Global classical solutions to the Cauchy problem for a nonlinear wave equation (Q1392705)

We prove the existence and uniqueness of a global classical solution to the Cauchy problem \[ {\partial^2u\over\partial t^2}- M\Biggl(\int_\Omega|\nabla u(x, t)|^2 dx\Biggr) \Delta u= 0,\quad u(x,0)= u_0(x),\quad {\partial u\over\partial t} (x,0)= u_1(x), \] where \(\Omega\) is a bounded or unbounded open set \(\mathbb{R}^n\), and \(M(\xi)\) is a locally Lipschitz function on \([0,+\infty[\).