A BLOWUP ALTERNATIVE RESULT FOR FRACTIONAL NON-AUTONOMOUS EVOLUTION EQUATION OF VOLTERRA TYPE

. In this article, we consider a class of fractional non-autonomous integro-diﬀerential evolution equation of Volterra type in a Banach space E , where the operators in linear part (possibly unbounded) depend on time t . Combining the theory of fractional calculus, operator semigroups and measure of noncompactness with Sadovskii’s ﬁxed point theorem, we ﬁrstly proved the local existence of mild solutions for corresponding fractional non-autonomous integro-diﬀerential evolution equation. Based on the local existence result and a piecewise extended method, we obtained a blowup alternative result for frac- tional non-autonomous integro-diﬀerential evolution equation of Volterra type. Finally, as a sample of application, these results are applied to a time frac- tional non-autonomous partial integro-diﬀerential equation of Volterra type with homogeneous Dirichlet boundary condition. This paper is a continua- tion of Heard and Rakin [13, J. Diﬀerential Equations, 1988] and the results obtained essentially improve and extend some related conclusions in this area.


(Communicated by Yuri Latushkin)
Abstract. In this article, we consider a class of fractional non-autonomous integro-differential evolution equation of Volterra type in a Banach space E, where the operators in linear part (possibly unbounded) depend on time t. Combining the theory of fractional calculus, operator semigroups and measure of noncompactness with Sadovskii's fixed point theorem, we firstly proved the local existence of mild solutions for corresponding fractional non-autonomous integro-differential evolution equation. Based on the local existence result and a piecewise extended method, we obtained a blowup alternative result for fractional non-autonomous integro-differential evolution equation of Volterra type. Finally, as a sample of application, these results are applied to a time fractional non-autonomous partial integro-differential equation of Volterra type with homogeneous Dirichlet boundary condition. This paper is a continuation of Heard and Rakin [13, J. Differential Equations, 1988] and the results obtained essentially improve and extend some related conclusions in this area.
1. Introduction. Evolution equations describe time dependent processes as they occur in physics, chemistry, economy, biology or other sciences. Mathematically, they appear in quite different forms, such as parabolic or hyperbolic partial differential equations, integro-differential equations, delay or difference differential equations or more general functional differential equations. When treating such classes of evolution equations, it is usually assumed that the partial differential operators depend on time t on account of this class of operators appears frequently in the applications, for the details please see Pazy [23] and Tanabe [25]. As a result, it is significant and interesting to investigate non-autonomous evolution equations, i.e., the differential operators in the main parts of the considered problems are dependent of time t. In 1988, Heard and Rakin [13] considered the following Volterra integro-differential equation γ µ ], · denotes the norm on Banach space E, · µ denotes the graph norm on E µ = D(A µ (0)), η, γ, ν are positive constants satisfying 0 < η, µ, γ < 1, and the nonlinear map g is Lipschitz continuous on the domain of A(0) into E, with respect to the graph norm of A(0). Also, the uniqueness of solution is proved under the restriction that the space E is a Hilbert space and γ = 1.
Fractional calculus is a mathematical topic more than 300 years. The concept of noninteger derivative and integral is a generalization of the traditional integerorder differential and integral calculus. Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. In recent years, fractional differential equations especially fractional evolution equations have attracted great interest because of their practical applications in many areas such as physics, chemistry, economics, social sciences, finance and other areas of science and engineering. For more details, see [2,4,6,8,9,11,12], [18]- [22], [28]- [32] and the references therein for more comments and citations.
Recently, in 2011, Rashid and Al-Omari [24] studied the local and global existence of mild solutions to a class of fractional semilinear impulsive Volterra type integrodifferential evolution equation in Banach space by using the fixed point technique, where C D α t is the standard Caputo's fractional time derivative of order α ∈ (0, 1], −A is assumed to be an infinitesimal generator of a compact C 0 -semigroup T (t) (t ≥ 0), the nonlinear maps f , g : I × E → E, I = [0, a), 0 < a ≤ ∞, are continuous, q : I → R and u 0 ∈ E. I k : E → E, 0 < t 1 < t 2 < · · · < t m < t m+1 <= a, ∆u| t=t k = u(t + k ) − u(t − k ), u(t + k ) and u(t − k ) represent respectively the right and left limits of u(t) at t = t k . In 2017, Gou and Li [12] generalized to the case that the C 0 -semigroup T (t) (t ≥ 0) generated by −A is non-compact, and obtained the local and global existence of mild solution for fractional impulsive Volterra type integro-differential evolution equation (1.2) with non-compact semigroup in Banach space E.
We noticed that the papers [12] and [24] are all devoted to investigating the local and global existence of mild solution for fractional Volterra type integro-differential evolution equation under the situation that the differential operators in the main parts are independent of time t, which means that the fractional Volterra type integro-differential evolution equations under considerations are autonomous. As we pointed out in the first paragraph, it is significant and interesting to investigate non-autonomous evolution equations, which means that the differential operators in the main parts of the considered problems are dependent of time t. In fact, fractional non-autonomous evolution equations have been studied by several authors in recent years. For example, El-Borai [8] investigated the existence and continuous dependence of fundamental solutions for a class of linear fractional non-autonomous evolution equations in 2004. In 2010, El-Borai, EI-Nadi and EI-Akabawy [9] give some conditions to ensure the existence of resolvent operator for a class of fractional non-autonomous evolution equations with classical Cauchy initial condition. In [22,32], the authors investigated the nonlinear fractional non-autonomous reactiondiffusion equation with delay by using appropriate fixed point theory.
Motivated by the above mentioned aspects, in this paper we will combine these earlier works and extend the study to the following fractional non-autonomous integro-differential evolution equation (FNEE) of Volterra type in Banach space E which is more general than those in many previous publications, where C D α t is the standard Caputo's fractional time derivative of order 0 < α ≤ 1, A(t) is a family of closed linear operators defined on a dense domain D(A) in Banach space E into E such that D(A) is independent of t, f , g : [0, +∞) × E → E are continuous nonlinear functions, the integral kernel function q : [0, T ) → E is locally integrable for 0 < T ≤ +∞, u 0 ∈ E. One can easily see that when α = 1, then the problem (1.3) will degrade into (1.1) studied by Heard and Rakin in [13].
Let us point out that the work of this paper has three wedges; Firstly, we extended the study of first order non-autonomous integro-differential evolution equation of Volterra type to fractional ones. Secondly, we extended the study of fractional autonomous integro-differential evolution equation of Volterra type to non-autonomous ones. Lastly, we proved that fractional non-autonomous integrodifferential evolution equation of Volterra type (1.3) exists a mild solution u ∈ C([0, T max ), E) on a maximal existence interval [0, T max ) and satisfies the blowup alternative under weaker conditions on the nonlinear items.
In this article, by using the famous Sadovskii's fixed point theorem, a new estimation technique of the measure of noncompactness and a piecewise extended method, we obtained a blowup alternative result for fractional non-autonomous integro-differential evolution equation of Volterra type (1.3). The results obtained in this paper are generalizations of related results. As the readers can see, the hypotheses on the nonlinear items in our theorems are reasonably weak and different from those in many previous papers such as [22,24,32], and the proofs provided are concise. One will see that even for the case α = 1, Theorem 4.1 below essentially extends the main results of Heard and Rakin [13]; as far as the mild solution of nonautonomous integro-differential evolution equation of Volterra type is concerned, by dropping the Hölder continuity of the nonlinear items from the hypotheses. Moreover, even for corresponding fractional autonomous integro-differential evolution equation of Volterra type, the results here are new.
The rest of this article is organized as follows. We provide in Section 2 some definitions, notations and necessary preliminaries. The local existence of mild solutions for fractional non-autonomous integro-differential evolution equation of Volterra type is obtained in Section 3. In Section 4, we proved that the fractional autonomous integro-differential evolution equation of Volterra type (1.3) exists a mild solution u ∈ C([0, T max ), E) on a maximal existence interval [0, T max ) and satisfies the blowup alternative. An concrete example is given to illustrate the feasibility of our main results in the last section.

