# Fractional Evolution Equations Governed by Coercive Differential Operators

Authors: Fu-Bo Li; Miao Li; Quan Zheng
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

This paper is concerned with evolution equations of fractional orderDαu(t)=Au(t);u(0)=u0,u′(0)=0, where A is a differential operator corresponding to a coercive polynomial taking values in a sector of angle less than π and 1<α<2. We show that such equations are well posed in the sense that there always exists an α-times resolvent family for the operator A.


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## Body

## 1. Introduction

It is well known that the abstract Cauchy problem of first order(1.1)u′(t)=Au(t),t>0;u(0)=x
is well posed if and only if A is the generator of a C0-semigroup. However, many partial differential operators (PDOs) such as the Schrödinger operator  iΔ  on  Lp(ℝn) (p≠2) cannot generate C0-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.LetX 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)∈Lloc1(ℝ+) if R(t)A⊂AR(t) and the resolvent equation(1.2)R(t)x=x+∫0t‍a(t-s)AR(s)xds,t≥0,x∈D(A)
holds. It is obvious that a C0-semigroup is a resolvent family for its generator with kernel a1(t)≡1;  a cosine function is a resolvent family for its generator with kernel a2(t)=t. If we define the  α-times resolvent family for A as being a resolvent family with kernel gα(t):=tα-1/Γ(α), then such resolvent families interpolate C0-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,
(1.3)Dαu(t)=Au(t),t>0;u(k)(0)=xk,k=0,1,…,m-1,
is well posed if and only if there exists an α-times resolvent family for A. Here Dα is the Caputo fractional derivative of order α>0 defined by
(1.4)Dαf(t):=∫0t‍gm-α(t-s)dmdsmf(s)ds,
where f∈Wm,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 C0-semigroups and cosine functions, this motivates us to consider the existence of fractional resolvent families for PDOs.There are several examples of the existence ofα-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. Section2 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.


## 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 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 ρC(A):={λ∈ℂ:λ-Ais injective andR(C)⊂R(λ-A)}. Let Σθ:={λ∈ℂ:|argλ|<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-timesC-regularized resolvent family forA if(a)
Sα(0)=C;(b)
Sα(t)A⊂ASα(t) for t≥0;(c)
C-1AC=A;(d)
forx∈D(A), Sα(t)x=Cx+∫0t‍((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)∥≤Meωt (t≥0) for some constants M≥1 and ω∈ℝ+, we will write A∈𝒞Cα(M,ω), and 𝒞Cα(ω):=∪{𝒞Cα(M,ω);M≥1}, 𝒞Cα:=∪{𝒞Cα(ω);ω≥0}.Define the operatorÃ by
(2.1)Ãx=C-1(limt↓0Γ(α+1)tα(Sα(t)x-Cx)),x∈D(Ã),
with
(2.2)D(Ã)={x∈X:limt↓0Sα(t)x-CxtαexistsandisinR(C)}.Proposition 2.2.
Suppose that there exists anα-times C-regularized resolvent family,  {Sα(t)}t≥0, for the operator A, and let  Ã be defined as above. Then A=Ã.Proof.
By the strong continuity ofSα(t), we have for every x∈X,
(2.3)∥gα+1(t)-1∫0t‍gα(t-s)Sα(s)xds-Cx∥≤gα+1(t)-1∫0t‍gα(t-s)∥Sα(s)x-Cx∥ds≤sup0≤s≤t∥Sα(s)x-Cx∥→0ast→0.
Thus for x∈D(A), by Definition 2.1,
(2.4)limt↓0gα+1(t)-1(Sα(t)x-Cx)=limt↓0gα+1(t)-1∫0t‍gα(t-s)Sα(s)Axds=CAx,
which means that x∈D(Ã) and Ãx=Ax. On the other hand, for x∈D(Ã), by the definition of Ã and Definition 2.1,
(2.5)CÃx=limt↓0gα+1(t)-1(Sα(t)x-Cx)=limt↓0gα+1(t)-1A∫0t‍gα(t-s)Sα(s)xds,
but limt→0gα+1(t)-1∫0t‍gα(t-s)Sα(s)xds=Cx, by (d) of Definition 2.1. Thus it follows from the closedness of A that Cx∈D(A) with ACx=CÃx. This implies that x∈D(C-1AC)=D(A), so we have Ã=A.The following generation theorem and subordination principle forα-times C-regularized resolvent families can be proved similarly as those for α-times resolvent families (see [10]).Theorem 2.3.
