aa r X i v : . [ m a t h . A P ] J un Fractional-Parabolic Systems
Anatoly N. Kochubei ∗ Institute of Mathematics,National Academy of Sciences of Ukraine,Tereshchenkivska 3, Kiev, 01601 UkraineE-mail: [email protected]
Abstract
We develop a theory of the Cauchy problem for linear evolution systems of partialdifferential equations with the Caputo-Dzhrbashyan fractional derivative in the time vari-able t . The class of systems considered in the paper is a fractional extension of the classof systems of the first order in t satisfying the uniform strong parabolicity condition.We construct and investigate the Green matrix of the Cauchy problem. While similarresults for the fractional diffusion equations were based on the H-function representa-tion of the Green matrix for equations with constant coefficients (not available in thegeneral situation), here we use, as a basic tool, the subordination identity for a modelhomogeneous system. We also prove a uniqueness result based on the reduction to anoperator-differential equation. Key words: parabolic systems; fractional derivative; fundamental solution; Levi method;subordination identity ∗ This work was supported in part by the Ukranian Foundation for Fundamental Research under Grant28.1/017. INTRODUCTION
Fractional diffusion equations of the form (cid:16) D ( α ) t u (cid:17) ( t, x ) − Au ( t, x ) = f ( t, x ) , ≤ t ≤ T, x ∈ R n , (1.1)where 0 < α < D ( α ) t is the Caputo-Dzhrbashyan fractional derivative, that is (cid:16) D ( α ) t u (cid:17) ( t, x ) = 1Γ(1 − α ) ∂∂t t Z ( t − τ ) − α u ( τ, x ) dτ − t − α u (0 , x ) ,A is a second order elliptic operator, are among the basic subjects in the theory of fractionaldifferential equations. The initial motivation came from physics – the equations of the abovetype were first used for modeling anomalous diffusion on fractals by Nigmatullin [32] and fora description of Hamiltonian chaos by Zaslavsky [47]. See the survey papers [14, 30] for adescription of the present status of this research area.The first mathematical works in this direction dealt either with the case of an abstract oper-ator A , that is with a kind of an abstract Cauchy problem [19] (see [18] for further references),or with the case where A = ∆ is a Laplacian. For the latter case, a fundamental solution of theCauchy problem (FSCP) is expressed via Fox’s H-function [20, 41]; uniqueness theorems wereproved in [20] for an equation with a general second order elliptic operator A ; see also [37].The first example of an initial-boundary value problem for this equation was considered in [46].Later the initial-boundary value problems for fractional diffusion equations were studied in [28],with an emphasis on the probabilistic aspects, and in [25]; for the probabilistic interpretationssee also [40, 29] and references therein.In [8] (see also [7]), Eidelman and the author constructed and investigated a FSCP forfractional diffusion equations with variable coefficients. We followed the classical parametrixmethod using an H-function representation for the parametrix kernel and the detailed informa-tion about asymptotic properties of the H-function available from [4, 17]. The fractional diffu-sion equation shares many essential properties with second order parabolic equations (thoughsome properties are different, like, for example, the singularity of the FSCP at x = 0 appearingfor n ≥ t = 0.Such a class of systems was first found by Petrowsky [34] in 1938. In fact, after Petrowsky’swork it was understood that a complete theory of partial differential equations must includea thorough study of systems of equations. Petrowsky introduced and investigated not onlyparabolic systems but also hyperbolic ones, systems with correct Cauchy problems etc. Forseveral decades, these subjects were central in the theory of partial differential equations; see, inparticular, [12, 35]. The class of parabolic systems was studied in the greatest detail [6, 7, 9, 23].So far only some special cases of partial fractional-differential systems have been considered[10, 11, 36, 44]. However it is the author’s opinion that the fractional calculus is mature enoughto initiate a general theory of systems of partial fractional-differential equations. In this paper2e follow the above line and find a fractional analog of the class of parabolic