Preliminaries.
We begin with this section by giving some notations. Let E be a Banach space with norm · . Throughout this work, we use I to denote the closed subset of the interval [0, +∞). We denote by C(I, E) the Banach space of all continuous functions from interval I into E equipped with the supremum norm u C = sup t∈I u(t) , and by L(E) the Banach space of all linear bounded operators from E to E endowed with the topology defined by operator norm. Let L 1 (I, E) be the Banach space of all E-value Bochner integrable functions defined on I with the norm u 1 = I u(t) dt.
Secondly, we introduce some basic definitions about the Riemann-Liouville integral and Caputo derivative of fractional order. 20]). The fractional integral of order α > 0 with the lower limit 0 for a function f ∈ L 1 ([0, +∞), R) is defined as Here and elsewhere Γ(·) denotes the Gamma function. 20]). The Caputo fractional derivative of order α with the lower limit zero for a function f : [0, +∞) → R, which is at least n-times differentiable can be defined as where n − 1 < α < n, n ∈ N. Thirdly, we give the proper definition of of mild solutions for FNEE (1.3). Throughout this paper, we assume that the linear operator −A(t) satisfies the following conditions: (A1) For any λ with Reλ ≥ 0, the operator λI + A(t) exists a bounded inverse operator λI + A(t) −1 in L(E) and where C is a positive constant independent of both t and λ; (A2) For any t, τ, s ∈ I, there exists a constant γ ∈ (0, 1] such that where the constants γ and C > 0 are independent of both t, τ and s. From Henry [15], Pazy [23] and Temam [26], we know that the assumption (A1) means that for each s ∈ I, the operator −A(s) generates an analytic semigroup e −tA(s) (t > 0), and there exists a positive constant C independent of both t and s such that where n = 0, 1, t > 0, s ∈ I. Furthermore, in assumption (A1), if we choose λ = 0 and t = 0, then there exists a positive constant C independent of both t and λ such that By the above discussion and [8, Theorem 2.6], we can get the definition of mild solutions for FNEE (1.3).
where the operators ψ(t, s), ϕ(t, η) and U (t) are defined by

