Abstract fractional linear pseudo-parabolic equations in Banach spaces: well-posedness, regularity, and asymptotic behavior

In this paper we study the well-posedness, regularity, and asymptotic behavior of the solutions to the abstract pseudo-parabolic equation ∂ _t^α u(t) = A u(t) + B∂ _t^β u(t) + f(t), where A , B are closed linear operators in a Banach space, and ∂ _t^γ u denotes the Caputo or Riemann–Liouville fractional derivative of order γ >0 .