We consider nonlinear wave equation as follows: $$\partial_t^2 \psi - \Delta \psi = \pm |\psi|^{2\sigma} \psi, \quad ( \psi(0, \cdot),\psi_t(0,\cdot))=(\psi_0,\psi_1)$$ where $\sigma \in \mathbb N, t\in \mathbb R, x\in \mathbb R^d, \psi_0\in H^s(\mathbb R^d), \psi_0\in H^{s-1}(\mathbb R^d)$ (Sobolev spaces)
My Questions: For which $s$ the above equation is locally well-posed in $H^s(\mathbb R^d)$? (I mean known results) Is it well-posed in $L^2$? (Or we have to take large $s$ or so?)
