Second order hyperbolic equations with time-dependent singularity or degeneracy
The author studies singular or degenerate hyperbolic problems \[ u''(t)+\Phi^ 2(t)\Lambda u(t)+\Psi (t)u'(t)+\Xi (t)u(t)=f(t), \] \[ u(t_ 0)=u_ 0,\quad | t|^{\nu}u'(t)=u_ 1\text{ for } t=t_ 0 \] on an interval \(t_ 0<t<s\). It is assumed that \(\Lambda\) is a nonnegative selfadjoint operator on a Hilbert space \({\mathcal H}\), that \(t^{\alpha}\Phi\), \(t^{\alpha -1}\Phi '\), \(t^{\alpha -2}\Phi ''\) are bounded, that \(\Psi\) (t)-\(\nu\) /t is locally integrable, where \(-(2\alpha +1)<\nu <1\), and that \(\Xi\) is bounded in a suitable operator norm. With some rather technical conditions upon \(\Xi\) and f (which may be disregarded if both vanish of sufficient order at \(t=0)\), wellposedness of the problem is proved for \((u,u_ 1)\in \Pi_{t_ 0}^{+\eta}\), \((u,| t|^{\nu}u')\in \Pi^{\nu}_ t\), \(t_ 0<t<s\), where \[ \Pi^{\eta}_ t=D_{+\eta}\times D_{\eta}\quad if\quad t\neq 0;\quad \Pi^{\eta}_ t=D_{\gamma +\eta}\times D_{\sigma +\eta}\quad if\quad t=0. \] Here \(\gamma =(\alpha +2-\nu)/(4(\alpha +1))\), \(\gamma +\sigma =\), and \(\eta\) is any fixed real. The space \(D_{\eta}\) is the domain of \(\Lambda^{\eta}\) with the graph norm if \(\eta\geq 0\), while it is the dual of \(D_{-\eta}\) if \(\eta <0\). The proof relies upon the study of an abstract evolution operator. The result extends that of the author \((=\) \textit{T. Shigeta}) [Commun. Partial Differ. Equations 12, 701- 776 (1987; Zbl 0621.35064)].
scientific article; zbMATH DE number 4169256