^{\theta_2})\,, \quad \text{as}\quad |u| \to \infty $$ for some \(\theta_1, \theta_2 > 0\) which depend on the covariance function \(f\). We show the global existence and uniqueness of the solution, based on a stopping argument. This is based on a joint work with Wenxuan Tao and Mickey Salins.

SASA | Le@Auburn, streamlined/detailed | Le@GitHub | © Le Chen, 2021, Auburn | Last Modified: