A logarithmic class of semilinear wave equations

We study the global existence, long-time behaviour and blow-up of classical solutions of the equation $\square u = u \textrm{ ln}^q(1 + u^2) \textrm{ in } (3 + 1)$-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 $\square u = - u|u|^{p-1}$. The global existence result for $0 < q \leq 2$ 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 $Keller \textrm{ and } J\ddot{o}rgens$ on the semilinear wave equation.