Letα∈(0,2]. Then the following statements are equivalent:(a)
A∈𝒞Cα(M,ω);(b)
A=C-1AC, (ωα,∞)⊆ρC(A) and
(2.6)∥dndλn(λα-1(λα-A)-1C)∥≤Mn!(λ-ω)n+1,λ>ω,n∈ℕ0:=ℕ∪{0};(c)
A=C-1AC, (ωα,∞)⊆ρC(A) and there exists a strongly continuous family {Sα(t)}t≥0⊂B(X) satisfying ∥Sα(t)∥≤Meωt such that
(2.7)λα-1(λα-A)-1Cx=∫0∞‍e-λtSα(t)xdt,λ>ω,x∈X.Theorem 2.4.
Suppose that0<α<β≤2, γ=α/β. If A∈𝒞Cβ(ω) then A∈𝒞Cα(ω1/γ) and the α-times C-regularized resolvent family for A, {Sα(t)}t≥0, can be extended analytically to Σmin{θ(γ),π}, where θ(γ):=(1/γ-1)π/2.


## 3. Coercive Operators and Mittag-Leffler Functions

We now introduce a functional calculus for generators of boundedC0-groups (cf. [14]), which will play a key role in our proof.LetiAj(1≤j≤n) be commuting generators of bounded C0-groups on a Banach space X. Write A=(A1,…,An) and Aμ=A1μ1⋯Anμn for μ=(μ1,…,μn)∈ℕ0n. Similarly, write Dμ=D1μ1⋯Dnμn, where Dj=-i∂/∂xj for j=1,…,n. For a polynomial P(ξ):=∑|μ|≤m‍aμξμ(ξ∈ℝn)(|μ|:=∑j=1n‍μj) with constant coefficients, we define P(A)=∑|μ|≤m‍aμAμ(ξ∈ℝn) with maximal domain. Then P(A) is closable. Let ℱ be the Fourier transform, that is, (ℱu)(η)=∫ℝn‍u(ξ)e-i(ξ,η)dξ for u∈L1(ℝn), where (ξ,η)=∑j=1n‍ξjηj. If u∈ℱL1(ℝn):={ℱv:v∈L1(ℝn)}, then there exists a unique function in L1(ℝn), written ℱ-1u, such that u=ℱ(ℱ-1u). In particular, ℱ-1u is the inverse Fourier transform of u if u∈𝒮(ℝn) (the space of rapidly decreasing functions on ℝn). We define u(A)∈B(X) by
(3.1)u(A)x=∫ℝn‍(ℱ-1u)(ξ)e-i(ξ,A)xdξ,x∈X,
where (ξ,A)=∑j=1n‍ξjAj.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)ℱL1(ℝn) is a Banach algebra under pointwise multiplication and addition with norm ∥u∥ℱL1:=∥ℱ-1u∥L1.
(b)u↦u(A) is an algebra homomorphism from ℱL1(ℝn) into B(X), and there exists a constant M>0 such that ∥u(A)∥≤M∥u∥ℱL1.
(c)E:={ϕ(A)x:ϕ∈𝒮(ℝn),x∈X}⊂∩μ∈ℕ0nD(Aμ), E¯=X, P(A)|E¯=P(A)¯ and ϕ(A)P(A)⊂P(A)ϕ(A)=(Pϕ)(A) for ϕ∈𝒮(ℝn).
