Consider the stochastic wave equation on $\mathbb{R}^d$, $d\in\{1,2,3\}$,

\begin{align}\label{wave-1}&\frac{\partial^2}{{\partial t}^2} u(t,x)- \frac{\partial^2}{{\partial x}^2}u(t,x) = b(u(t,x)) + \sigma(u(t,x)) \dot{W}(t,x),\ t\in(0,T],\notag\\&u(0,x)=u_0(x), \quad \frac{\partial}{\partial t} u(0,x)=v_0(x),\end{align}

For $d=1$, $\dot W$ is a space-time white noise, while for $d=2, 3$,$\dot W$ is a white noise in time and correlated in space. The functions $b$ and $\sigma$ are such that

\begin{equation}\label{coef} \vert\sigma(x)| \le \sigma_1+\sigma_2 |x| \big(\ln_+(|x])\big)^a,\quad |b(x)| \le \theta_1 + \theta_2 |x| \big(\ln_+(|x])\big)^\delta,\end{equation}

where $\theta_i, \sigma_i\in\mathbb{R}_+$, $i=1,2$, $\sigma_2 \ne0$, $\delta, a >0$, with $b$ {\em dominating} over $\sigma$. We prove that for any fixed $T>0$, there exists a random field solution to, this solution is unique and satisfies $\sup_{(t,x)\in[0,T]\times \mathbb{R}^d}|u(t,x)| < \infty$, a.s.

The research is motivated by the recent work [R. Dalang, D. Khoshnevisan, T. Zhang, {\em AoP, 2019}] on a one-dimensional reaction-diffusion equation with super-linear coefficients satisfying. We see that the $L^\infty$ method used by these authors can be successfully applied in the case of wave equations, and it can also be used when the noise has a spatial covariance given by Riesz, Bessel, and fractional type kernels. This is ongoing joint work with A. Millet (U. Paris 1, Panthéon-Sorbonne).