Positive Solutions for the Initial Value Problem of Fractional Evolution Equations

Abstract

By using the fixed point theorems and the theory of analytic semigroup, we investigate the existence of positive mild solutions to the Cauchy problem of Caputo fractional evolution equations in Banach spaces. Some existence theorems are obtained under the case that the analytic semigroup is compact and noncompact, respectively. As an example, we study the partial differential equation of the parabolic type of fractional order.

1. Introduction

The differential equations involving fractional derivatives in time have recently been studied extensively. One can see, for instance, the monographs [1–5] and the survey [6–8]. In particular, there has been a significant development in fractional evolution equations. Existence of solutions for fractional evolution equations has been studied by many authors during recent years. Many excellent results are obtained in this field; see [9–19] and the references therein. In [9, 10], El-Borai first constructed the type of mild solutions to fractional evolution equations in terms of a probability density. And then the author investigated the existence, uniqueness, and regularity of solutions of fractional integrodifferential equations in [11, 12]. Recently, this theory was developed by Zhou et al. [13–16]. Particularly, they studied the existence and controllability of mild solution of fractional delay integrodifferential equations with a compact analytic semigroup in [16]. In [17–19], the authors studied the existence of mild solutions of fractional impulsive delay or impulsive evolution equations. But as far as we know, there are no results on the existence of positive solutions of fractional evolution equations.

In this paper, by using the fixed point theorems combined with the theory of analytic semigroup, we investigate the existence of positive mild solutions for the initial value problem (IVP) of fractional evolution equations in Banach space 𝑋 as 𝐷^𝑞₀𝑢(𝑡)(𝑡)((𝑡𝑢(0)+𝐴𝑢=𝑓𝑡,𝑢)),𝑡>0,=𝑢₀, (1) where 𝐷^𝑞₀ denotes the Caputo fractional derivative of order 𝑞∈(0,1) with the lower limits zero, −𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is the infinitesimal generator of an analytic semigroup 𝑆(𝑡)(𝑡≥0) of uniformly bounded linear operators, and 𝑓 is the nonlinear term and will be specified later.

The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional power of the generator of the analytic semigroup and the definition of mild solutions of IVP(1). In Section 3, we study the existence of positive mild solutions for the IVP(1). In Section 4, an example is given to illustrate the applicability of abstract results obtained in Section 3.

2. Preliminaries

In this section, we introduce some basic facts about the fractional power of the generator of analytic semigroup and the fractional calculus that are used throughout this paper.

Let 𝑋 be a Banach space with norm ‖⋅‖. Throughout this paper, we assume that −𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is the infinitesimal generator of an analytic semigroup 𝑆(𝑡)(𝑡≥0) of uniformly bounded linear operator in 𝑋; that is, there exists 𝑀≥1 such that ‖𝑆(𝑡)‖≤𝑀 for all 𝑡≥0. Without loss of generality, let 0∈𝜌(−𝐴), where 𝜌(−𝐴) is the resolvent set of −𝐴. Then for any 𝛼>0, we can define 𝐴^−𝛼 by 𝐴^−𝛼1∶=Γ(𝛼)^∞₀𝑡^𝛼−1𝑆(𝑡)𝑑𝑡. (2) Then 𝐴^𝛼 can be defined by 𝐴^𝛼∶=(𝐴^−𝛼)⁻¹ because 𝐴^−𝛼 is one to one. It can be shown that each 𝐴^𝛼 has dense domain and that 𝐷(𝐴^𝛽)⊂𝐷(𝐴^𝛼) for 0≤𝛼≤𝛽. Moreover, 𝐴^𝛼+𝛽𝑥=𝐴^𝛼𝐴^𝛽𝑥=𝐴^𝛽𝐴^𝛼𝑥 for every 𝛼,𝛽∈ℝ and 𝑥∈𝐷(𝐴^𝜇) with 𝜇∶=max{𝛼,𝛽,𝛼+𝛽}, where 𝐴⁰=𝐼,𝐼 is the identity in 𝑋 (for proofs of these facts we refer to the literature [20–22]).

We denote by 𝑋_𝛼 the Banach space of 𝐷(𝐴^𝛼) equipped with norm ‖𝑥‖_𝛼=‖𝐴^𝛼𝑥‖ for 𝑥∈𝐷(𝐴^𝛼), which is equivalent to the graph norm of 𝐴^𝛼. Then we have 𝑋_𝛽↪𝑋_𝛼 for 0≤𝛼≤𝛽≤1 (with 𝑋₀=𝑋), and the embedding is continuous. Moreover, 𝐴^𝛼 has the following basic properties.

