Fractional Evolution Equations Governed by Coercive Differential Operators

Abstract

This paper is concerned with evolution equations of fractional order D α u t Au t ; u 0 u 0 , u 0 0, where A is a differential operator corresponding to a coercive polynomial taking values in a sector of angle less than π and 1 < α < 2. We show that such equations are well posed in the sense that there always exists an α-times resolvent family for the operator A.

On a class of time-fractional differential equations

“…See also her recent work [5]. For the fractional differential equations governed by some concrete partial differential operators, we refer to [20,21,22,23,25]. Let us recall the definition of the α-times resolvent families for (1.2).…”

Perturbation Theory for Abstract Volterra Equations

“…The above result is applied to abstract timefractional equations considered in [20,21] and to differential operators in the spaces of Hölder continuous functions (von Wahl [22]). Possible applications of Corollary 8 and Theorem 7 can be also made to coercive differential operators considered by Li et al [23,Section 4] and by the author [24]. In the remainder of the third section, we reconsider and slightly improve results of Arendt and Batty [25] and Desch et al [26] on rank-1 perturbations.…”

“…If ( ) = ∫ 0 ( ) , ∈ [0, ), where ∈ 1 loc ([0, )) and ̸ = 0, then we obtain the unification concept for (local) -convolutedsemigroups and cosine functions [1]. We refer the reader to [23,28,32,37,38] for some applications of ( , )-regularized -resolvent families in the study of the following abstract time-fractional equation with > 0 :…”

Fractional Abstract Cauchy Problems

“…Li et al [13] considered fractional order evolution equation D α u(t) = They gave basic properties and analyticity criteria of fractional resolvent operator functions and discussed the relations between integrated resolvent families and resolvent families.…”