In this paper of Caicedo and Castro https://www.aimsciences.org/journals/displayArticles.jsp?paperID=4083 they prove that for the seminilinear wave equation $$\square u + \lambda u + h(u) = c\sin(x+t)$$ subject to double periodic conditions $u(x,t)=u(x,t+2\pi)=u(x+2\pi)=0$ there is no continuous solutions for |c| large enough. Here $h$ could be any continuous function with compact support and $-\lambda\notin\sigma(\square)$. It is not very hard to prove the existence of weak solutions. The thing here is that the data $c\sin(x+t)$ is smooth (actually analytic) but there is no regularity at all.

In this paper of Caicedo and Castro https://www.aimsciences.org/journals/displayArticles.jsp?paperID=4083 they prove that for the seminilinear wave equation $$\square u + \lambda u + h(u) = c\sin(x+t)$$ subject to double periodic conditions $u(x,t)=u(x,t+2\pi)=u(x+2\pi)$ there is no continuous solutions for |c| large enough. Here $h$ could be any continuous function with compact support and $-\lambda\notin\sigma(\square)$. It is not very hard to prove the existence of weak solutions. The thing here is that the data $c\sin(x+t)$ is smooth (actually analytic) but there is no regularity at all.