Lemma 1 (see [23]). 𝐴^𝛼 has the following properties.(i)𝑆(𝑡)∶𝑋→𝑋_𝛼 for each 𝑡>0 and 𝛼≥0. (ii)𝐴^𝛼𝑆(𝑡)𝑥=𝑆(𝑡)𝐴^𝛼𝑥 for each 𝑥∈𝐷(𝐴^𝛼) and 𝑡≥0. (iii)For every 𝑡>0, 𝐴^𝛼𝑆(𝑡) is bounded in 𝑋 and there exists 𝑀_𝛼>0 such that 𝐴^𝛼𝑆(𝑡)≤𝑀_𝛼𝑡^−𝛼. (3)

Let 𝐽 be a closed interval on ℝ⁺=[0,∞). In the following we denote by 𝐶(𝐽,𝑋_𝛼) the Banach space of all continuous functions from 𝐽 into 𝑋_𝛼 endowed with supnorm given by ‖𝑢‖_𝐶=sup_𝑡∈𝐽‖𝑢(𝑡)‖_𝛼 for 𝑢∈𝐶(𝐽,𝑋_𝛼). For any 𝑡≥0, denote by 𝑆_𝛼(𝑡) the restriction of 𝑆(𝑡) to 𝑋_𝛼. From Lemma 1(i) and (ii), for any 𝑥∈𝑋_𝛼, we have ‖𝑆(𝑡)𝑥‖_𝛼=𝐴^𝛼⋅𝑆(𝑡)𝑥=𝑆(𝑡)⋅𝐴^𝛼𝑥≤‖𝑆(𝑡)‖⋅𝐴^𝛼𝑥=‖𝑆(𝑡)‖⋅‖𝑥‖_𝛼,‖𝑆(𝑡)𝑥−𝑥‖_𝛼=𝐴^𝛼⋅𝑆(𝑡)𝑥−𝐴^𝛼𝑥=𝑆(𝑡)⋅𝐴^𝛼𝑥−𝐴^𝛼𝑥→0 (4) as 𝑡→0. Therefore, 𝑆(𝑡)(𝑡≥0) is a strongly continuous semigroup in 𝑋_𝛼, and ‖𝑆_𝛼(𝑡)‖_𝛼≤‖𝑆(𝑡)‖ for all 𝑡≥0. To prove our main results, the following lemma is also needed.

Lemma 2 (see [24]). If 𝑆(𝑡)(𝑡≥0) is a compact semigroup in 𝑋, then 𝑆_𝛼(𝑡)(𝑡≥0) is a compact semigroup in 𝑋_𝛼, and hence it is norm continuous.

Let us recall the following known definitions in fractional calculus. For more details, see [9, 13–16, 18, 19].

Definition 3. The fractional integral of order 𝜎>0 with the lower limits zero for a function 𝑓 is defined by 𝐼^𝜎₀𝑓(𝑡)=1Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1𝑓(𝑠)𝑑𝑠,𝑡>0, (5) where Γ is the gamma function.
The Riemann-Liouville fractional derivative of order 𝑛−1<𝜎<𝑛 with the lower limits zero for a function 𝑓 can be written as ^𝐿𝐷^𝜎₀𝑓(𝑡)=1Γ()𝑑𝑛−𝜎𝑑𝑡^𝑛𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝜎−1}𝑓(𝑠)𝑑𝑠,𝑡>0,𝑛∈ℕ. (6) Also the Caputo fractional derivative of order 𝑛−1<𝜎<𝑛 with the lower limits zero for a function 𝑓∈𝐶^𝑛[0,∞) can be written as 𝐷^𝜎₀𝑓(𝑡)=1Γ(𝑛−𝜎)^𝑡₀(𝑡−𝑠)^{𝑛−𝜎−1}𝑓^(𝑛)(𝑠)𝑑𝑠,𝑡>0,𝑛∈ℕ. (7)

Remark 4. (1) The Caputo derivative of a constant is equal to zero.
(2) If 𝑓 is an abstract function with values in 𝑋, then integrals which appear in Definition 3 are taken in Bochner's sense.

Lemma 5 (see [14]). A measurable function ℎ∶𝐽→𝑋 is Bochner integrable if ‖ℎ‖ is Lebesgue integrable.