(d) Letu∈Cj(ℝn)(j>n/2). Suppose that there exist constants L, M0, a>0, and b∈[-1,2a/n-1) such that
(3.2)|Dku(ξ)|≤{M0|k||ξ|b|k|-a,for|ξ|≥L,|k|≤j,M0|k|,for|ξ|<L,|k|≤j,
where k∈ℕ0n, then u∈ℱL1(ℝn) and ∥u∥ℱL1≤MM0n/2 for some constant M>0.Recall that the Mittag-Leffler function (see [15, 16]) is defined by
(3.3)Eα,β(z):=∑n=0∞‍znΓ(αn+β)=12πi∫𝒯‍μα-βeμμα-zdμ,α,β>0,z∈ℂ,
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
(3.4)∫0∞‍e-λttβ-1Eα,β(ωtα)dt=λα-βλα-ω,Reλ>ω1/α,ω>0
and with their asymptotic expansion as z→∞. If 0<α<2,  β>0, then
(3.5)Eα,β(z)=1αz(1-β)/αexp(z1/α)+εα,β(z),|argz|≤12απ,(3.6)Eα,β(z)=εα,β(z),|arg(-z)|<(1-12α)π,
where
(3.7)εα,β(z)=-∑n=1N-1‍z-nΓ(β-αn)+O(|z|-N)
as z→∞, and the O-term is uniform in argz if |arg(-z)|≤(1-α/2-ϵ)π. Note that for β>0,
(3.8)|Eα,β(z)|≤Eα,β(|z|),z∈ℂ.The following two lemmas are about derivatives of the Mittag-Leffler functions.Lemma 3.2.
(3.9)Eα,β′(z)=1αEα,α+β-1(z)-β-1αEα,α+β(z).Proof.
By the definition ofEα,β(z),
(3.10)Eα,β′(z)=(∑n=0∞‍znΓ(αn+β))′=∑n=1∞‍nzn-1Γ(αn+β)=∑n=1∞‍nzn-1Γ(αn+β-1+1)=∑n=1∞‍zn-1·(αn+β-1)α(αn+β-1)Γ(αn+β-1)-∑n=1∞‍β-1α·zn-1Γ(αn+β)=1αEα,α+β-1(z)-β-1αEα,α+β(z),
as we wanted to show.For short,Eα(z):=Eα,1(z).Lemma 3.3.
Suppose that1<α<2. For every n∈ℕ and ϵ>0 there exist constants M>0 and L>0 such that for k=0,…,n,
(3.11)|Eα(k)(z)|≤M|z|,if|z|≥L,|arg(-z)|≤(1-α2-ϵ)π.Proof.
First note thatEα′(z)=(1/α)Eα,α(z), and by induction on k one can prove that
(3.12)Eα(k)(z)=∑j=1k‍ajEα,αk-(k-j)(z),
where aj 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 fixedr>0, a polynomial P(ξ) is called  r-coercive if |P(ξ)|-1=O(|ξ|-r) as |ξ|→∞. In the sequel, M is a generic constant independent of t which may vary from line to line.Lemma 3.4.
Suppose thatP(ξ) is an r-coercive polynomial of order m and  {P(ξ):ξ∈ℝn}⊂ℂ∖Σα′π/2, where 1<α′<2. Let k0=[n/2]+1. Then for 1<α<α′, γ>0, a∈Σα′π/2, there exist constants M,L≥0 such that
(3.13)|Dμ[Eα(tαP)(a-P)-γ]|≤M(1+tα|μ|)|ξ|(m-1)|μ|-rγ,|ξ|≥L,|μ|≤k0,t≥0.Proof.
Suppose that for|ξ|≥L, (3.11) holds up to order k0 and
(3.14)|P(ξ)|≥M|ξ|r,|a-P(ξ)|≥M|ξ|r.
By induction, one can show that
(3.15)DμEα(tαP)=∑j=1|μ|‍tαjEα(j)(tαP)Qj,
where degQj≤mj-|μ|. Thus if |ξ|≥L and |tαP|≥L,
(3.16)|DμEα(tαP)|≤M(1+tα(|μ|-1))|ξ|m|μ|-|μ|-r≤M(1+tα|μ|)|ξ|(m-1)|μ|-r,
and if |ξ|≥L with |tαP|≤L, by (3.8) and (3.12) we know that
(3.17)|DμEα(tαP)|≤M(tα+tα|μ|)|ξ|(m-1)|μ|.
Altogether, we have
(3.18)|DμEα(tαP)|≤M(1+tα|μ|)|ξ|(m-1)|μ|,|ξ|≥L.
And by
(3.19)|Dμ(a-P)-γ|≤M|ξ|(m-r-1)|μ|-rγ,|ξ|≥L
and Leibniz's formula we have
(3.20)|Dμ(Eα(tαP)(a-P)-γ)|≤M(1+tα|μ|)|ξ|(m-1)|μ|-rγ,|ξ|≥L.Lemma 3.5.
