LAUSR

In this paper, we give necessary and sufficient conditions for the $$L^p$$ -well-posedness (resp. $$B_{p,q}^s$$ -well-posedness) for the third order degenerate differential equation with finite delay: $$(Mu)'''(t) + (Nu)''(t)= Au(t) + Bu'(t) + Gu''_t + Fu'_t + Hu_t + f(t)$$ on $$[0,2\pi ])$$ with periodic boundary conditions $$(Mu)(0) = (Mu)(2\pi )$$ , $$(Mu)'(0) = (Mu)'(2\pi )$$ , $$(Mu)''(0) = (Mu)''(2\pi )$$ , where A, B, M and N are closed linear operators on a Banach space X satisfying $$D(A)\cap D(B)\subset D(M)\cap D(N)$$ , the operators G, F and H are bounded linear from $$L^p([-2\pi ,0];X)$$ (resp. $$B_{p,q}^s([-2\pi ,0];X)$$ ) into X.