Well-posedness of second-order degenerate differential equations with finite delay on $$$ {L}^p\left(\mathbb{R};X\right) $$$

Using operator-valued $$$ {L}^p $$$-Fourier multiplier theorem, weighted Sobolev spaces, and the Carleman transform, we characterize the well-posedness of second-order degenerate differential equations with finite delay $$$ {(Mu)}^{\prime \prime }(t)= Au(t)+F{u}_t+G{u}_t^{\prime }+f(t) $$$ on $$$ {L}^p\left(\mathbb{R};X\right) $$$, where $$$ A:D(A)\subseteq X\to X $$$ and $$$ M:D(M)\subseteq X\to X $$$ are closed linear operators defined on a Banach space $$$ X $$$, the operators $$$ F $$$ and $$$ G $$$ are in $$$ B\left({L}^p\left(\left[-\tau, 0\right];X\right);X\right) $$$ for some fixed $$$ \tau >0 $$$, and $$$ {u}_t(s)=u\left(t+s\right),{u}_t^{\prime }(s)={u}^{\prime}\left(t+s\right) $$$ when $$$ t\in \mathbb{R} $$$ and $$$ s\in \left[-\tau, 0\right] $$$. These results are used to study the well-posedness of the associated second-order neutral degenerate differential equations.