This proves (3.13). Suppose that the assumptions of Lemma3.4 are satisfied. Let γ>nm/2r. Then Eα(tαP)(a-P)-γ∈ℱL1(ℝn) and
(3.21)∥Eα(tαP)(a-P)-γ∥ℱL1≤M(1+tαn/2),t≥0.
The same result holds with Eα(tαP) replaced by Eα,α(tαP).Proof.
By Lemma3.1(d), it remains to prove that for |ξ|≤L,
(3.22)|Dμ(Eα(tαP)(a-P)-γ)|≤M(1+tα|μ|),|μ|≤k0,t≥0.
To show this we can use (3.15) and then give the estimates according to the values tαP. For |ξ|≤L with |tαP|≥L the estimate (3.8) can be applied, and for |ξ|≤L with |tαP|≤L note that all the functions Eα(j)(tαP) are uniformly bounded.
For the second part of the lemma, note thatEα,α(z)=αEα′(z).


## 4. Existence ofα-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 thatP is an r-coercive polynomial of order m, and {P(ξ):ξ∈ℝn}⊂ℂ∖Σα′π/2, where 1<α′<2. Then for 1<α<α′,  a∈Σα′π/2, γ>nm/2r,  C=(a-P)-γ(A), there exists an analytic α-times C-regularized resolvent family Sα(t) for P(A)¯, and Sα(t)=(Eα(tαP)(a-P)-γ)(A) with
(4.1)∥Sα(t)∥≤M(1+tαn/2),t≥0.Proof.
Letut=Eα(tαP)(a-P)-γ, t≥0. By Lemma 3.5, ut∈ℱL1 and ∥ut∥ℱL1≤M(1+tαn/2). Define Sα(t)=ut(A). Then by Lemma 3.1(b), Sα(t)∈B(X),  ∥Sα(t)∥≤M(1+tαn/2), and in particular C=Sα(0)∈B(X). To check the strong continuity of Sα(t), take ϕ∈𝒮(ℝn). Then for t,t+h≥0, by Lemma 3.5(4.2)∥Sα(t+h)ϕ(A)-Sα(t)ϕ(A)∥≤M∥(Eα((t+h)αP)-Eα(tαP))(a-P)-γϕ∥ℱL1≤M∥∫tt+h‍sα-1Eα,α(sαP)(a-P)-γPϕds∥ℱL1≤M∫tt+h‍sα-1(1+sαn/2)ds·∥Pϕ∥ℱL1→0,ash→0.
Since the set E of Lemma 3.1 is dense in X, we have done. Next we will show that
(4.3)λα-1(λα-P(A)¯)-1C=∫0∞‍e-λtSα(t)dt,λ>0.
In fact, for ϕ∈𝒮(ℝn), by Lemma 3.1(b) and (c) we have
(4.4)P(A)Sα(t)ϕ(A)=(Putϕ)(A)=Sα(t)P(A)ϕ(A).
Since ℱL1(ℝn) is a Banach algebra, it follows that ut,ut(λ-P)ϕ∈ℱL1. Thus by Lemmas 3.1,   3.5, (3.4), and Fubini's theorem one obtains that for x∈X, λ>0,
(4.5)∫0∞‍e-λtSα(t)(λα-P(A))ϕ(A)xdt=∫0∞‍e-λt(ut(λα-P)ϕ)(A)xdt=(∫0∞‍e-λtutdt(λα-P)ϕ)(A)x=λα-1Cϕ(A)x.
This implies that
(4.6)∫0∞‍e-λtSα(t)(λα-P(A)¯)xdt=λα-1Cx,x∈D(P(A)¯),
once again by the density of the set E of Lemma 3.1. A similar argument works to get
(4.7)(λα-P(A)¯)∫0∞‍e-λtSα(t)xdt=λα-1Cx,x∈X.
Therefore, we have proved (4.3). And it is routine to show that C-1P(A)¯C=P(A)¯, thus by Theorem 2.3 we know that Sα(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 Sα(t) is analytic.We can extend this result to a more general case.Theorem 4.2.
LetP(ξ) be an r-coercive polynomial of order m such that {P(ξ):ξ∈ℝn}⊂ℂ∖(ω+Σα′π/2) for some ω≥0, and let α′>1. Then for 1<α<α′, a∈ω+Σα′π/2, γ>nm/2r, and C=(a-P)-γ(A), there exists an analytic α-times C-regularized resolvent family Sα(t) for P(A)¯ with
(4.8)∥Sα(t)∥≤M(1+tαn/2)exp(ω1/αt),t≥0.Proof.