3)
and ξ α is a probability density function defined on [0, +∞) such that it's Laplace transform is given by The following properties about the operators ψ(t, s), ϕ(t, η) and U (t) will be needed in our argument.

5)
where C is a positive constant independent of both t and η. Furthermore,

6)
and Next, we introduce the definition for Kuratowski measure of noncompactness, which will be used in the proof of our main results.
The following properties about the Kuratowski measure of noncompactness are well known. 3,7]). Let E be a Banach space and U , V ⊂ E be bounded. The following properties are satisfied : In this article, we denote by µ(·) and µ C (·) the Kuratowski measure of noncompactness on the bounded set of E and C(I, E) respectively. For any B ⊂ C(I, E) and t ∈ I, set . For more details about the properties of Kuratowski measure of noncompactness, we refer to the monographs [3] and [7].
The following lemmas are needed in our argument.
is a countable set and there exists a function m ∈ L 1 (I, R + ) such that for every n ∈ N, a.e. t ∈ I.
Then µ(D(t)) is Lebesgue integrable on I, and  3. Local existence of mild solutions. In this section, we prove the local existence of mild solutions for fractional non-autonomous integro-differential evolution equation (FNEE) of Volterra type with initial time t 0 ∈ [0, ∞) and initial value is a constant will be specified later.
Then for every Proof. Define an operator Q : By direct calculus we know that the operator Q is well defined. From Definition 2.4 and (3.2), it is easy to verify that the mild solution of FNEE (3.1) on J is equivalent to the fixed point of the operator Q defined by (3.2). Denote is the Beta function. We firstly prove that the operator Q defined by (3.2) maps the bounded closed convex set Ω = {u ∈ C(J, E) : For any u ∈ Ω and t ∈ J, by Lemma 2.5, (3.2) and (3.3), we get that which means that Qu ∈ Ω. Therefore, we have proved that the operator Q maps Ω to Ω. Secondly, we prove that Q : Ω → Ω is a continuous operator. To this end, let {u n } ∞ n=1 ⊂ Ω be a sequence such that lim n→+∞ u n = u in Ω. By the continuity of the second variable for the nonlinear functions f and g, we get that for any t ∈ J lim n→+∞ f (t, u n (t)) − f (t, u(t)) = 0, lim n→+∞ g(t, u n (t)) − g(t, u(t)) = 0. (3.4) (3.4) combined with the boundedness of nonlinear functions f and g, one gets that g(s, u n (s)) − g(s, u(s)) → 0 as n → +∞.
From the boundedness of nonlinear functions f and g, we know that for every t ∈ J and t 0 ≤ η ≤ t By again the boundedness of nonlinear functions f and g combined with proper integral transformation and the definition of Beta function, we get that for every t ∈ J, t 0 ≤ η ≤ t and t 0 ≤ s ≤ η and t ∈ J, combined with (3.5)-(3.8) and the Lebesgue dominated convergence theorem, we know that for every t ∈ J, which implies that Qu n − Qu C → 0 as n → ∞.
In what follows, we prove that the operator Q : Ω → Ω is equicontinuous. For any u ∈ Ω and t 0 ≤ t 1 < t 2 ≤ t 0 + h, we get from (3.2) that which means that Therefore, in order to prove that the operator Q : Ω → Ω is equicontinuous, we only need to check I i tend to 0 independently of u ∈ Ω when t 2 −t 1 → 0, i = 1, 2, · · · , 6. For I 1 , from Lemma 2.5 and the fact that the function η → (t 2 − η) α−1 (1 + η γ ) is Lebesgue integrable we get that For t 1 = t 0 and t 0 < t 2 ≤ t 0 +h, it is easy to see that I 2 = 0. For t 1 > t 0 and > 0 small enough, by Lemma 2.5 and the facts that the function η → is Lebesgue integrable and operator-valued function ψ(t − η, η) is continuous in uniform topology about the variables t and η for t 0 < t 2 ≤ t 0 + h and t 0 ≤ η ≤ t − , we have For I 3 , by Lemma 2.5 and the boundedness of nonlinear functions f and g, one get that For t 1 = t 0 and t 0 < t 2 ≤ t 0 +h, it is easy to see that I 4 = 0. For t 1 > t 0 and > 0 small enough, from the boundedness of nonlinear functions f and g, Lemma 2.5 and the facts that the function η → [(t 2 − η) α−1 + (t 1 − η) α−1 ] is Lebesgue integrable and operator-valued function ψ(t − η, η) is continuous in uniform topology about the variables t and η for t 0 ≤ t ≤ t 0 + h and t 0 ≤ η ≤ t − , we have For I 5 , by the boundedness of nonlinear functions f and g, Lemma 2.5 and the fact that the function η → (t 2 − η) α−1 (η − t 0 ) γ is Lebesgue integrable, we know that For t 1 = t 0 and t 0 < t 2 ≤ t 0 + h, it is easy to see that I 6 = 0. For t 1 > t 0 and > 0 small enough, by Lemma 2.5, the boundedness of nonlinear functions f and g, the facts that the function η → [(t 2 − η) α−1 + (t 1 − η) α−1 ]I γ η 1 is Lebesgue integrable as well as the operator-valued function ψ(t − η, η) is continuous in uniform topology about the variables t and η for t 0 ≤ t ≤ t 0 + h and t 0 ≤ η ≤ t − , we know that →0 as t 2 − t 1 → 0 and → 0.
Let B = co Q(Ω), where co means the closure of convex hull. Then one can easily to verify that the operator Q maps B into itself and B ⊂ C(J, E) is equicontinuous. Next, we prove that the operator Q : B → B is a condensing operator. For any D ⊂ B, by Lemma 2.8, we know that there exists a countable set D 0 = {u n } ⊂ D, such that µ C (Q(D)) ≤ 2µ C (Q(D 0 )). (3.9) By the equicontinuity of B, we know that D 0 ⊂ B is also equicontinuous. Therefore, by (3.2), Lemma 2.9 and the assumption (H f ) one get that Because Q(D 0 ) ⊂ D is bounded and equicontinuous, from Lemma 2.10 we know that µ C (Q(D 0 )) = max t∈[t0,t0+h] µ(Q(D 0 )(t)). (3.11) Therefore, by (3.3) and (3.9)-(3.11), we get that which means that Q : Ω → Ω is a condensing operator. It follows from the famous Sadovskii's fixed point theorem that the operator Q has at least one fixed point u ∈ Ω, which is just a mild solution of FNEE (3.        For 0 < t < t < T max and > 0 small enough, by (3.2), (4.2), Lemma 2.5 and the fact that the operator-valued function ψ(t − η, η) is continuous in uniform topology about the variables t and η for 0 ≤ t < T max and 0 ≤ η ≤ t − , we know that →0 as t , t → T − max and → 0.
Then the time fractional non-autonomous partial integro-differential equation of Volterra type with homogeneous Dirichlet boundary condition (5.1) can be transformed into the abstract form of time fractional non-autonomous integro-differential evolution equation of Volterra type (1.3).