systems. Notethat simple examples of such fractional parabolic systems appear as linearized two-componentfractional reaction-diffusion systems used in the study of self-organization phenomena; see[10, 11] and references therein.We consider systems of the form (1.1) where u = ( u , . . . , u N ) is a vector-valued function, A = A ( x, D x ) = A ( x, D x ) + A ( x, D x ) , (1.2)is a differential operator of even order 2 b with matrix-valued coefficients,( A ( x, D x ) u ) i = N X j =1 X | β | =2 b a ijβ ( x ) D βx u j , i = 1 , . . . , N, (1.3)( A ( x, D x ) u ) i = N X j =1 X | β | < b a ijβ ( x ) D βx u j , i = 1 , . . . , N, (1.4) D βx = D β x · · · D β n x n , D x j = 1 √− ∂∂x j , | β | = β + · · · + β n . We assume that all the coefficients a ijβ ( x ) are bounded and satisfy the global H¨older condition (cid:12)(cid:12) a ijβ ( x ) − a ijβ ( y ) (cid:12)(cid:12) ≤ C | x − y | γ (below the letters C, c will denote various positive constants while γ > | · | norms of all finite vectors and matrices). Wealso assume the uniform strong parabolicity condition: for all η ∈ R n , z ∈ C N ,Re h A ( x, η ) z, z i ≤ − δ | η | b | z | , δ > . (1.5)In fact, the key ingredients in the construction of a FSCP for a problem with variablecoefficients are precise estimates for the model problem (cid:16) D ( α ) t u ( t, x ) (cid:17) = A ( y, D x ) u ( t, x ) (1.6)containing only the homogeneous highest order differential operator in x , with “frozen” coeffi-cients depending on a parameter point y . As the first step, one has to consider the case wherethe coefficients a ijβ ( x ), | β | = 2 b , are constant. Already in this case, the study of a FSCP is farfrom trivial. The approach used in [8, 7] based on the H-function representation, does not workfor systems.Instead, we use the subordination representation [3, 1] expressing the FSCP for the modelsystem via the FSCP for the first-order (in t ) parabolic system. At the first sight, it looksan easy approach to all fractional problems. However the subordination identity involves theintegration in t over the half-axis (0 , ∞ ), while usually a FSCP for a parabolic equation orsystem is constructed only on a finite time interval. Nevertheless, for our model case of constantcoefficients the subordination method works efficiently giving, by the way, new proofs of theestimates known for fractional diffusion equations. Note also that the probabilistic side of3ubordination, not touched here, is also an important subject of fractional analysis; see [2, 21,31, 33].We also prove a uniqueness theorem for general systems (1.1). Again, the method of provinguniqueness in [20, 7] (based on a kind of the maximum principle) is applicable only for secondorder equations. Here we use the reduction to an abstract equation from [19] and the regularizedresolvent estimate from [16].The main results of this paper are collected in Section 2. Section 3 contains miscellaneousauxiliary results used subsequently. Proofs of the estimates for the estimates for the Greenmatrix of a homogeneous system with constant coefficients are given in Section 4 and arecomplemented in Section 5 with some considerations regarding the parametrix kernels. InSection 6, these results are used to substantiate the Levi method in our situation. The proofof the uniqueness theorem is given in Section 7.The author is grateful to the anonymous referee for helpful comments and suggestions. In this section we introduce basic notions and formulate principal results. The proofs will begiven in subsequent sections. . Let us consider systems of the form (cid:16) D ( α ) t u ( t, x ) (cid:17) = A ( D x ) u ( t, x ) (2.1)where ( A ( D x ) u ) i = N X j =1 X | µ | =2 b a ijµ D µx u j , i = 1 , . . . , N,a ijµ ∈ C , and for any η ∈ R n , z ∈ C N ,Re h A ( η ) z, z i ≤ − δ | η | b | z | , δ > . (2.2)Under the assumption (2.2) (in fact, even under a weaker assumption of parabolicity in thesense of Petrowsky), the differential expression A ( D ) defines on the space L ( R n , C N ) of squareintegrable vector-functions with values in C N , an infinitesimal generator A of a C -semigroup S ( t ) = e t A (see [22]).By the subordination theorem (see Theorem 3.1 in [3]), the system (2.1) interpreted asan equation in L ( R n , C N ), possesses a solution operator S α ( t ), such that for any element u = u ( x ) from the domain D ( A ), the function u ( t, x ) = ( S α ( t ) u )( x ), t ≥ x ∈ R n , is asolution of the equation (2.1) satisfying the initial condition u (0 , x ) = u ( x ). In addition, S α ( t ) = ∞ Z ϕ t,α ( s ) S ( s ) ds, t ≥ , (2.3)where ϕ t,α ( s ) = t − α Φ α ( st − α ), Φ α ( ζ ) = ∞ X k =0 ( − ζ ) k k !Γ( − αk + 1 − α ) ,
