The authors consider the abstract degenerate equation of the type \[ u_{tt}(t) +a(t)Au(t) =f\bigl(t,u(t)\bigr), \quad t\in[0,T], \tag{1} \]
\[ u(0)=u_0, \quad u_t(0)= u_1,\tag{2} \]
\[ u(t)\in D(A),\quad t\in[0,T],\tag{3} \] where \(u\) maps \([0,T]\) into a separable Hilbert space \(H\), \(0\leq T\leq\infty\), and \(A\) is a nonnegative selfadjoint operator.
Firstly, the global well-posedness of the abstract linearized system (1)–(3), \(f=h(t)\), is investigated under the assumption
\((H_1)\) there exists a sequence \((t_j)_j \subset [0, \infty)\) decreasing (or increasing) to some \(t_0\in [0,\infty)\) such that \[ a\in C^1\bigl([0, \infty)\subset \cup_j\{t_j\}, \mathbb{R}\bigr), \quad a\geq 0, \] and \[ \sum_{j \in\mathbb{N}} \int^{t_{j+1}}_{t_j} {\bigl|a'(r)\bigr |\over a(r)+ \varepsilon} dr\leq M|\log \varepsilon |,\quad \varepsilon\in(0, \varepsilon_0]\text{ for some }\varepsilon_0 \in(0,1), \] and respectively
\((H_2)\) \(a\geq 0\) and \(a\in C^1([0,\infty),\mathbb{R})\), \(a'\geq 0\) or \(a'\leq 0\).
It is proved that the linearized system has a unique solution which satisfies a certain estimate.
For the nonlinear problem (1)–(3) it is shown a local existence theorem using a fixed point argument. More exactly, the authors prove that there exists a unique solution to (1)–(3) in the hypothesis \((H_2)\) and assuming that \(f:[0,T]\times H\to H\) is smooth and for \(s>0\), there is a nondecreasing function \(\varphi_s:[0,\infty)\to[0,\infty)\) and a function \(\psi_s: [0,\infty) \times[0,\infty) \to[0,\infty)\) nondecreasing in each component, such that for \(B:=C^0 ([0,T],D(A^s))\) we have
(4) \(\forall w\in B\), \(\forall t\in[0,T]: \|f(t,w(t)) \|_{D(A^s)} \leq\varphi_s (\|w(t) \|_{D(A^s)}) \cdot\|w(t)\|_{D (A^s)}\),
(5) \(\forall u,w\in B:\|f(\cdot,u)- f(\cdot,w) \|_B\leq\psi_s (\|u\|_B,\|w\|_B)\cdot \|u-w \|_B\).
These results are applied to the weakly hyperbolic initial-boundary value problem \[ u_{tt}(t,x)-a(t)\sum^n_{i,k=1} \partial_i a_{ik} (x)\partial_k u(t,x)=f(t,x, u(t,x)), \]
\[ u(0,x)= u_0(x),\;u_t(0,x)= u_1(x), \quad x\in \Omega, \]
\[ u(t,\cdot) |_{\partial \Omega}=0\quad \text{resp. }\nu_i(\cdot) a_{ik}(\cdot) \partial_ku(t,\cdot) |_{\partial \Omega} =0,\;t\geq 0. \] The paper finishes with remarks concerning the case when \(A\) has additional, negative eigenvalues.