Oscillation of solutions of nonlinear wave equations (Q676751)

Let \({\mathcal L}\) be a formally selfadjoint second order linear elliptic operator, and \[ {\mathcal N}u= -\gamma(t) {\mathcal L} \{A(x,t,;u)u\} +b(x,t;u)u. \] Let \(u\) be the solution of the initial boundary value problem \[ \partial_t \bigl(\alpha(t) \partial_tu\bigr) +\beta(t) \partial_tu+ {\mathcal N} u=0 \quad\text{in } \Omega\times \mathbb{R}^+. \] \[ u(x,t)= 0\quad \text{on } \partial \Omega\times \mathbb{R}^+, \qquad u(x,t_0)= u_0(x),\quad \partial_tu(x,t_0) =u_1(x) \quad\text{in } \Omega. \] Suppose that \[ \int^\infty_h {1\over\alpha(t)} dt=\infty, \quad \lim_{t\rightarrow \infty} \left\{ {\beta (t) \over 2\alpha(t)} +\int^t_h \left[\gamma(s)- {\beta(s)^2 \over 4\alpha(s)} \right] \right\}ds= \infty \] for any \(h>0\). A typical example is the Kirchhoff equation \(\partial^2_tu -\text{const.} (1+|\nabla u|^2) \Delta u=0\). Under some assumptions on the signs of the minimum eigenvalue of \({\mathcal L}\), \(A(x,t;u)\) and \(b(x,t;u)\), the author shows that, unless the solution \(u\) vanishes identically, there exist two points \((x_1,t_1)\) and \((x_2,t_2)\) such that \(u(x_1,t_1) u(x_2,t_2) <0\). It is also shown that an analogous result holds when \({\mathcal N}\) is a higher order operator.