4o that Φ α can be written as the Wright functionΦ α ( ζ ) = Ψ h − (1 − α, − α ) (cid:12)(cid:12)(cid:12) − ζ i . (2.4)See [18] for general information regarding the definition and properties of the Wright functions;see also Section 3.3 below.The function Φ α is a probability density:Φ α ( t ) ≥ , t > ∞ Z Φ α ( t ) dt = 1 . It is connected also with the Mittag-Leffler function E α ( ζ ) = ∞ X k =0 ζ k Γ(1 + αk ) , ζ ∈ C , (2.5)via the Laplace transform identity E α ( − ζ ) = ∞ Z Φ α ( t ) e − ζt dt, ζ ∈ C . (2.6)By the classical theory of parabolic equations, the semigroup S ( t ) possesses the integralrepresentation ( S ( t ) η )( x ) = Z R n Z ( t, x − ξ ) η ( ξ ) dξ, η ∈ L ( R n , C N ) , in terms of the FSCP Z ( t, x ) of the parabolic system ∂u∂t = A ( D x ) u . It follows from theestimates of Φ α and Z (see below) that, for example, if η ∈ S ( R n ), then( S α ( t ) η )( x ) = Z R n Z α ( t, x − ξ ) η ( ξ ) dξ where Z α ( t, x ) = ∞ Z ϕ t,α ( s ) Z ( s, x ) ds, x = 0 (2.7)(as we have seen for the diffusion equations [8, 7], Z α may have a singularity at x = 0).The kernel Z α is a FSCP for the system (2.1).In order to obtain an integral representation of a solution u ( t, x ) of the inhomogeneousequation (cid:16) D ( α ) t u ( t, x ) (cid:17) − A ( D x ) u ( t, x ) = f ( t, x ) , u (0 , x ) = u ( x ) , in the form u ( t, x ) = Z R n Z α ( t, x − ξ ) u ( ξ ) dξ + t Z dτ Z R n Y α ( t − τ, x − y ) f ( τ, y ) dy Y α ( t, x ) = (cid:16) D (1 − α ) t Z α (cid:17) ( t, x ) , x = 0 . As we will see, Y α ( t, x ) = ∞ Z ψ t,α ( s ) Z ( s, x ) ds, x = 0 (2.8)where ψ t,α ( s ) = t − Ψ h − (0 , − α ) (cid:12)(cid:12)(cid:12) − st − α i . (2.9)Note that ϕ t,α is the law of the inverse to a α -stable subordinator; see [29]. In the contextof fractional calculus models of physical phenomena, the functions ϕ t,α and ψ t,α appeared forthe first time in [26].Using properties of the functions Z and Φ α we get the integral identities Z R n Z α ( t, x ) dx = 1 , Z R n Y α ( t, x ) dx = t α − Γ( α ) . Theorem 1.
The matrix-functions Z α ( t, x ) , Y α ( t, x ) are infinitely differentiable for t > , x = 0 , and satisfy the following estimates. Denote R = t − α | x | b , ρ ( t, x ) = (cid:0) t − α | x | b (cid:1) b − α . (i) If R ≥ , then (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α ( n + | β | )2 b exp( − σρ ( t, x )) , σ >
0; (2.10) (cid:12)(cid:12) D βx Y α ( t, x ) (cid:12)(cid:12) ≤ Ct − α ( n + | β | )2 b + α − exp( − σρ ( t, x )) . (2.11) (ii) If R ≤ , n + | β | < b , then (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α ( n + | β | )2 b , (2.12) (cid:12)(cid:12) D βx Y α ( t, x ) (cid:12)(cid:12) ≤ Ct − α ( n + | β | )2 b + α − . (2.13) (iii) If R ≤ , n + | β | > b , then (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α | x | − n +2 b −| β | , (2.14) (cid:12)(cid:12) D βx Y α ( t, x ) (cid:12)(cid:12) ≤ Ct − | x | − n +2 b −| β | . (2.15) (iv) If R ≤ , n + | β | = 2 b , then (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α , if n = 1; (2.16) (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α [ | log( t − α | x | b ) | + 1] , if n ≥
2; (2.17) (cid:12)(cid:12) D βx Y α ( t, x ) (cid:12)(cid:12) ≤ Ct − . (2.18)6 v) If R ≥ , then (cid:12)(cid:12)(cid:12)(cid:12) ∂Z α ( t, x ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ct − αn b − exp( − σρ ( t, x )) . (2.19) (vi) If R ≤ , n < b , then (cid:12)(cid:12)(cid:12)(cid:12) ∂Z α ( t, x ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ct − αn b − . (2.20) If R ≤ , n > b , then (cid:12)(cid:12)(cid:12)(cid:12) ∂Z α ( t, x ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ct − α − | x | − n +2 b . (2.21) If R ≤ , n = 2 b , then (cid:12)(cid:12)(cid:12)(cid:12) ∂Z α ( t, x ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ct − α − [ | log( t − α | x | b ) | + 1] . (2.22) In all the above estimates, the constants depend only on
