I'm interested in solving the following equation, on the torus $\theta\in[0,2\pi]$,
$$\left\{\begin{array}{ll} \ \ \ \ \ \ \ \phi_t=f(\theta)|\phi_{\theta}|,\\ \phi(\theta,0)=\phi_0(\theta). \end{array}\right.$$
Here, every function appearing is a $C^1$ function on the torus, i.e., a $2\pi$-periodic $C^1$ function.
My goal is to give an example of $f,\phi_0$ such that the solution $\phi=\phi(\theta,t)$ exists, but the spatial derivative $\phi_{\theta}$ is NOT uniformly bounded in time.
Any help would be appreciated!