Inhomogeneous Fractional Evolutionary Equation in the Sectorial Case
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INHOMOGENEOUS FRACTIONAL EVOLUTIONARY EQUATION IN THE SECTORIAL CASE UDC 517.955.1, 517.986.7
V. E. Fedorov and E. A. Romanova
Abstract. In this paper, we prove the unique solvability of the Cauchy problem for a linear inhomogeneous equation in a Banach space that is solved with respect to the Gerasimov–Caputo fractional derivative. We assume that the operator acting on the unknown function in the equation generates a set of resolving operators of the corresponding homogeneous equation, which is exponentially bounded and analytic in a sector containing the positive semiaxis. The general form of solutions to the Cauchy problem is obtained. The general results are applied to the study of the unique solvability of a certain class of initial-boundary-value problems for partial differential equations solvable with respect to the Gerasimov–Caputo fractional derivative by time, containing in the simplest case initial-boundary-value problems for fractional diffusion and diffusion-wave equations. Keywords and phrases: Gerasimov–Caputo fractional derivative, evolutionary equation, resolving family of operators, initial-boundary-value problem, diffusion-wave equation. AMS Subject Classification: 35R11, 34G10
1. Introduction. Let A : DA → Z be a linear closed operator whose domain DA is dense in a Banach space Z. For the equation Dtα z(t) = Az(t) + f (t),
t ∈ [0, T ),
(1)
Dtα
is the fractional Gerasimov–Caputo derivative (see [2, 7]), f : [0, T ) → Z, and where α > 0, T ∈ (0, ∞], consider the Cauchy problem u(k) (0) = uk ,
k = 0, 1, . . . , m − 1.
(2)
For α = 1, one of the most important cases is the case where in a sector containing the positive semiaxis, there exists an analytic operator semigroup that solves the equation z(t) ˙ = Az(t) (see [9, 12, 15]). The operator A generating this semigroup is called the sectorial operator (see [3]). Conditions that generalize the sectorial property, i.e., conditions of the existence of an analytic, exponentially bounded in a sector, resolving family of operators for the homogeneous equation (1) are stated in [1, Theorem 2.14] as a consequence of a more general theorem on the solvability of evolutionary integral equations (see [11, Theorem 2.1] and [4]). However, these papers do not contain results on the unique solvability and the form of solutions of the Cauchy problem for the nonhomogeneous problem (1), (2). These questions are discussed in the present paper. The results obtained here allow one to examine a certain class of initial-boundary-value problems for partial differential equations solved with respect to the Gerasimov–Caputo fractional derivative by time. The simplest examples of this class are initial-boundary-value problems for the fractional diffusion and wave-diffusion equations, which were studied earlier by many authors (see, e.g., the survey [8]). Note that if an operator A generates an exponentially bounded resolving family of operators for the homogeneous equation (1) for α > 2, then it is bounded (see [1, Theorem 2.6]), and the study of the problem (1), (2) becom
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