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

Abstract

We consider the well-posedness of the second order degenerate integrodifferential equations (P2): , (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), (Mu′)(0) = (Mu′)(2π), in periodic Lebesgue-Bochner spaces L^p(????, X), periodic Besov spaces (????, X) and and periodic Triebel-Lizorkin spaces (????, X), where A and M are closed linear operators on a Banach space X satisfying D(A) ⊂ D(M), a ∊ L¹ (ℝ₊) and α is 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.