N, n , max (cid:12)(cid:12) a ijµ (cid:12)(cid:12) , and the strongparabolicity constant δ . The estimates (2.10)-(2.22) agree with their counterparts for the fractional diffusion equa-tions [8, 7], though in the latter case, for some values of n , there are more precise estimates of Z α and Y α .Note also that the fractional derivative D ( α ) t Z α satisfies the same estimate as the derivatives D βx Z α , | β | = 2 b . . As stated in Introduction, we consider the system (1.1)–(1.4) withbounded H¨older continuous coefficients, under the uniform strong parabolicity condition (1.5).We call a vector-function u ( t, x ), 0 ≤ t ≤ T , x ∈ R n , a classical solution of the system (1.1),with the initial condition u (0 , x ) = u ( x ) , x ∈ R n , (2.23)if: (i) u ( t, x ) is continuously differentiable in x up to the order 2 b , for each t > (ii) for each x ∈ R n , u ( t, x ) is continuous in t on [0 , T ], and its fractional integral (cid:0) I − α u (cid:1) ( t, x ) = 1Γ(1 − α ) t Z ( t − τ ) − α u ( τ, x ) dτ (2.24)is continuously differentiable in t for 0 ≤ t ≤ T . (iii) u ( t, x ) satisfies the equation (1.1) and the initial condition (2.23).7 classical solution u ( t, x ) is called a uniform classical solution , if it is continuous in t uniformly with respect to x ∈ R n , and the first derivative of the fractional integral (2.24) existsuniformly with respect to x ∈ R n .Our main task is to construct a Green matrix for the problem (1.1), (2.23), that is such apair (cid:8) Z (1) α ( t, x ; ξ ) , Y (1) α ( t, x ; ξ ) (cid:9) , t ∈ [0 , T ] , x, ξ ∈ R n , that for any bounded function f , jointly continuous in ( t, x ) and locally H¨older continuous in x uniformly with respect to t , and any bounded locally H¨older continuous function u , thefunction u ( t, x ) = Z R n Z (1) α ( t, x ; ξ ) u ( ξ ) dξ + t Z dλ Z R n Y (1) α ( t − λ, x ; y ) f ( λ, y ) dy. (2.25)is a classical solution of the problem (1.1),(2.23).Denote by Z (0) α ( t, x − ξ ; y ) and Y (0) α ( t, x − ξ ; y ) the kernels defined just as Z α ( t, x − ξ ) and Y α ( t, x − ξ ), but for the system (1.6) with the coefficients a ijβ , | β | = 2 b , “frozen” at a point y ∈ R n , and other coefficients set equal to zero; in (1.6), y appears as a parameter. Theorem 2. (a) There exists a Green matrix n Z (1) α ( t, x ; ξ ) , Y (1) α ( t, x ; ξ ) o of the form Z (1) α ( t, x ; ξ ) = Z (0) α ( t, x − ξ ; ξ ) + V Z ( t, x ; ξ ) ,Y (1) α ( t, x ; ξ ) = Y (0) α ( t, x − ξ ; ξ ) + V Y ( t, x ; ξ ) , where the kernels Z (0) α ( t, x ; ξ ) , Y (0) α ( t, x ; ξ ) satisfy the estimates listed in Theorem 1 with co-efficients independent on the parameter point ξ . The functions V Z , V Y satisfy the followingestimates. (i) If n + | β | < b , then (cid:12)(cid:12) D βx V Z ( t, x ; ξ ) (cid:12)(cid:12) ≤ Ct − α b ( | β | + γ ) | x − ξ | − n + γ − γ e − σρ ( t,x − ξ ) , < γ < γ, σ >
0; (2.26) (cid:12)(cid:12) D βx V Y ( t, x ; ξ ) (cid:12)(cid:12) ≤ Ct − α − α | β | b | x − ξ | − n + γ e − σρ ( t,x − ξ ) . (2.27) (ii) If n + | β | ≥ b , | β | < b, then (cid:12)(cid:12) D βx V Z ( t, x ; ξ ) (cid:12)(cid:12) ≤ Ct − α + αγ b | x − ξ | − n +2 b −| β | + γ − γ e − σρ ( t,x − ξ ) ; (2.28) (cid:12)(cid:12) D βx V Y ( t, x ; ξ ) (cid:12)(cid:12) ≤ Ct − αγ b | x − ξ | − n ++2 b −| β | + γ − γ e − σρ ( t,x − ξ ) . (2.29) (iii) If | β | = 2 b, then (cid:12)(cid:12) D βx V Z ( t, x ; ξ ) (cid:12)(cid:12) ≤ Ct − α + µ | x − ξ | − n + µ e − σρ ( t,x − ξ ) ; (2.30) (cid:12)(cid:12) D βx V Y ( t, x ; ξ ) (cid:12)(cid:12) ≤ Ct − µ | x − ξ | − n + µ e − σρ ( t,x − ξ ) (2.31) where µ , µ > . b) If the functions u ( x ) , f ( t, x ) are bounded and globally H¨older continuous (for f , uni-formly with respect to t ), and f is continuous in t uniformly with respect to x ∈ R n , then thesolution (2.25) is a uniform classical solution. All its derivatives in x , up to the order b , arebounded and globally H¨older continuous, uniformly with respect to t ∈ [0 , T ] . Note that the estimates in Theorem 2 can be written in a variety of ways. For example, in(2.26) we may write t − α b | β | = (cid:0) t − α b | x − ξ | (cid:1) | β | | x − ξ | −| β | and obtain, taking 0 < σ ′ < σ , that (cid:12)(cid:12) D βx V Z ( t, x ; ξ ) (cid:12)(cid:12) ≤ Ct − α b γ | x − ξ | − n −| β | + γ − γ e − σ ′ ρ ( t,x − ξ ) . This kind of transformation is often used in proofs of various estimates in this paper. . Here we maintain the same assumptions as in Theorem 2.
