Darboux-Protter problems for some class of multidimensional singular hyperbolic equations (Q2713902)

Let \(D\) be a~bounded domain of variables \((x_1,\dots,x_m,t)\) in \(\mathbb R^{m+1}\), \(m\geq 2\), whose boundary consists of the characteristic surfaces \(|x|=t\) and \(|x|=1-t\) and a~part of the plane \(t=0\). The author considers in \(D\) the multidimensional singular hyperbolic equation NEWLINE\[NEWLINE \Delta_x U-U_{tt}+ \sum^m_{i=1} a_i(x,t)U_{x_i} +b(x,t)U_t-{\alpha\over t}U_t+c(x,t)U=0. NEWLINE\]NEWLINE For this equation the author studies the weight analogs of Darboux--Protter problems which have been investigated for the wave equation or for hyperbolic equations that degenerate at \(t=0\).NEWLINENEWLINENEWLINEThe author shows that the problems under study have infinitely many solutions. It is worth noting that for \(\alpha=0\) the author's results correspond to the available information on the properties of solutions to Darboux--Protter problems for the multidimensional wave equation.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].