For 𝑥∈𝑋, we define two families {𝑈(𝑡)}_𝑡≥0 and {𝑉(𝑡)}_𝑡≥0 of operators by 𝑈(𝑡)𝑥=^∞₀𝜂_𝑞(𝜃)𝑆𝑡^𝑞𝜃𝑉(𝑡)𝑥𝑑𝜃,𝑥=𝑞^∞₀𝜃𝜂_𝑞(𝜃)𝑆𝑡^𝑞𝜃𝑥𝑑𝜃,0<𝑞<1, (8) where 𝜂_𝑞(𝜃)=1𝑞𝜃^{−1−1/𝑞}𝜌_𝑞𝜃^−1/𝑞,𝜌_𝑞(𝜃)=1𝜋^∞_𝑛=1(−1)^𝑛−1𝜃^{−𝑞𝑛−1}Γ𝑛𝑞+1(),𝑛!sin𝑛𝜋𝑞,𝜃∈0,∞ (9) where 𝜂_𝑞 is a probability density function defined on (0,∞), which has properties 𝜂_𝑞(𝜃)≥0 for all 𝜃∈(0,∞) and ∫^∞₀𝜂_𝑞(𝜃)𝑑𝜃=1. It is not difficult to verify (see [14]) that for 𝜇∈[0,1], we have ^∞₀𝜃^𝜇𝜂_𝑞(𝜃)Γ𝑑𝜃=1+𝜇Γ1+𝑞𝜇. (10) Clearly, if the semigroup 𝑆(𝑡)(𝑡≥0) is positive, then, by the definitions, the operators 𝑈(𝑡) and 𝑉(𝑡) are also positive for all 𝑡≥0.

The following lemma follows from the results in [14, Lemma 2.9] and [15, Lemmas 3.2–3.5].

Lemma 6. The operators 𝑈(𝑡) and 𝑉(𝑡) have the following properties.(i)For any fixed 𝑡≥0 and any 𝑥∈𝑋_𝛼, one has ‖𝑈(𝑡)𝑥‖_𝛼≤𝑀‖𝑥‖_𝛼,‖𝑉(𝑡)𝑥‖_𝛼≤𝑞𝑀Γ‖𝑥‖1+𝑞^𝛼=𝑀Γ𝑞‖𝑥‖_𝛼. (11)(ii)The operators 𝑈(𝑡) and 𝑉(𝑡) are strongly continuous for all 𝑡≥0. (iii)If the semigroup 𝑆(𝑡)(𝑡≥0) is compact, then 𝑈(𝑡) and 𝑉(𝑡) are compact operators in 𝑋 for 𝑡>0.(iv)If the semigroup 𝑆_𝛼(𝑡)(𝑡≥0) is norm continuous, then the restriction of 𝑈(𝑡) to 𝑋_𝛼 and the restriction of 𝑉(𝑡) to 𝑋_𝛼 are uniformly continuous for 𝑡>0.

Definition 7 (see [25, 26]). Let 𝐵 be a bounded set of a real Banach space 𝐸. Set 𝛽(𝐵)=inf{𝛿>0: 𝐵 can be expressed as the union of a finite number of sets such that the diameter of each set does not exceed 𝛿; that is, 𝐵=∪^𝑚_𝑖=1𝐵_𝑖 with diam(𝐵_𝑖)≤𝛿,𝑖=1,2,…,𝑚}. 𝛽(𝐵) is called the Kuratowski measure of noncompactness of set 𝐵.

It is clear that 0≤𝛽(𝐵)<∞. For the Kuratowski measure of noncompactness, we have the following well-known results.

Lemma 8 (see [26]). If 𝐷∈𝐶(𝐽,𝐸) is bounded and equicontinuous, then 𝛽(𝐷)=𝛽(𝐷(𝐽))=max_𝑡∈𝐽𝛽(𝐷(𝑡)), (12) where 𝐷(𝐽)={𝑥(𝑡)∶𝑥∈𝐷,𝑡∈𝐽}.

