Fractional Evolution Equations Governed by Coercive Differential Operators

Abstract

This paper is concerned with evolution equations of fractional order where is a differential operator corresponding to a coercive polynomial taking values in a sector of angle less than and . We show that such equations are well posed in the sense that there always exists an -times resolvent family for the operator .

1. Introduction

It is well known that the abstract Cauchy problem of first order is well posed if and only if is the generator of a -semigroup. However, many partial differential operators (PDOs) such as the Schrödinger operator on () cannot generate -semigroups. It was Kellermann and Hieber [1] who first showed that some elliptic differential operators on some function spaces generate integrated semigroups, and their results are improved and developed in [2, 3]. Because of the limitations of integrated semigroups, the results in [1–3] are confined to elliptic differential operators with constant coefficients. One of the limitations is that the resolvent sets of generators must contain a right half-plane; however, it is known that there are many nonelliptic operators whose resolvent sets are empty (see, e.g., [4]). On the other hand, the resolvent sets of the generators of regularized semigroups need not be nonempty; this makes it possible to apply the theory of regularized semigroups to nonelliptic operators, such as coercive operators and hypoelliptic operators (see [5–8]). Moreover, for second-order equations, Zheng [9] considered coercive differential operators with constant coefficients generating integrated cosine functions. The aim of this paper is to consider fractional evolution equations associated with coercive differential operators.

Let be a Banach space, and let be a closed linear unbounded operator with densely defined domain . A family of strongly continuous bounded linear operators on , , is called a resolvent family for with kernel if and the resolvent equation holds. It is obvious that a -semigroup is a resolvent family for its generator with kernel ; a cosine function is a resolvent family for its generator with kernel . If we define the -times resolvent family for as being a resolvent family with kernel , then such resolvent families interpolate -semigroups and cosine functions.

Recently Bazhlekova studied classes of such resolvent families (see [10]). Let , and let be the smallest integer greater than or equal to . It was shown in [10] that the fractional evolution equation of order , is well posed if and only if there exists an -times resolvent family for . Here is the Caputo fractional derivative of order defined by where for every interval . The hypothesis on can be relaxed; see [10] for details. Fujita in [11] studied (1.3) for the case that , the Laplacian on , which interpolates the heat equation and the wave equation. Since -times resolvent families interpolate -semigroups and cosine functions, this motivates us to consider the existence of fractional resolvent families for PDOs.

There are several examples of the existence of -times resolvent families for concrete PDOs in [10], but Bazhlekova did not develop the theory of -times resolvent families for general PDOs. The authors showed in [12] that there exist fractional resolvent families for elliptic operators. In this paper we will consider coercive operators. Since -times resolvent families are not sufficient for applications we have in mind, we first extend, in Section 2, such a notion to the setting of -regularized resolvent families which was introduced in [13]. To do this, we use methods of the Fourier multiplier theory.

This paper is organized as follows. Section 2 contains the definition and some basic properties of -times regularized resolvent families. Section 3 prepares for the proof of the main result of this paper. Our main result, Theorem 4.1, shows that there are -times regularized resolvent families for PDOs corresponding to coercive polynomials taking values in a sector of angle less than . Some examples are also given in Section 4.

2. -Times Regularized Resolvent Family

Throughout this paper, is a complex Banach space, and we denote by the algebra of all bounded linear operators on . Let be a closed densely defined operator on , let and be its domain and range, respectively, and let , be injective. Define . Let be the open sector of angle in the complex plane, where arg is the branch of the argument between and .

Definition 2.1. A strongly continuous family is called an -times -regularized resolvent family for if(a);(b) for ;(c);(d)for , . is called analytic if it can be extended analytically to some sector .

If () for some constants and , we will write , and , .

Define the operator by with

Proposition 2.2. Suppose that there exists an -times -regularized resolvent family, , for the operator , and let be defined as above. Then .

Proof. By the strong continuity of , we have for every , Thus for , by Definition 2.1, which means that and . On the other hand, for , by the definition of and Definition 2.1, but , by (d) of Definition 2.1. Thus it follows from the closedness of that with . This implies that , so we have .

The following generation theorem and subordination principle for -times -regularized resolvent families can be proved similarly as those for -times resolvent families (see [10]).

Theorem 2.3. Let . Then the following statements are equivalent:(a);(b), and (c), and there exists a strongly continuous family satisfying such that

Theorem 2.4. Suppose that , . If then and the -times -regularized resolvent family for , , can be extended analytically to , where .

3. Coercive Operators and Mittag-Leffler Functions

We now introduce a functional calculus for generators of bounded -groups (cf. [14]), which will play a key role in our proof.