We only consider the area above thex-axis, the lower area can be treated similarly.
Let Ray 1:={ω+ρeiα′π/2:0≤ρ<∞},  and let Ray 2 :={ρeiαπ/2:0≤ρ<∞}, where 1<α<α′<2. Let G be the point (ω,0), and set B to denote the intersection point of the two above rays. Let Ω denote the region to the left side of Ray 1 and 2 (see Figure 1).
IfP(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 DμEα(tαP(ξ)) for such values P(ξ), we use (3.12), (3.15), and (3.5) to obtain
(4.9)|DμEα(tαP(ξ))|≤M(1+tα|μ|)exp(ω1/αt)
if Re((ρeiθ)1/α)<ω1/α, where ρeiθ denotes an arbitrary point on the line segment from G to B.
Since(4.10)ρsin(α′π/2)=ωsin(α′π/2-θ),
we have
(4.11)Re((ρeiθ)1/α)=ω1/α(sin(α′π/2)sin(α′π/2-θ))1/αcos(θ/α).
Thus, to show that Re((ρeiθ)1/α)<ω1/α (0<θ≤απ/2) one needs to check that
(4.12)cosα(θ/α)<cosθ+sinθ·tan(α′-12π),0<θ≤απ2;
and this is true if
(4.13)cos(θ/α)≤cosθ+sinθ·tan(α-12π),0<θ≤απ2,
since 1<α<α′<2.
We first consider the case whenπ/2≤θ≤απ/2. Let g(θ)=cosθ+sinθ·tan(((α-1)/2)π)-cos(θ/α), then g′(θ)=-sinθ+(1/α)sin(θ/α)+cosθ·tan(((α-1)/2)π)≤0 since sinθ>(1/α)sin(θ/α) and cosθ≤0 for π/2≤θ≤απ/2. So g(θ) decreases with respect to θ, which means that g(θ)≥0 since g(απ/2)=0.
For0<θ<π/2, we will show that
(4.14)cos(θ/α)≤cosθ+sinθ·α-12π,
which implies (4.13). Now for fixed θ∈(0,π/2), denote by h(α)=cosθ+sinθ·((α-1)/2)π-cos(θ/α). Since α>1, we have h′(α)=(π/2)sinθ-(θ/α2)sin(θ/α)>0; it thus follows that h(α)≥h(1)=0. Therefore we have proved (4.14).
Now by (3.19) and (4.9) one obtains, for |ξ|≥L,
(4.15)|Dμ(Eα(tαP)(a-P)-γ)|≤M(1+tα|μ|)|ξ|(m-1)|μ|-rγexp(ω1/αt),
and for |ξ|≤L,
(4.16)|Dμ(Eα(tαP)(a-P)-γ)|≤M(1+tα|μ|)exp(ω1/αt).
An argument similar to that one of the proof of Theorem 4.1 gives our claim.Figure 1In the following theorem, we do not assume thatP is coercive, but the choice of C is different.Theorem 4.3.
Suppose thatP(ξ) is a polynomial of order m, and {P(ξ):ξ∈ℝn}⊂ℂ∖(ω+Σα′π/2), where 1<α′<2. Then for 1<α<α′<2, β>n/2, C=(1+|A|2)-mβ/2 (which is defined by (3.1) with u(x)=(1+|x|2)-mβ/2), there exists an analytic α-times C-regularized resolvent family Sα(t) for P(A)¯ such that
(4.17)∥Sα(t)∥≤M(1+tαn/2)exp(ω1/αt),t≥0.Proof.
From (4.9) and
(4.18)|Dμ(1+|ξ|2)-β/2|≤M|ξ|-|μ|-β,|ξ|≥L,μ∈ℕ0n,
we have for |ξ|≥L,
(4.19)|Dμ(Eα(tαP)(1+ξ|2)-β/2)|≤M(1+tα|μ|)|ξ|(m-1)|μ|-βexp(ω1/αt),
and for |ξ|≤L,
(4.20)|Dμ(Eα(tαP)(1+|ξ|2)-β/2)|≤M(1+tα|μ|)exp(ω1/αt).
