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

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.

KEYWORDS:

• Fractional-order equation
• the Riemann–Liouville derivative
• Cauchy-type problem
• semilinear evolution equation
• initial-boundary value problem
• fractional partial differential equations

AMS SUBJECT CLASSIFICATIONS:

• 35R11
• 34G20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The reported study was funded by Act 211 of Government of the Russian Federation, contract 02.A03.21.0011; and by the Russian Foundation for Basic Research, project number 19-31-90008.