Unique existence of evolution equations of hyperbolic type with countably many singular or degenerate points (Q1119787)

The author studies the unique solvability of the abstract singular evolution equation of the hyperbolic type \(u''(t)+\Phi (t)\wedge u(t)=0\) on \((t_ 0,1)\), \(u(t_ 0)=u_ 0\), \(u'(t_ 0)=u_ 1\), where \(\Phi\) is a nonnegative function on (-1,1) with countably many degenerate or singular points t accumulating at 0. As an application to partial differential equations, the \(H^{\infty}(\Omega)\) well-posedness of the initial boundary value problem for the equation \(\partial^ 2u/\partial t^ 2+\Phi (t)P(x,D_ x)u=0\) in \(\Omega \times (t_ 0,1)\), where \(\Omega\) is a domain in \(R^ n\) with smooth boundary, and \(P(x,D_ x)\) is a 2mth order formally self-adjoint uniformly strongly elliptic differential operator with smooth coefficients. An example is given to illustrate the main result.