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(𝕋, X), periodic Besov spaces (𝕋, X) and and periodic Triebel-Lizorkin spaces (𝕋, 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.

Mathematics Subject Classification (2010):

• 34G10
• 47D06
• 35K65
• 34K13

Keywords:

• Well-posedness
• Fourier multiplier
• degenerate integro-differential equation
• Lebesgue-Bochner spaces
• Besov spaces
• Triebel-Lizorkin spaces