Theorem 3.
Let u ( t, x ) , ≤ t ≤ T , x ∈ R n , be a uniform classical solution of the problem(1.1), (2.23) with f ( t, x ) ≡ , u ( x ) ≡ . Suppose that the function u ( t, x ) and all its derivativesof orders ≤ b are bounded and globally H¨older continuous. Then u ( t, x ) equals zero identically. The rest of the paper is devoted to the proofs of the above results. Some of them canbe extended easily to more general situations. For example, the coefficients of the subordinateoperator A may depend on t . Some of the estimates of Theorem 1 (except the case n + | β | = 2 b )remain valid for systems generalizing in an obvious way the first order systems parabolic in thesense of Petrowsky. However for the whole range of the above results we need the condition(1.5). . Let B be a complex N × N matrix. TheMittag-Leffler function E α of the matrix B is defined by substituting B into the power series(2.5): E α ( B ) = ∞ X k =0 B k Γ(1 + αk ) , < α < . (3.1)Note that the analytic function E α of a matrix does not coincide with the matrix formed byvalues of the function E α on matrix elements.For a class of matrices, we find a matrix analog of the asymptotic representation of theMittag-Leffler function (see e.g. [5]). Proposition 1.
Suppose that for any z ∈ C N , Re h Bz, z i ≤ − δ | z | , δ > . (3.2) Then E α ( B ) = − − α ) B − + H (3.3) where the matrix H is such that | H | ≤ Cδ − (the constant C does not depend on B ). roof . Denote by γ ( r, ω ) the contour in the complex plane oriented in the direction of theincrease of arg ζ and consisting of the following parts: the rays γ ± = { ζ ∈ C : arg ζ = ± ω, | ζ | ≥ r } and the arc { ζ ∈ C : − ω < arg ζ < ω, | ζ | = r } . Here r > π < ω ≤ π .Let us use Hankel’s integral representation1Γ( s ) = 12 πi Z γ ( r,ω ) e ζ ζ − s dζ , Re s > ζ = η /α . Since r > s ) = 12 πiα Z γ ( r,β ) e η /α η − sα + α − dη, (3.5)for any β with πα < β ≤ πα . For our purposes, we will assume that πα < β < min( π , πα ), sothat the contour γ ( r, ω ) is located in the right half-plane, and cos βα < B − λI ) − exists for Re λ > − δ , and | ( B − λI ) − | ≤ λ + δ (see Lemma V.6.1 in [13]). In particular, | B − | ≤ δ − . Substituting (3.4) into (3.1) we findthat E α ( B ) = 12 πiα ∞ X k =0 Z γ ( r,β ) e η /α η − k − dη B k = 12 πiα Z γ ( r,β ) e η /α η − ( ∞ X k =0 ( Bη − ) k ) dη = − πiα Z γ ( r,β ) e η /α ( B − ηI ) − dη, if r > | B | . Note that on the rays contained in γ ( r, β ), (cid:12)(cid:12)(cid:12) e η /α (cid:12)(cid:12)(cid:12) = exp (cid:0) cos βα · | η | /α (cid:1) wherecos βα < r = 1. The function B Z γ (1 ,β ) e η /α ( B − ηI ) − dη is an analytic function of the matrix B on the open set { B : Re h Bz, z i < − δ | z | , ∀ z ∈ C N \{ }} coinciding with E α ( B ) on its intersection with the open set { B : | B | < } (see [24] regardinganalytic functions on matrices; an analytic function on N × N matrices is in fact an analyticfunction of N complex variables). Therefore E α ( B ) = − πiα Z γ (1 ,β ) e η /α ( B − ηI ) − dη (3.6)10or any matrix B satisfying (3.2).Next we use the identity( B − ηI ) − = B − + ηB − ( B − ηI ) − . Substituting it into (3.6) and using (3.5) we get E α ( B ) = − − α ) B − − B − πiα Z γ (1 ,β ) e η /α η ( B − ηI ) − dη, which implies the required representation (3.3). (cid:4) We will need also and estimate of E α,α ( B ) where the Mittag-Leffler type function E α,α isdefined by the series E α,α ( ζ ) = ∞ X k =0 ζ k Γ( α + αk ) . (3.7) Proposition 2.
