Well-posedness of second order degenerate integro-differential equations in vector-valued function spaces.

We consider the well-posedness of the second order degenerate integrodifferential equations (P2):, (0 ≤ t ≤ 2π) with periodic boundary conditionsu(0) =u(2π), (Mu′)(0) = (Mu′)(2π), in periodic Lebesgue-Bochner spacesLp(𝕋,X), periodic Besov spaces(𝕋,X) and and periodic Triebel-Lizorkin spaces(𝕋,X), whereAandMare closed linear operators on a Banach spaceXsatisfyingD(A) ⊂D(M),a∊L1(ℝ+) andαis a scalar number. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for the well-posedness of (P2) in the above three function spaces. [ABSTRACT FROM PUBLISHER]