Well-posedness of equations with fractional derivative
The author studies the well-posedness for the equation \[ D^\alpha u(t)= Au(t)+ f(t),\quad 0\leq t\leq 2\pi, \] where \(A\) is a closed operator in a Banach space \(X\) and \(D^\alpha\), \(\alpha> 0\), is the fractional derivative in the sense of Weyl. Using known results on \(L^p\) multiplier theorems, the author gives necessary and/or sufficient conditions for the \(L^p\) well-posedness of the problem. The conditions involve the resolvent of \(A\) and the Rademacher boundedness. Corresponding results on the well-posedness of this problem are also obtained in periodic Besov spaces, periodic Triebel-Lizorkin spaces and periodic Hardy spaces.
scientific article; zbMATH DE number 5790096