If a matrix B satisfies (3.2), then | E α,α ( B ) | ≤ Cδ − (3.8) where C does not depend on B .Proof . As in the proof of Proposition 1, we obtain the representation (with the samenotations) E α,α ( B ) = − πiα Z γ (1 ,β ) e η /α η − α ( B − ηI ) − dη, from which we get that E α,α ( B ) = − πiα B − Z γ (1 ,β ) e η /α η − α dη − πiα B − Z γ (1 ,β ) e η /α η − α ( B − ηI ) − dη. (3.9)In the representation (3.5), we may pass to the limit, as s →
0. As a result, the first integralin (3.9) equals 0. Estimating the second integral we come to (3.8). (cid:4)
In another result of this kind, we deal with the Mittag-Leffler type function E α, ( ζ ) = ∞ X k =0 ζ k Γ( αk ) . Proposition 3.
Under the assumptions of Proposition 1, E α, ( B ) = − − α ) B − + H, where | H | ≤ Cδ − , and the constant does not depend on B . . We will need, as technical tools, estimates ofderivatives D β G ( x ) of the Green matrix of the elliptic operator A ( D ) − I , and also estimatesof differences D β ( G ( x ; y ′ ) − G ( x ; y ′′ )) of derivatives of the Green matrices of the operators A ( y ′ , D x ) − I and A ( y ′′ , D x ) − I , with coefficients “frozen” at the points y ′ and y ′′ . Specifically,we need to consider the case where n + | β | = 2 b . The presence of the term − I is essential – thematrix A ( η ) − I ( η ∈ R n ) has, under the assumption (2.2), all the eigenvalues with nonzeroreal parts. For this class of elliptic systems with constant coefficients, Eidelman [6] found anintegral representation of Green matrices, and the estimate (cid:12)(cid:12) D β G ( x ) (cid:12)(cid:12) ≤ C (cid:18) | x | (cid:19) , | x | ≤ , (3.10)where it is assumed that n + | β | = 2 b (of course, other cases were considered in [6] too). It alsofollows from the constructions in [6] that G ∈ C ∞ ( R n \ { } ).However the estimates for the differences are not given in [6], and for completeness we havenow to apply the method from [6] to this situation.Let N ( y, t, x ) be a FSCP of the parabolic system ∂u ( t, x ) ∂t = ( A ( D x ) − I ) u ( t, x ) . Then [6] G ( x ; y ) = ∞ Z N ( y, t, x ) dt, so that D βx [ G ( x ; y ′ ) − G ( x ; y ′′ )] = ∞ Z D βx [ N ( y ′ , t, x ) − N ( y ′′ , t, x )] dt. For the FSCP N ( y, t, x ) we have the representation N ( y, t, x ) = (2 π ) − n Z R n e ix · ξ e ( A ( y,ξ ) − I ) t dξ = (2 π ) − n t − n b e − t Z R n e it − / b x · ξ e A ( y,ξ ) dξ. Therefore, if n + | β | = 2 b , then D βx [ N ( y ′ , t, x ) − N ( y ′′ , t, x )] = (2 π ) − n t − e − t Z R n ξ β e it − / b x · ξ h e A ( y ′ ,ξ ) − e A ( y ′′ ,ξ ) i dξ. (3.11)Following [6] we use the identity e A ( y ′ ,ξ ) t − e A ( y ′′ ,ξ ) t = t Z e A ( y ′′ ,ξ )( t − τ ) [ A ( y ′ , ξ ) − A ( y ′′ , ξ )] e A ( y ′ ,ξ ) τ dτ, (cid:12)(cid:12)(cid:12) e A ( y ′ ,ξ ) − e A ( y ′′ ,ξ ) (cid:12)(cid:12)(cid:12) ≤ C | y ′ − y ′′ | γ e − δ | ξ | b . We can consider inequalities of this kind containing, instead of ξ ∈ R n , a point ξ + iη , η ∈ R n . By a lemma from [6] (Chapter 1), the inequality (1.5) implies the inequalityRe h A ( y, ξ + iη ) z, z i ≤ (cid:0) − δ | ξ | b + µ | η | b (cid:1) | f | ( δ , µ > (cid:12)(cid:12) e A ( y,ξ + iη ) (cid:12)(cid:12) ≤ Ce − δ | ξ | b + µ | η | b ( δ , µ > (cid:12)(cid:12)(cid:12) ξ β h e A ( y ′ ,ξ + iη ) − e A ( y ′′ ,ξ + iη ) i(cid:12)(cid:12)(cid:12) ≤ C | y ′ − y ′′ | γ e − δ | ξ | b + µ | η | b (3.12)( δ , µ > (cid:12)(cid:12) D βx [ N ( y ′ , t, x ) − N ( y ′′ , t, x )] (cid:12)(cid:12) ≤ Ct − e − t | y ′ − y ′′ | γ e − ct − b − | x | b b − , so that (cid:12)(cid:12) D βx [ G ( x, y ′ ) − G ( x, y ′′ )] (cid:12)(cid:12) ≤ C | y ′ − y ′′ | γ ∞ Z t − e − t exp (cid:16) − ct − b − | x | b b − (cid:17) dt = C | y ′ − y ′′ | γ ∞ Z s − exp (cid:16) − s | x | b − s − b − (cid:17) ds = C | y ′ − y ′′ | γ [ I ( x ) + I ( x )]where I ( x ) = Z s − exp (cid:16) − s | x | b − s − b − (cid:17) ds ≤ Z s − exp (cid:16) − s − b − (cid:17) ds ≤ C,I ( x ) = ∞ Z s − exp (cid:16) − s | x | b − s − b − (cid:17) ds ≤ ∞ Z s − e − s | x | b ds = | x | b ∞ Z log s · e − s | x | b ds ≤ x | b ∞ Z | log s | · e − s | x | b ds = ∞ Z (cid:12)(cid:12) log( | x | − b σ ) (cid:12)(cid:12) e − σ dσ ≤ C (1 + | log | x | ) . This results in the required estimate (cid:12)(cid:12) D βx [ G ( x, y ′ ) − G ( x, y ′′ )] (cid:12)(cid:12) ≤ C | y ′ − y ′′ | γ (cid:18) | x | (cid:19) , | x | ≤ , n + | β | = 2 b. (3.13)13 .3. Properties of the functions ϕ t,α and ψ t,α . The function Φ α (see (2.4)) involved in thesubordination representation (2.3) is such the Φ α ( t ) ≥ t > α ( t ) ∼ Ct − / e − ct − α , t → + ∞ ( C, c > ≤ ϕ t,α ( s ) ≤ Ct − α e − cs − α t − α − α , s > . (3.14)Note that ϕ t,α ( s ) →
0, as t →
0, for each s >
0. It is also important that ∞ Z Φ α ( s ) ds = 1 . (3.15)In order to study the kernel Y α (see (2.8)), we need the function ψ t,α ( s ) = D − αt ϕ t,α ( s ) . Making a change of variables in (2.6) we can write E α ( − ζ ) = ∞ Z ϕ t,α ( σ ) e − ζσt − α dσ, whence E α ( − ζ t α ) = ∞ Z ϕ t,α ( σ ) e − ζσ dσ. (3.16)The identity (3.16) is written in part as a warning of a complicated nature of the subordina-tion identities. As t → +0, the left-hand side of (3.16) tends to 1, while in the right-hand side, ϕ t,α ( σ ) → σ >
0. Thus, it is impossible to interchange the integration and takingthe limit. It will be reasonable here to use, instead of the Caputo-Dzhrbashyan derivative, theRiemann-Liouville derivative (cid:0) D − α u (cid:1) ( t ) = 1Γ( α ) ddt t Z ( t − τ ) − α u ( τ ) dτ coinciding with D − α u wherever u (0) = 0. In particular, if s >
0, then ψ t,α ( s ) = D − α ,t ϕ t,α ( s ) . (3.17)By (2.4), we have the Wright function representation ϕ t,α ( s ) = t − α Ψ h − (1 − α, − α ) (cid:12)(cid:12)(cid:12) − st − α i . ϕ t,α ( s ) = 12 πi γ + i ∞ Z γ − i ∞ Γ( λ )Γ(1 − α + αλ ) s − λ t αλ − α dλ, (3.18) γ > γ = α + να , ν = 0 , , , . . . .It is known (see, for example, [18]) that D − α transforms t αλ − α into Γ(1 − α + αλ )Γ( αλ ) t αλ − .Now we get from (3.17) and (3.18) that ψ t,α ( s ) = 12 πi γ + i ∞ Z γ − i ∞ Γ( λ )Γ( αλ ) s − λ t αλ − dλ (the legitimacy of this transformation follows from the asymptotics of Gamma function). Usingagain the equivalence of the series and integral representations of the Wright functions ((12.41)in [4]) we prove the representaton (2.9): ψ t,α ( s ) = t − Ψ h − (0 , − α ) (cid:12)(cid:12)(cid:12) − st − α i , (3.19)so that ψ t,α ( s ) = t − ∞ X k =1 ( − st − α ) k k !Γ( − αk ) . (3.20)Using the asymptotics of the Wright function found in [45] (see also [4], Theorem 25) weget the estimate | ψ t,α ( s ) | ≤ Ct − exp n − c ( st − α ) − α o , (3.21)with c, C > ψ t,α with the Mittag-Leffler type functions. Apply the operator D − α ,t to both sides of the equality (3.16). It is known(see (2.1.54) in [18]) that in the left-hand side we obtain t α − E α,α ( − ζ t α ). Thus we come to theidentity E α,α ( − ζ t α ) = t − α ∞ Z Ψ h − (0 , − α ) (cid:12)(cid:12)(cid:12) − st − α i e − ζs ds, ζ > . (3.22)Another transformation kernel of the above kind is ν t,α ( s ) = ∂∂t ϕ t,α ( s ) . (3.23)Repeating the above reasoning we find that ν t,α ( s ) = t − α − Ψ h − ( − α, − α ) (cid:12)(cid:12)(cid:12) − st − α i (3.24)15here the Wright function has the form Ψ h − ( − α, − α ) (cid:12)(cid:12)(cid:12) z i = ∞ X k =0 ( − z ) k k !Γ( − α − αk ) . Using the asymptotics from [4, 45] we obtain, for s > t >
