Fractional Evolution Equations Governed by Coercive Differential Operators

and Applied Analysis 3 2. α-Times Regularized Resolvent Family Throughout this paper,X is a complex Banach space, and we denote by B X the algebra of all bounded linear operators onX. LetA be a closed densely defined operator onX, letD A and R A be its domain and range, respectively, and let α ∈ 0, 2 , C ∈ B X be injective. Define ρC A : {λ ∈ C : λ − A is injective and R C ⊂ R λ − A }. Let Σθ : {λ ∈ C : |arg λ| < θ} be the open sector of angle 2θ in the complex plane, where arg is the branch of the argument between −π and π . Definition 2.1. A strongly continuous family {Sα t }t≥0 ⊂ B X is called an α -times C regularized resolvent family for A if a Sα 0 C; b Sα t A ⊂ ASα t for t ≥ 0; c C−1AC A; d for x ∈ D A , Sα t x Cx ∫ t 0 t − s α−1/Γ α Sα s Axds. {Sα t }t≥0 is called analytic if it can be extended analytically to some sector Σθ. If ‖Sα t ‖ ≤ Me t ≥ 0 for some constants M ≥ 1 and ω ∈ R , we will write A ∈ CαC M,ω , and C α C ω : ∪{C α C M,ω ;M ≥ 1}, C α C : ∪{C α C ω ;ω ≥ 0}. Define the operator Ã by Ãx C−1 ( lim t↓0 Γ α 1 tα Sα t x − Cx ) , x ∈ D ( Ã ) , 2.1


Introduction
It is well known that the abstract Cauchy problem of first order u t Au t , t > 0; u 0 x 1.1 is well posed if and only if A is the generator of a C 0 -semigroup.However, many partial differential operators PDOs such as the Schr ödinger operator iΔ on L p R n p / 2 cannot generate C 0 -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 X be a Banach space, and let A be a closed linear unbounded operator with densely defined domain D A .A family of strongly continuous bounded linear operators on X, {R t } t≥0 , is called a resolvent family for A with kernel a t ∈ L 1 loc R if R t A ⊂ AR t and the resolvent equation holds.It is obvious that a C 0 -semigroup is a resolvent family for its generator with kernel a 1 t ≡ 1; a cosine function is a resolvent family for its generator with kernel a 2 t t.If we define the α -times resolvent family for A as being a resolvent family with kernel g α t : t α−1 /Γ α , then such resolvent families interpolate C 0 -semigroups and cosine functions.
Recently Bazhlekova studied classes of such resolvent families see 10 .Let 0 < α ≤ 2, and let m be the smallest integer greater than or equal to α.It was shown in 10 that the fractional evolution equation of order α, is well posed if and only if there exists an α-times resolvent family for A.Here D α is the Caputo fractional derivative of order α > 0 defined by where f ∈ W m,1 I for every interval I.The hypothesis on f can be relaxed; see 10 for details.Fujita in 11 studied 1.3 for the case that A Δ, the Laplacian ∂/∂x 2 on R, which interpolates the heat equation and the wave equation.Since α-times resolvent families interpolate C 0 -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 α-times resolvent families for concrete PDOs in 10 , but Bazhlekova did not develop the theory of α-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 α-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 C-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 α-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 α-times regularized resolvent families for PDOs corresponding to coercive polynomials taking values in a sector of angle less than π.Some examples are also given in Section 4.

α-Times Regularized Resolvent Family
Throughout this paper, X is a complex Banach space, and we denote by B X the algebra of all bounded linear operators on X.Let A be a closed densely defined operator on X, let D A and R A be its domain and range, respectively, and let α ∈ 0, 2 , C ∈ B X be injective.Define be the open sector of angle 2θ in the complex plane, where arg is the branch of the argument between −π and π.

Definition 2.1. A strongly continuous family
can be extended analytically to some sector Σ θ .
If S α t ≤ Me ωt t ≥ 0 for some constants M ≥ 1 and ω ∈ R , we will write Proposition 2.2.Suppose that there exists an α-times C-regularized resolvent family, {S α t } t≥0 , for the operator A, and let A be defined as above.Then A A.
Proof.By the strong continuity of S α t , we have for every x ∈ X,

2.3
Thus for x ∈ D A , by Definition 2.1, which means that x ∈ D A and Ax Ax.On the other hand, for x ∈ D A , by the definition of A and Definition 2.1, The following generation theorem and subordination principle for α-times Cregularized resolvent families can be proved similarly as those for α-times resolvent families see 10 .

