Fractional Evolution Equations Governed by Coercive Differential Operators
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 .
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  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., ). 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  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 ). Let , and let be the smallest integer greater than or equal to . It was shown in  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  for details. Fujita in  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 , but Bazhlekova did not develop the theory of -times resolvent families for general PDOs. The authors showed in  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 . 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 ).
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. ), 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 .
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.
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 ,
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.
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
Proof. From (4.9) and we have for , and for , It thus follows from Lemma 3.1 that when , . Similarly as in the proof of Theorem 4.1 we can show that there is an analytic -times -regularized resolvent family for .
From now on will be or . The partial differential operator defined by with is closed and densely defined on . Since is the generator of the bounded -group given by on , we can apply the above results to on . It is remarkable that when ( these results can be improved. In fact, if , then the functions 's in the proofs of the above theorems give rise to Fourier multipliers on having norm of polynomial growth at infinity, where . For details we refer to [3, 8]). We summarize these conclusions in the following two theorems.
Theorem 4.4. Suppose that the assumptions of Theorem 4.2 are satisfied. (a)For or , , where , there exists an analytic -times -regularized resolvent family for and (b)For , , where , , there exists an analytic -times -regularized resolvent family for and
Theorem 4.5. Suppose that the assumptions of Theorem 4.3 are satisfied.(a)For or , , where , there exists an analytic -times -regularized resolvent family for and (b)For , , where , there exists an analytic -times -regularized resolvent family for and
We end this paper with some examples to demonstrate the applications of our results.
Example 4.6. (a) The polynomial corresponding to the Laplacian on () is . By Theorem 4.4, for every there exists an analytic -times -regularized resolvent family for the operator , where .(b) Consider on () with Then . We claim that is -coercive. Indeed, if , then If , then for some proper constant , as desired. By Theorems 4.4 and 4.5, for every there exists an analytic -times -regularized resolvent family for , where with or with . We remark that if and , then (see ). Since , it follows from [18, Theorem 1] that . Consequently, in this case there is no -times resolvent family for for any .
The authors are very grateful to the referees for many helpful suggestions to improve this paper. The first and second authors were supported by the NSF of China (Grant no. 10501032) and NSFC-RFBR Programm (Grant no. 108011120015), and the third by TRAPOYT and the NSF of China (Grant no. 10671079).
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