Well‐posedness of degenerate fractional integro‐differential equations in vector‐valued functional spaces

Shangquan Bu and Gang Cai

Mathematische Nachrichten, 2020, vol. 293, issue 10, 1931-1946

Abstract: We study the well‐posedness of the fractional degenerate integro‐differential equations (Pα):Dα(Mu)(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+∫−∞tb(t−s)Bu(s)ds+f(t),(t∈T:=[0,2π]),in Lebesgue–Bochner spaces Lp(T;X) and Besov spaces Bp,qs(T;X), where A, B and M are closed linear operators on a Banach space X satisfying D(A)∩D(B)⊂D(M), D(A)∩D(B)≠{0}, α>0 and a,b∈L1(R+). We completely characterize the well‐posedness of (Pα) in the above vector‐valued function spaces on T by using operator‐valued Fourier multiplier. We also give an example that our abstract results may be applied.

Date: 2020
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