An abstract degenerate hyperbolic equation with application to mixed problems (Q1587240)

The Cauchy problem \[ u''+ \sum_{|\alpha|= 2m} a_\alpha(t)\cdot B^\alpha u= f(t),\quad u(0)= u_0,\quad u'(0)= u_1 \] in a Hilbert space \(H\) is considered. Here \(B= (B_1,B_2,\dots, B_n)\) is a \(n\)-tuple of selfadjoint operators on \(H\) with (dense) domain \(D(B_j)\), \(B^\alpha= B^{\alpha_1}_1\circ\cdots\circ B^{\alpha_n}_n\) for any multiindex \(\alpha= (\alpha_1,\dots, \alpha_n)\), \(a_\alpha(t)\) are real analytic on \([0,T]\). The main result is: If the resolvent operators \(R(i, B_j)\), corresponding to \(B_j\), commute and \[ \sum_{|\alpha|= 2m} a_\alpha(t)\cdot \xi^\alpha\geq 0\quad\text{for any }\xi\in\mathbb{R}^n, \] then there exists an integer \(s_0\) such that, for all \(s\geq 2m\), \(u_0,u_1\in H^{s+ s_0}\) and \(f\in C([0, T]; H^{s+ s_0})\), the Cauchy problem has a unique solution in \(C^2([0, T]; H^s)\) with \(H^s= \bigcap_{1\leq j_i\leq n} D(B_{j_1}\circ\cdots\circ B_{j_s})\). Applications to mixed initial-boundary value problems are also presented.