I am looking for the analytic solution to the initial value problem of a semi-linear wave equation of the form
\begin{align} u_{tt} - \Delta u + au_{t} + f(x,u) &= 0 \qquad \text{on } \Omega\times[0,T] \\ u(\cdot, t=0) &= u_{0} \quad \ \ \ \text{on } \Omega \\ u_{t}(\cdot, t=0) &= u_{1} \quad \ \ \ \text{on } \Omega \\ u(\cdot, t) &= 0 \qquad \text{on } \partial \Omega \times [0,T] \end{align}
where $\Omega\subset\mathbb{R}^d, T>0, a>0, u_{0}, u_{1} \in L^{2}(\Omega)$ and $f\in C^{3}$ with
$$\bigg\lvert D^i \left(\frac{\partial}{\partial u} \right)^jf(x,u) \bigg\rvert \leq k \lvert u \rvert^{(r+1-j)} \qquad \text{for } 0 \leq i+j\leq 3$$
where $r \geq 1$ and $k > 0$. Here, $\Omega, u_{0}, u_{1}, a$ and $f$ can be chosen pretty much freely, though I think it makes the most sense to look for solutions on a low-dimensional $\Omega$ with a very simple geometry. Ideally, I'm looking for a solution for something like $f(x,u) = u^{2}$.
It's easy to find solutions for the case $f(x,u) \equiv 0$ and $\Omega = [-c,c]^2$ but I was wondering if someone knows how to get an analytic solution when $f$ is non-zero.
For the equation with $f(x,u) = u^2$ in one space dimension, the differential equation has a family of steady-state solutions $$ u(x,t) = 6 \mathcal{P}\! \left(x - x_0 ;0, c \right)$$ where $\mathcal{P}$ is the Weierstrass P function. You may be able to choose $x_0$ and $c$ to match the boundary conditions $u = 0$ on $\partial \Omega$.