Lemma 9 (see [27]). Let 𝐷 be a countable set of strongly measurable function 𝑥∶𝐽→𝐸 such that there exists an 𝑀∈𝐿(𝐽,ℝ⁺) such that ‖𝑥(𝑡)‖≤𝑀(𝑡)𝑎.𝑒.,𝑡∈𝐽 for all 𝑥∈𝐷. Then 𝛽(𝐷(𝑡))∈𝐿(𝐽,ℝ⁺) and

𝛽_𝐽𝑥(𝑡)𝑑𝑡∶𝑥∈𝐷≤2_𝐽𝛽(𝐷(𝑡))𝑑𝑡. (13)

Lemma 10 (see [25] Mönch fixed point theorem). Let B be a closed and convex subset of 𝐸 and 𝑦₀∈𝐵. Assume that the continuous operator 𝐴∶𝐵→𝐵 has the following property: 𝐷⊂𝐵 is countable, and 𝐷⊂Co({𝑦₀}∪𝐴(𝐷))→𝐷 is relatively compact. Then 𝐴 has a fixed point in 𝐵.

Based on an overall observation of the previous related literature, in this paper we adopt the following definition of mild solution of IVP(1).

Definition 11. By a mild solution of the IVP(1), one means a function 𝑢∈𝐶(𝐽,𝑋_𝛼) satisfying 𝑢(𝑡)=𝑈(𝑡)𝑢₀+^𝑡₀(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠 (14) for all 𝑡∈𝐽.

3. Existence of Positive Mild Solutions

In this section, we introduce the existence theorems of positive mild solutions of the IVP(1). The discussions are based on fractional calculus and fixed point theorems.

Let 𝜆₁ be the smallest positive real eigenvalue of the linear operator 𝐴, and let 𝑒₁∈𝐷(𝐴) be the positive eigenvector corresponding to 𝜆₁. For any 𝑇>0 and 𝑟>0, we write Ω_𝑟∶=𝑢∈𝐶[0,𝑇],𝑋_𝛼∶‖𝑢(𝑡)‖_𝛼≤𝑟,𝑢(𝑡)≥𝜎𝑒₁,𝑡∈[0,𝑇], (15) where 𝜎>0 is a constant. Our main results are as follows.

Theorem 12. Let −𝐴∶𝐷(𝐴)⊂𝑋→𝑋 be the infinitesimal generator of a positive and compact analytic semigroup 𝑆(𝑡)(𝑡≥0) of uniformly bounded linear operators. Assume that 𝑓∈𝐶(ℝ⁺×𝑋_𝛼,𝑋) satisfies the following conditions.(𝐻₁) For any 𝑢∈Ω_𝑟, one has 𝑓(𝑡,𝑢(𝑡))≥𝑓𝑡,𝜎𝑒₁≥𝜆₁𝜎𝑒₁,𝑡∈[0,𝑇]. (16) (𝐻₂)𝑓 maps bounded sets of ℝ⁺×𝑋_𝛼 into bounded sets of 𝑋.If 𝑢₀∈𝑋_𝛼 with 𝑢₀≥𝜎𝑒₁ and 𝛼𝑞<1/2 for some 1/2<𝑞<1, then the IVP(1) has at least one positive mild solution 𝑢∈𝐶([0,𝑇),𝑋_𝛼). And if 𝑇<∞, one has lim_𝑡∈𝑇⁻‖𝑢(𝑡)‖_𝛼=∞.

