Well-posedness of degenerate differential equations with fractional derivative in vector-valued functional spaces

Abstract

In this paper, we study the well-posedness of the degenerate differential equations with fractional derivative Dα(Mu)(t)=Au(t)+f(t),(0≤t≤2π) in Lebesgue–Bochner spaces Lp(T;X), periodic Besov spaces Bp,qs(T;X) and periodic Triebel–Lizorkin spaces Fp,qs(T;X), where A and M are closed linear operators in a complex Banach space X satisfying D(A)⊂D(M), α>0 and Dα is the fractional derivative in the sense of Weyl. Using known operator-valued Fourier multiplier results, we completely characterize the well-posedness of this problem in the above three function spaces by the R-bounedness (or the norm boundedness) of the M-resolvent of A.