Coercive Operators and Mittag-Leffler Functions
We now introduce a functional calculus for generators of bounded C 0 -groups cf.14 , which will play a key role in our proof.
Let iA j 1 ≤ j ≤ n be commuting generators of bounded C 0 -groups on a Banach space X.Write A A 1 , . . ., A n and , where D j −i∂/∂x j for j 1, . . ., n.For a polynomial P ξ : In particular, F −1 u is the inverse Fourier transform of u if u ∈ S R n the space of rapidly decreasing functions on R n .We define u A ∈ B X by where ξ, A n j 1 ξ j A j .We will need the following lemma, in which the statements a and b are wellknown, c and d can be found in 14 and 6 , respectively.Lemma 3.1.a FL 1 R n is a Banach algebra under pointwise multiplication and addition with norm u FL 1 : where k ∈ N n 0 , then u ∈ FL 1 R n and u FL 1 ≤ MM n/2 0 for some constant M > 0.
Recall that the Mittag-Leffler function see 15, 16 is defined by where the path T is a loop which starts and ends at −∞ and encircles the disc |t| ≤ |z| 1/α 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 z → ∞.If 0 < α < 2, β > 0, then The following two lemmas are about derivatives of the Mittag-Leffler functions.
Proof.By the definition of E α,β z ,

3.10
as we wanted to show.
Lemma 3.3.Suppose that 1 < α < 2. For every n ∈ N and > 0 there exist constants M > 0 and L > 0 such that for k 0, . . ., n, Proof.First note that E α z 1/α E α,α z , and by induction on k one can prove that where a j only depend on α and k.Since α > 1 we have that αk − k − j > 0 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 r > 0, a polynomial as |ξ| → ∞.In the sequel, M is a generic constant independent of t which may vary from line to line.Lemma 3.4.Suppose that P ξ is an r-coercive polynomial of order m and {P ξ :

Existence of α-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 P is an r-coercive polynomial of order m, and {P ξ :

there exists an analytic α-times C-regularized resolvent family S α t for P A , and S α t E α t α P a − P
Proof.Let u t E α t α P a − P −γ , t ≥ 0. By Lemma 3.5, u t ∈ FL 1 and u t FL 1 ≤ M 1 t αn/2 .Define S α t u t A .Then by Lemma 3.1 b , S α t ∈ B X , S α t ≤ M 1 t αn/2 , and in particular C S α 0 ∈ B X .To check the strong continuity of S α t , take φ ∈ S R n .Then for t, t h ≥ 0, by Lemma 3.5 Since the set E of Lemma 3.1 is dense in X, we have done.Next we will show that In fact, for φ ∈ S R n , by Lemma 3.1 b and c we have Since FL 1 R n is a Banach algebra, it follows that u t , u t λ − P φ ∈ FL 1 .Thus by Lemmas 3.1, 3.5, 3.4 , and Fubini's theorem one obtains that for x ∈ X, λ > 0,

4.5
This implies that once again by the density of the set E of Lemma 3.1.A similar argument works to get Therefore, we have proved 4.3 .And it is routine to show that C −1 P A C P A , thus by Theorem 2.3 we know that S α t is the α-times C-regularized resolvent family for P A .Moreover, since α < α is arbitrary, by the subordination principle Theorem 2.4 we know that S α t is analytic.
We can extend this result to a more general case.
If P ξ falls into Ω, 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 P ξ within the triangle ΔGOB.To estimate D μ E α t α P ξ for such values P ξ , we use 3.12 , 3.15 , and 3.5 to obtain 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 P is coercive, but the choice of C is different.
Theorem 4.3.Suppose that P ξ is a polynomial of order m, and {P ξ : (which is defined by 3.1 with u x 1 |x| 2 −mβ/2 ), there exists an analytic α-times C-regularized resolvent family S α t for P A such that S α t ≤ M 1 t αn/2 exp ω 1/α t , t ≥ 0. 4.17 Proof.From 4.9 and From now on X will be The partial differential operator P D defined by is closed and densely defined on X.Since iD j ∂/∂x j 1 ≤ j ≤ n is the generator of the bounded C 0 -group {T j t } t∈R given by T j t f x 1 , . . ., x n f x 1 , . . ., x j−1 , x j t, x j 1 , . . ., x n t ∈ R 4.23 on X, we can apply the above results to P D on X.It is remarkable that when X L p R n 1 < p < ∞ these results can be improved.In fact, if A D D 1 , . . ., D n , then the functions u t 's in the proofs of the above theorems give rise to Fourier multipliers on L p R n having norm of polynomial growth t n p at infinity, where n p n|1/2 − 1/p|.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.