Well‐posedness of second‐order degenerate differential equations with finite delay on Lp(ℝ;X) L^p(R;X) . (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)^′ ′ (t)= Au(t)+Fu_t+Gu_t^′ +f(t) on L p ( ℝ ; X ) L^p(R;X) , where A : D ( A ) ⊆ X → X A:D(A)X→ X and M : D ( M ) ⊆ X → X M:D(M)X→ 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(L^p([-τ, 0];X);X) for some fixed τ > 0 τ >0 , and u t ( s ) = u ( t + s ), u t ′ ( s ) = u ′ ( t + s ) u_t(s)=u(t+s), u_t^′ (s)=u^′(t+s) when t ∈ ℝ tR and s ∈ [ − τ, 0 ] s[-τ, 0] . These results are used to study the well‐posedness of the associated second‐order neutral degenerate differential equations.