Some oscillatory properties of the wave equation in several space dimensions (Q1109975)

We establish various oscillation properties of the non-zero solutions to wave equations with Dirichlet boundary condition of the form \(u_{tt}- \Delta u+g(t,x,u)=0,\quad (t,x)\in {\mathbb{R}}^+\times \Omega;\quad u=0,\quad (t,x)\in {\mathbb{R}}^+\times \partial \Omega.\)Here, \(\Omega\) is a bounded open subset of \({\mathbb{R}}^ n\), \(n\geq 2\) and g: \({\mathbb{R}}^+\times \Omega \times {\mathbb{R}}\to {\mathbb{R}}\) verifies ug(t,x,u)\(\geq 0\). Specifically, - the solutions must change sign in any truncated cylinder (0,T)\(\times \Omega\) for T sufficiently large; - if \(\Omega\) is a ball, then radial solutions cannot remain positive near the boundary of \(\Omega\) during a time interval larger than the diameter of \(\Omega\) ; - a similar result holds in an arbitrary domain \(\Omega\) and for any solution of the linear equation.