Proof. For any 𝑡₀≥0 and 𝑥₀∈𝑋_𝛼 with 𝑥₀≥𝜎𝑒₁, we first prove that the initial value problem (IVP) of fractional evolution equations 𝐷^𝑞_𝑡0𝑢(𝑡)+𝐴𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡>𝑡₀,𝑢𝑡₀=𝑥₀ (17) has at least one positive mild solution on 𝐽=[𝑡₀,𝑡₀+ℎ_𝑡0], where ℎ_𝑡0 is a positive constant and will be given later.
Let 𝑅_𝑡0∶=2𝑀(‖𝑥₀‖_𝛼+1)+𝜎𝑒₁>0. Denote Ω_𝑅𝑡0∶=𝑢∈𝐶𝐽,𝑋_𝛼∶‖𝑢(𝑡)‖_𝛼≤𝑅_𝑡0,𝑢(𝑡)≥𝜎𝑒₁,𝑡∈𝐽. (18) Then Ω_𝑅𝑡0⊂𝐶(𝐽,𝑋_𝛼) is a nonempty bounded convex closed set. The assumption (𝐻₂) implies that there is a constant 𝐶=𝐶(𝑡₀)>0 such that 𝑓(𝑡,𝑢)≤𝐶 (19) for any 𝑡∈𝐽 and 𝑢∈Ω_𝑅𝑡0.
Define an operator 𝑄 by (𝑄𝑢)(𝑡)=𝑈𝑡−𝑡₀𝑥₀+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠,𝑡∈𝐽. (20) By the continuity of 𝑓, it is not difficult to prove that 𝑄∶𝐶(𝐽,𝑋_𝛼)→𝐶(𝐽,𝑋_𝛼) is continuous. By the positivity of the semigroup 𝑆(𝑡)(𝑡≥0), the assumption (𝐻₁), and (20), we easily see that (𝑄𝑢)(𝑡)≥(𝑄𝜎𝑒₁)(𝑡). Clearly, the positive mild solution of the IVP(17) on 𝐽 is equivalent to the fixed point of operator 𝑄 in Ω_𝑅𝑡0. We will use Schauder fixed point theorem to prove that 𝑄 has fixed points in Ω_𝑅𝑡0.
We first prove that 𝑄∶Ω_𝑅𝑡0→Ω_𝑅𝑡0 is continuous. Let ℎ_𝑡0≤[𝑀(1−𝛼)Γ(1+𝑞(1−𝛼))(‖𝑥₀‖_𝛼+1)/𝑀_𝛼𝐶Γ(2−𝛼)]^{1/𝑞(1−𝛼)}. For any 𝑢∈Ω_𝑅𝑡0 and 𝑡∈𝐽, by Lemma 6, (10), (19), and (20), we have ‖(𝑄𝑢)(𝑡)‖_𝛼≤𝑈(𝑡−𝑡₀)𝑥_0𝛼+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠_𝛼≤𝑀𝑥_0𝛼+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝐴^𝛼𝑉(⋅𝑓((𝑠𝑥𝑡−𝑠)𝑠,𝑢))𝑑𝑠≤𝑀_0𝛼+𝑞𝑀_𝛼𝐶^𝑡_𝑡0(𝑡−𝑠)^{𝑞(1−𝛼)−1}𝑑𝑠⋅^∞₀𝜃^1−𝛼𝜂_𝑞(𝜃)𝑥𝑑𝜃=𝑀_0𝛼+𝑀_𝛼𝐶Γ(2−𝛼)()Γ()ℎ1−𝛼1+𝑞1−𝛼^{𝑡𝑞(1−𝛼)0}≤𝑅_𝑡0. (21)
Let 𝑣₀≡𝜎𝑒₁. Then 𝑣₀(𝑡)=𝜎𝑒₁ for any 𝑡∈𝐽 and 𝜙(𝑡)≜𝐷^𝑞_𝑡0𝑣₀(𝑡)+𝐴𝑣₀(𝑡)=𝜆₁𝜎𝑒₁≤𝑓𝑡,𝜎𝑒₁,𝑡∈𝐽. (22) By the positivity of the semigroup 𝑆(𝑡)(𝑡≥0), assumption (𝐻₁), and (20), for any 𝑡∈𝐽, we have 𝜎𝑒₁=𝑣₀(𝑡)=𝑈𝑡−𝑡₀𝑣₀𝑡₀+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝜙(𝑠)𝑑𝑠≤𝑈𝑡−𝑡₀𝑥₀+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓𝑠,𝜎𝑒₁=𝑑𝑠𝑄𝜎𝑒₁(𝑡)≤(𝑄𝑢)(𝑡). (23) Thus, 𝑄∶Ω_𝑅𝑡0→Ω_𝑅𝑡0 is continuous.
By using a similar argument as in the proof of Theorem 3.1 in [14], we can prove that 𝑄∶Ω_𝑅𝑡0→Ω_𝑅𝑡0 is a compact operator. Hence by Schauder fixed point theorem, the operator 𝑄 has at least one fixed point 𝑢^∗ in Ω_𝑅𝑡0, which satisfies 𝑢^∗(𝑡)≥𝜎𝑒₁>0 for all 𝑡∈𝐽. Hence 𝑢^∗ is a positive mild solution of the IVP(1) on 𝐽.
Therefore, there exists [0,ℎ₀] such that the IVP(1) has at least one positive mild solution 𝑢^∗∈𝐶([0,ℎ₀],𝑋_𝛼). Now, by the standard proof method of extension theorem of initial value problem, 𝑢^∗ can be extended to a saturated solution 𝑢∈𝐶([0,𝑇),𝑋_𝛼) of the IVP(1), whose existence interval is [0,𝑇), and if 𝑇<∞, we have lim_𝑡→𝑇⁻‖𝑢(𝑡)‖_𝛼=∞.

