Well‐posedness of second‐order degenerate differential equations with finite delay on Lp(ℝ;X)$$ L^p\left(\mathbb{R};X\right) $$. (14th December 2022) - Bu, Shangquan; Zhong, Yuchen
Abstract : Using operator‐valued L p $$ {L}^p $$ ‐Fourier multiplier theorem, weighted Sobolev spaces, and the Carleman transform, we characterize the well‐posedness of second‐order degenerate differential equations with finite delay ( M u ) ′ ′ ( t ) = A u ( t ) + F u t + G u t ′ + f ( t ) $$ {(Mu)}^{\prime \prime }(t)= Au(t)+F{u}_t+G{u}_t^{\prime }+f(t) $$ on L p ( ℝ ; X ) $$ {L}^p\left(\mathbb{R};X\right) $$, where A : D ( A ) ⊆ X → X $$ A:D(A)\subseteq X\to X $$ and M : D ( M ) ⊆ X → X $$ M:D(M)\subseteq X\to X $$ are closed linear operators defined on a Banach space X $$ X $$, the operators F $$ F $$ and G $$ G $$ are in B ( L p ( [ − τ, 0 ] ; X ) ; X ) $$ B\left({L}^p\left(\left[-\tau, 0\right];X\right);X\right) $$ for some fixed τ > 0 $$ \tau >0 $$, and u t ( s ) = u ( t + s ), u t ′ ( s ) = u ′ ( t + s ) $$ {u}_t(s)=u\left(t+s\right), {u}_t^{\prime }(s)={u}^{\prime}\left(t+s\right) $$ when t ∈ ℝ $$ t\in \mathbb{R} $$ and s ∈ [ − τ, 0 ] $$ s\in \left[-\tau, 0\right] $$ . These results are used to study the well‐posedness of the associated second‐order neutral degenerate differential equations.