It thus follows from Lemma 3.1 that when β>nm/2, Eα(tαP)(1+|ξ|2)-β/2∈ℱL1(ℝn). Similarly as in the proof of Theorem 4.1 we can show that there is an analytic α-times C-regularized resolvent family for P(A)¯.From now onX will be Lp(ℝn)(1≤p<∞) or C0(ℝn):={f∈C(ℝn):lim|x|→∞f(x)=0}. The partial differential operator P(D) defined by
(4.21)P(D)f=ℱ-1(Pℱf)
with
(4.22)D(P(D))={f∈X:ℱ-1(Pℱf)∈X}
is closed and densely defined on X. Since iDj=∂/∂xj(1≤j≤n) is the generator of the bounded C0-group {Tj(t)}t∈ℝ given by
(4.23)Tj(t)f(x1,…,xn)=f(x1,…,xj-1,xj+t,xj+1,…,xn)t∈ℝ
on X, we can apply the above results to P(D) on X. It is remarkable that when X=Lp(ℝn) (1<p<∞) these results can be improved. In fact, if A=D=(D1,…,Dn), then the functions ut's in the proofs of the above theorems give rise to Fourier multipliers on Lp(ℝn) having norm of polynomial growth tnp at infinity, where  np=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 Theorem4.2 are  satisfied. (a)
ForX=L1(ℝn) or C0(ℝn), C=(a-P)-γ(D), where γ>nm/2r, there exists an analytic α-times C-regularized resolvent family Sα(t) for P(D) and
(4.24)∥Sα(t)∥≤M(1+tαn/2)exp(ω1/αt),t≥0.(b)
ForX=Lp(ℝn), C=(a-P)-γ(D), where γ>npm/r, np=n|1/2-1/p|, there exists an analytic α-times C-regularized resolvent family Sα(t) for P(D) and(4.25)∥Sα(t)∥≤M(1+tαnp)exp(ω1/αt),t≥0.Theorem 4.5.
Suppose that the assumptions of Theorem4.3 are satisfied.(a)
ForX=L1(ℝn) or C0(ℝn), C=(1-Δ)-mβ/2, where β>n/2, there exists an analytic α-times C-regularized resolvent family Sα(t) for P(D) and
(4.26)∥Sα(t)∥≤M(1+tαn/2)exp(ω1/αt),t≥0.(b)
ForX=Lp(ℝn), C=(1-Δ)-mβ/2, where β>np, there exists an analytic α-times C-regularized resolvent family Sα(t) for P(D) and
(4.27)∥Sα(t)∥≤M(1+tαnp)exp(ω1/αt),t≥0.We end this paper with some examples to demonstrate the applications of our results.Example 4.6.
(a) The polynomial corresponding to the LaplacianΔ on Lp(ℝn) (n>1,p≠2) is P(ξ)=-|ξ|2. By Theorem 4.4, for every 1<α<2 there exists an analytic α-times (1-Δ)-γ-regularized resolvent family for the operator Δ, where γ>np.(b)
ConsiderP(D) on Lp(ℝ2)  (1<p<∞) with
(4.28)P(ξ)=-(1+ξ12)(1+(ξ2-ξ1l)2)(l∈ℕ).Then P(ξ)≤-1(ξ∈ℝ2). We claim that P is (2/l)-coercive. Indeed, if |ξ2|≥2|ξ1l|, then
(4.29)|P(ξ)|≥(1+ξ12)(1+14ξ22)≥14|ξ|2.
If |ξ2|<2|ξ1l|, then
(4.30)|P(ξ)|≥1+|ξ1|2≥c|ξ|2/lfor|ξ|≥1,
for some proper constant c, as desired. By Theorems 4.4  and 4.5, for every 1<α<2 there exists an analytic α-times C-regularized resolvent family for P(D), where C=(1-P)-γ(D) with γ>2(l2+l)|1/2-1/p| or C=(1-Δ)-(l+1)β with β>2|1/2-1/p|. We remark that if l≥5 and |1/2-1/p|≥1/4+1/l, then 0∈σ(P(D)) (see [17]). Since 0∈P(ℝ2), it follows from [18, Theorem 1] that ρ(P(D))=∅. Consequently, in this case there is no α-times resolvent family for P(D) for any α.



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*Source: 438690-2009-06-22.xml*