For any 𝑇>0 and 𝑟>0, define Ω_𝑟 as in (15). If 𝑓(𝑡,𝑢) is increasing in Ω_𝑟, that is, 𝑓(𝑡,𝑢) satisfies the condition(𝐻₁)^∗ for any 𝑢₁,𝑢₂∈Ω_𝑟 with 𝑢₁(𝑡)≤𝑢₂(𝑡) for all 𝑡∈[0,𝑇], we have 𝑓𝑡,𝑢₁(𝑡)≤𝑓𝑡,𝑢₂(𝑡),𝑡∈[0,𝑇], (24)then we have 𝑓(𝑡,𝑢(𝑡))≥𝑓(𝑡,𝜎𝑒₁) for any 𝑢∈Ω_𝑟 and 𝑡∈[0,𝑇]. On the other hand, if 𝑓(𝑡,𝑢) satisfies linear growth condition, then it maps the bounded sets into the bounded sets. Hence by Theorem 12, we have the following existence result.

Corollary 13. Let −𝐴∶𝐷(𝐴)⊂𝑋→𝑋 be the infinitesimal generator of a positive and compact analytic semigroup 𝑆(𝑡)(𝑡≥0) of uniformly bounded linear operators. Assume that 𝑓∈𝐶(ℝ⁺×𝑋_𝛼,𝑋) satisfies condition (𝐻₁)* and (𝐻₂)* there exists a constant 𝑎_𝑓>0 such that 𝑓(𝑡,𝑥)≤𝑎_𝑓1+‖𝑥‖_𝛼 (25)for all 𝑡∈[0,𝑇] and 𝑥∈𝑋_𝛼.
If 𝑓(𝑡,𝜎𝑒₁)≥𝜆₁𝜎𝑒₁ for all 𝑡∈[0,𝑇], 𝑢₀∈𝑋_𝛼 with 𝑢₀≥𝜎𝑒₁ and 𝛼𝑞<1/2 for some 1/2<𝑞<1, then the IVP(1) has at least one positive mild solution 𝑢∈𝐶([0,𝑇),𝑋_𝛼). And if 𝑡<∞, one has lim_𝑡→𝑇⁻‖𝑢(𝑡)‖_𝛼=∞.

Since the analytic semigroup is norm continuous, it follows that we can delete the compactness condition on the analytic semigroup 𝑆(𝑡)(𝑡≥0) and obtain the following existence result.

Theorem 14. Assume that −𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is the infinitesimal generator of a positive analytic semigroup 𝑆(𝑡)(𝑡≥0) of uniformly bounded linear operators, and that 𝑓∈𝐶(ℝ⁺×𝑋_𝛼,𝑋) satisfies the condition (𝐻₁) and(𝐻₃) for any 𝑇>0 and 𝑟>0, 𝑓(𝑡,Ω_𝑟)∶={𝑓(𝑡,𝑢)∶𝑢∈Ω_𝑟} is relatively compact in 𝑋_𝛼 for all 𝑡∈[0,𝑇], where Ω_𝑟 is defined as in (15).If 𝑢₀∈𝑋_𝛼 with 𝑢₀≥𝜎𝑒₁ and 𝛼𝑞<1/2 for some 1/2<𝑞<1, then the IVP(1) has at least one positive mild solution 𝑢∈𝐶([0,𝑇),𝑋_𝛼). And if 𝑡<∞, one has lim_𝑡→𝑇⁻‖𝑢(𝑡)‖_𝛼=∞.