Let be commuting generators of bounded -groups on a Banach space . Write and for . Similarly, write , where for . For a polynomial with constant coefficients, we define with maximal domain. Then is closable. Let be the Fourier transform, that is, for , where . If , then there exists a unique function in , written , such that . In particular, is the inverse Fourier transform of if (the space of rapidly decreasing functions on ). We define by where .

We will need the following lemma, in which the statements (a) and (b) are well-known, (c) and (d) can be found in [14] and [6], respectively.

Lemma 3.1. (a) is a Banach algebra under pointwise multiplication and addition with norm .
(b) is an algebra homomorphism from into , and there exists a constant such that .
(c) , , and for .
(d) Let . Suppose that there exist constants , , , and such that where , then and for some constant .

Recall that the Mittag-Leffler function (see [15, 16]) is defined by where the path is a loop which starts and ends at and encircles the disc in the positive sense. The most interesting properties of the Mittag-Leffler functions are associated with their Laplace integral and with their asymptotic expansion as . If , , then where as , and the -term is uniform in if . Note that for ,

The following two lemmas are about derivatives of the Mittag-Leffler functions.

Lemma 3.2.

Proof. By the definition of , as we wanted to show.

For short, .

Lemma 3.3. Suppose that . For every and there exist constants and such that for ,

Proof. First note that , and by induction on one can prove that where only depend on and . Since we have that whence, by the asymptotic formula for Mittag-Leffler functions (3.6), we obtain (3.11).

Now let us recall the definition of coercive polynomials. For fixed , a polynomial is called -coercive if as . In the sequel, is a generic constant independent of which may vary from line to line.

Lemma 3.4. Suppose that is an -coercive polynomial of order and , where . Let . Then for , , , there exist constants such that

Proof. Suppose that for , (3.11) holds up to order and By induction, one can show that where . Thus if and , and if with , by (3.8) and (3.12) we know that Altogether, we have And by and Leibniz's formula we have

Lemma 3.5. This proves (3.13). Suppose that the assumptions of Lemma 3.4 are satisfied. Let . Then and The same result holds with replaced by .

Proof. By Lemma 3.1(d), it remains to prove that for , To show this we can use (3.15) and then give the estimates according to the values . For with the estimate (3.8) can be applied, and for with note that all the functions are uniformly bounded.
For the second part of the lemma, note that .

4. Existence of -Times Regularized Resolvents for Operator Polynomials

In this section, we will construct the fractional regularized resolvent families for coercive differential operators on Banach spaces.

Theorem 4.1. Suppose that is an -coercive polynomial of order , and , where . Then for , , , , there exists an analytic -times -regularized resolvent family for , and with

Proof. Let , . By Lemma 3.5, and . Define . Then by Lemma 3.1(b), , , and in particular . To check the strong continuity of , take . Then for , by Lemma 3.5 Since the set of Lemma 3.1 is dense in , we have done. Next we will show that In fact, for , by Lemma 3.1(b) and (c) we have Since is a Banach algebra, it follows that . Thus by Lemmas 3.1, 3.5, (3.4), and Fubini's theorem one obtains that for , , This implies that once again by the density of the set of Lemma 3.1. A similar argument works to get Therefore, we have proved (4.3). And it is routine to show that , thus by Theorem 2.3 we know that is the -times -regularized resolvent family for . Moreover, since is arbitrary, by the subordination principle (Theorem 2.4) we know that is analytic.

We can extend this result to a more general case.

Theorem 4.2. Let be an -coercive polynomial of order such that for some , and let . Then for , , , and , there exists an analytic -times -regularized resolvent family for with

Proof. We only consider the area above the -axis, the lower area can be treated similarly.
Let Ray 1 , and let Ray 2 , where . Let be the point , and set to denote the intersection point of the two above rays. Let denote the region to the left side of Ray 1 and 2 (see Figure 1).
If falls into , the asymptotic formula (3.6) can be applied to get estimates similarly as in the proof of Theorem 4.1. It remains to consider the values within the triangle . To estimate for such values , we use (3.12), (3.15), and (3.5) to obtain if Re, where denotes an arbitrary point on the line segment from to .
Since we have Thus, to show that Re () one needs to check that and this is true if since .
We first consider the case when . Let , then since and for . So decreases with respect to , which means that since .
For , we will show that which implies (4.13). Now for fixed , denote by . Since , we have ; it thus follows that . Therefore we have proved (4.14).
Now by (3.19) and (4.9) one obtains, for , and for , An argument similar to that one of the proof of Theorem 4.1 gives our claim.

Figure 1 

In the following theorem, we do not assume that is coercive, but the choice of is different.

Theorem 4.3. Suppose that is a polynomial of order , and , where . Then for , , (which is defined by (3.1) with ), there exists an analytic -times -regularized resolvent family for such that