Semilinear fractional-order evolution equations of Sobolev type in the sectorial case

Abstract

ABSTRACT The local unique solvability of the Cauchy-type problem to a semilinear equation in a Banach space, which is solved with respect to the highest order Riemann–Liouville derivative, is proved. A linear unbounded operator at the unknown function in the equation generates an analytic in a sector resolving the family of operators of the linear homogeneous fractional-order equation. This result is applied to the study of initial-boundary value problems for a class of nonlinear partial differential equations, in particular, containing a nonlinear superdiffusion equation. Besides, it is used for the investigation of the local unique solvability of the Showalter–Sidorov type problem to a semilinear Sobolev-type equation in a Banach space with a sectorial pair of operators and with Riemann–Liouville derivatives. For this aim, we essentially use two types of condition on the nonlinear operator: the condition on the image of this operator or the condition of its independence of the elements of the degeneration subspace. These abstract results are demonstrated on examples of initial-boundary value problems for Sobolev-type nonlinear partial differential equations.