Proof. For any 𝑡₀≥0 and 𝑥₀∈𝑋_𝛼 with 𝑥₀≥𝜎𝑒₁, we first prove that the IVP(17) has at least one positive mild solution on 𝐽=[𝑡₀,𝑡₀+ℎ_𝑡0], where ℎ_𝑡0>0 is a constant and will be specified later. Define an operator 𝑄 as in (20). Let 𝑅_𝑡0=2𝑀(‖𝑥₀‖_𝛼+1)+𝜎𝑒₁. Write Ω_𝑅𝑡0 as in (18). The condition (𝐻₃) implies that 𝑓(𝑡,Ω_𝑅𝑡0) is bounded for any 𝑡∈𝐽, that is, there is a positive constant 𝐶=𝐶(𝑡₀) such that 𝑓(𝑡,𝑢(𝑡))≤𝐶,𝑡∈𝐽,𝑢∈Ω_𝑅𝑡0. (26) Let ℎ_𝑡0≤[𝑀(1−𝛼)Γ(1+𝑞(1−𝛼))(‖𝑥₀‖_𝛼+1)/𝑀_𝛼𝐶Γ(2−𝛼)]^{1/𝑞(1−𝛼)}. A similar argument as in the proof of Theorem 12 shows that 𝑄∶Ω_𝑅𝑡0→Ω_𝑅𝑡0 is continuous and 𝑄Ω_𝑅𝑡0 is equicontinuous.
Thus, for any 𝐷⊂Ω_𝑅𝑡0, let 𝐷(𝑡)∶={𝑢(𝑡)∶𝑢∈𝐷},𝑡∈𝐽. Since 𝑄𝐷⊂𝑄Ω_𝑅𝑡0⊂Ω_𝑅𝑡0 is equicontinuous and bounded, by Lemma 8, we have 𝛽(𝑄𝐷)=max_𝑡∈𝐽𝛽((𝑄𝐷)(𝑡)). (27)
Now, let 𝐷={𝑢_𝑚∶𝑚=1,2,…}⊂Ω_𝑅𝑡0 with 𝐷⊂Co({𝑦₀}∪𝑄𝐷) for some 𝑦₀∈Ω_𝑅𝑡0. It is obvious that (𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))_𝛼≤𝐶𝑀_𝛼𝑞Γ(2−𝛼)Γ()()1+𝑞1−𝛼𝑡−𝑠^𝑞−1∈𝐿𝐽,ℝ⁺. (28) Hence by Lemma 9 and (20), we have 𝛽𝑡𝑈((𝑄𝐷)())=𝛽𝑡−𝑡₀𝑥₀+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓(𝑠,𝐷(𝑠))𝑑𝑠=𝛽^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓(𝑠,𝐷(𝑠))𝑑𝑠≤2^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝐴^𝛼𝑉(𝑓((𝑠≤𝑡−𝑠)⋅𝛽𝑠,𝐷))𝑑𝑠2𝑀_𝛼𝑞Γ(2−𝛼)Γ1+𝑞(1−𝛼)^𝑡_𝑡0(𝑡−𝑠)^{𝑞(1−𝛼)−1}⋅𝛽𝑓(𝑠,𝐷(𝑠))𝑑𝑠=0. (29) It follows that 𝛽((𝑄𝐷)(𝑡))=0 for all 𝑡∈𝐽. By Lemma 8 and (27), we have 𝛽(𝑄𝐷)=max_𝑡∈𝐽𝛽((𝑄𝐷)(𝑡))=0. Thus, we have 𝛽(𝐷)≤𝛽Co𝑦₀𝑦∪𝑄𝐷=𝛽₀∪𝑄𝐷=𝛽(𝑄𝐷)=0. (30) This implies that 𝐷 is relatively compact. Therefore, by Mönch fixed point theorem, the operator 𝑄 has at least one fixed point 𝑢^∗∈Ω_𝑅𝑡0, which satisfies 𝑢^∗(𝑡)≥𝜎𝑒₁>0 for all 𝑡∈𝐽. Hence 𝑢^∗ is a positive mild solution of the IVP(17) on 𝐽.
Therefore, there exists [0,ℎ₀] such that the IVP(1) has at least one positive mild solution 𝑢^∗∈𝐶([0,ℎ₀],𝑋_𝛼). 𝑢^∗ can be extended to a saturated solution 𝑢∈𝐶([0,𝑇),𝑋_𝛼) of IVP(1), whose existence interval is [0,𝑇) and when 𝑡≤∞, we have lim_𝑡→𝑇⁻‖𝑢(𝑡)‖_𝛼=∞.

4. Positive Mild Solutions of Parabolic Equations

Let Ω⊂ℝ^𝑁 be a bounded domain with a sufficiently smooth boundary 𝜕Ω. Let 𝐴(𝑥,𝐷)𝑢=−^𝑁_𝑖,𝑗=1𝜕𝜕𝑥_𝑖𝑎_𝑖𝑗(𝑥)𝜕𝑢𝜕𝑥_𝑗+𝑎₀(𝑥)𝑢 (31) be a uniformly elliptic differential operator of divergence form in Ω, where the coefficients 𝑎_𝑖𝑗∈𝐶^1+𝜇(Ω)(𝑖,𝑗=1,2,…,𝑁) and 𝑎₀∈𝐶^𝜇(Ω) for some 𝜇∈(0,1). We assume that [𝑎_𝑖𝑗(𝑥)]_𝑁×𝑁 is a positive define symmetric matric for every 𝑥∈Ω, and there exists a constant 𝜈>0 such that ^𝑁_𝑖,𝑗=1𝑎_𝑖𝑗(𝑥)𝜉_𝑖𝜉_𝑗≥𝜈𝜉²,∀𝜉=𝜉₁,𝜉₂,…,𝜉_𝑁∈ℝ^𝑁,𝑥∈Ω. (32)

