Global solutions and self-similar solutions of semilinear wave equation (Q1601830)

The authors study the global Cauchy problem for the semilinear wave equation \[ \begin{cases} \partial^2_tu- \Delta u=-\lambda |u|^{\alpha-1} u,\\ u|_{t=0}=f,\;\partial_t u|_{t=0}=g, \end{cases} \] where \(\lambda \in\mathbb{R}\), \(\alpha>1\), \(u=u(t,x)\) is a complex-valued function. They prove existence, uniqueness and regularity results. In particular, they prove the existence of a regular self-similar solution, and set up some energy estimates.