Quasilinear Fractional Order Equations and Fractional Powers of Sectorial Operators

Abstract

The fractional powers of generators for analytic operator semigroups are used for the proof of the existence and uniqueness of a solution of the Cauchy problem to a first order semilinear equation in a Banach space. Here, we use an analogous construction of fractional powers Aγ for an operator A such that −A generates analytic resolving families of operators for a fractional order equation. Under the condition of local Lipschitz continuity with respect to the graph norm of Aγ for some γ∈(0,1) of a nonlinear operator, we prove the local unique solvability of the Cauchy problem to a fractional order quasilinear equation in a Banach space with several Gerasimov–Caputo fractional derivatives in the nonlinear part. An analogous nonlocal Lipschitz condition is used to obtain a theorem of the nonlocal unique solvability of the Cauchy problem. Abstract results are applied to study an initial-boundary value problem for a time-fractional order nonlinear diffusion equation.