Let 𝑎₀(𝑥)≥0 on Ω. We use (𝑥,𝑡,𝜂) to denote a generic point of Ω×ℝ⁺×ℝ, where ℝ=[0,+∞) and ℝ=(−∞,+∞). Let 𝐹∶Ω×ℝ⁺×ℝ→ℝ be a continuous function. We discuss the existence of positive mild solutions for the parabolic initial boundary value problem (IBVP) 𝜕^𝑞𝜕𝑡^𝑞𝑢(𝑥,𝑡)+𝐴(𝑥,𝐷)𝑢(𝑥,𝑡)=𝐹(𝑥,𝑡,𝑢(𝑥,𝑡))inΩ×ℝ⁺,𝑢|_𝜕Ω𝑢()(𝑥)=0,𝑥,0=𝜑inΩ, (33) where 0<𝑞<1 is a constant.

Let 𝜆₁ be the smallest positive real eigenvalue of elliptic operator 𝐴(𝑥,𝐷) under the Dirichlet boundary condition 𝑢|_𝜕Ω=0. It is well known (cf. Amann [22, 28]) that 𝜆₁>0. Let 𝑒₁(𝑥) be the positive eigenvector corresponding to 𝜆₁. Assume that 𝐹∶Ω×ℝ⁺×ℝ→ℝ is continuous and satisfies the following conditions.(𝐹₁) For any 𝑇>0 and 𝑟>0, there exists a constant 𝜎>0 such that 𝐹𝑥,𝑡,𝜂≥𝐹𝑥,𝑡,𝜎𝑒₁(𝑥)≥𝜆₁𝜎𝑒₁(𝑥),𝑡∈[0,𝑇],𝑥∈Ω, (34)where 𝜂∈ℝ with 𝜎𝑒₁(𝑥)≤𝜂≤𝑟.(𝐹₂) For any 𝑇>0, there exists a constant 𝑎_𝑓>0 such that 𝐹𝑥,𝑡,𝜂≤𝑎_𝑓1+𝜂,𝑥∈Ω,𝑡∈[0,𝑇]. (35)

Let 𝑋=𝐿²(Ω). Define an operator 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 by 𝐷(𝐴)=𝐻²(Ω)∩𝐻¹₀(Ω),𝐴𝑢=𝐴(𝑥,𝐷)𝑢. (36) It is well known (cf. Li [29]) that −𝐴 generates a compact analytic semigroup 𝑆(𝑡)(𝑡≥0) and 𝐷(𝐴^1/2)=𝐻¹₀(Ω). By the maximum principle of the equation of the parabolic type, it is easy to prove that 𝑆(𝑡)(𝑡≥0) is also a positive semigroup in 𝑋. The assumptions (𝐹₁) and (𝐹₂) imply that the mapping 𝑓∶ℝ⁺×𝑋_1/2→𝑋 defined by 𝑓(𝑡,𝑢)(⋅)=𝐹(⋅,𝑡,𝑢(⋅)),𝑡∈ℝ⁺,𝑢∈𝐻¹₀(Ω), (37) is continuous and satisfies the conditions (𝐻₁) and (𝐻₂). Thus, the IBVP(33) can be rewritten into the abstract form of IVP(1). By Theorem 12, we have the following existence result for the IBVP(33).

Theorem 15. Assume that 𝐹∶Ω×ℝ⁺×ℝ→ℝ is continuous and satisfies conditions (𝐹₁) and (𝐹₂). If 𝜑∈𝐻¹₀(Ω) with 𝜑(𝑥)≥𝜎𝑒₁(𝑥) for any 𝑥∈Ω and 1/2<𝑞<1, then the IBVP(33) has at least one positive mild solution 𝑢  that satisfies 𝑢(𝑥,𝑡)≥𝜎𝑒₁(𝑥) for any 𝑥∈Ω and 𝑡∈[0,𝑇]. And if 𝑇<+∞, one has lim_𝑡→𝑇⁻|𝑢(𝑡)|=+∞.

Acknowledgments

This research was supported by the NNSF of China (Grant no. 11261053), the Fundamental Research Funds for the Gansu Universities, and the Project of NWNU-LKQN-11-3.