Local properties of solutions to the Sobolev equation (Q2719467)

The authors study the local behavior of solutions to the equation \(D_t^2\Delta u + D_{x_3}^2u = 0\) as \(t\to\infty\). It is proved that for some estimates the solution of the equation at every internal point of the domain is either oscillating, i.e. it changes its sign an infinite number of times in any neighborhood of \(t=\infty\), or is a function summable in \(t\) on the interval~\((0,\infty)\).