A logarithmic class of semilinear wave equations

Abstract

We study the global existence, long-time behaviour and blow-up of classical solutions of the equation $◻u=u{\text{ ln}}^q(1+u^2)\text{ in }(3+1)$-space-time with arbitrary big initial data. Thus we have a case of a repellent potential energy term in the relevant energetic identity, contrary to the attractive energy case described by the well-known equation $◻u=-u|u{|}^{p-1}$. The global existence result for $0<q\le 2$ is first established. Then special "counterdecay" (for $0<q<2)$ and blow-up effects (for $q>2$) are found, which show that $q=2$ is a "critical" value. In this way it is answered, in particular, to a question that has arisen already in the pioneering works of $Keller\text{ and }J\ddot{o}rgens$ on the semilinear wave equation.