Global existence and non-existence theorems for nonlinear wave equations

In this article we focus on the global well-posedness of an initial-boundary value problem for a nonlinear wave equation in all space dimensions. The nonlinearity in the equation features the damping term /u/(k)/u(t)/(m) sgn(u(t)) and a source term of the form /u/(p-1)u, where k, p greater than or equal to 1 and 0 < m < 1. In addition, if the space dimension n greater than or equal to 3, then the parameters k, m and p satisfy p, k/(1 - m) less than or equal to n/(n - 2). We show that whenever k + m greater than or equal to p, then local weak solutions are global. On the other hand, we prove that whenever p > k + m and the initial energy is negative, then local weak solutions blow-up in finite time, regardless of the size of the initial data.