FRACTIONAL NON-AUTONOMOUS EVOLUTION EQUATION OF VOLTERRA TYPE 1991
Proof. By the definitions of nonlinear functions f and g one can easily to verify that the nonlinear functions f , g : [0, +∞) × L 2 ([0, π], R) → L 2 ([0, π], R) are continuous, map bounded sets in [0, +∞) × L 2 ([0, π], R) into bounded sets in L 2 ([0, π], R). Furthermore, from the definition of nonlinear sunctions f and g, we know that f (t, u) and g(t, v) is Lipschitz continuous about the variables u and v with Lipschitz constants k f = 1/4 and k g = 1, respectively. Therefore, by Lemma 2.7 (vii) we know that the assumption (H f ) is satisfied with positive constants In addition, the definition of q(t − s) means that q : [0, +∞) → L 2 ([0, π], R) is locally integrable. Therefore, all the assumptions of Theorem 4.1 are satisfied. Hence, from Theorem 4.1 we know that the time fractional non-autonomous partial integro-differential equation of Volterra type with homogeneous Dirichlet boundary condition (5.1) exists a mild solution u ∈ C([0, T ), L 2 ([0, π], R)) on a maximal existence interval [0, T ) for T > 0, and if T < +∞ then lim t→T − u(t) 2 = +∞. This completes the proof of Theorem 5.1.