0, the estimate | ν t,α ( s ) | ≤ Ct − α − exp n − c ( st − α ) − α o , c > . (3.25)It is known ([18], formula (1.10.2)) that ∂∂t E α ( − ζ t α ) = t − E α, ( − ζ t α )where E α, ( ζ ) = ∞ P k =0 ζ k Γ( αk ) . Differentiating both sides of the identity (3.16) we find, afterelementary transformations, that E α, ( − ζ ) = ∞ Z Ψ h − ( − α, − α ) (cid:12)(cid:12)(cid:12) − s i e − ζs ds, ζ > . (3.26) . Let us use the identity (2.7) which implies a representation of spatialderivatives, D βx Z α ( t, x ) = ∞ Z ϕ t,α ( s ) D βx Z ( s, x ) ds, x = 0 . (4.1)Below we use both the integral representation for Z , Z ( s, x ) = (2 π ) − n Z R n e ix · ξ e sA ( ξ ) dξ, (4.2)and its estimates (cid:12)(cid:12) D βx Z ( t, x ) (cid:12)(cid:12) ≤ Ct − n + | β | b exp n − c | x | b b − t − b − o (4.3)( c >
0) valid for all x ∈ R n , t > Z α ( t, x ) satisfies the system (2.1) in a weaksense (a strong solution is obtained after a convolution in spatial variables with a test function;see Section 2.1). It will be seen from the investigation of the spatial derivatives below that A ( D x ) Z α exists in the classical sense for t > x = 0. Therefore (cid:16) D ( α ) t Z α (cid:17) ( t, x ) coin-cides almost everywhere with A ( D x ) Z α ( t, x ) ( t > , x = 0) and does not require a separateinvestigation. 16t follows from the inequalities (3.14) and (4.3) that Z α ( t, x ) is infinitely differentiable in x = 0, and (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α ∞ Z s − n + | β | b exp n − c | x | b b − s − b − o exp n − cs − α t − α − α o ds (for t >
0, the second exponential factor on the right guarantees the convergence of the integralat infinity; for x = 0, the first exponential factor gives the convergence at the origin). Afterthe change of variables s − b − = σ , we find that (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α ∞ Z σ b − b ( n + | β | ) − b exp n − cσ | x | b b − o exp n − cσ − b − − α t − α − α o dσ. (4.4)In order to obtain an estimate for R ≥
1, we use the asymptotics of the integralΩ( ζ ) = ∞ Z e − ζt e − dt − κ t λ dt, ζ → ∞ , (4.5)where d > κ > λ ∈ R , found in [38] (formula (12.80)). Namely,Ω( ζ ) ∼ a ( d κ ζ − ) λ +1 κ +1 exp (cid:20) − (1 + 1 κ ) ρ (cid:21) ρ − / (4.6)where ρ = ( d κ ζ κ ) κ , a = 2 (cid:0) κ (cid:1) / Γ( ).Making in (4.4) the change of variables σ = t − α b − η we get, after easy calculations, theinequality (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α n + | β | b ∞ Z η b − b ( n + | β | ) − b exp (cid:16) − cη − b − − α (cid:17) exp (cid:16) − cR b − η (cid:17) dη where the integral has the form of (4.5) with ζ = R b − , d = c , κ = 2 b − − α , λ = 2 b − b ( n + | β | ) − b . Using (4.6) and ignoring powers of R (changing, if necessary, the constant in theexponential factor) we obtain the inequality (2.10).Suppose that R ≤ n + | β | > b . We use again the inequality (4.4), but make the changeof variables σ = τ | x | − b b − and replace the exponential factor containing | x | by 1. We find that (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α | x | − n −| β | +2 b ∞ Z τ b − b ( n + | β | ) − b e − cτ dτ where 2 b − b ( n + | β | ) − b > −
1. This implies (2.14).The inequalities (2.11) and (2.15) are proved similarly, on the basis of the estimate (3.21).17et R ≤ n + | β | < b . Performing a change of variables we can rewrite (4.2) in the form Z ( s, x ) = (2 π ) − n s − n b Z R n e is − / b x · ξ e A ( ξ ) dξ, and by virtue of (4.1), D βx Z α ( t, x ) = (2 π ) − n t − α ∞ Z Φ α ( st − α ) s − n + | β | b ds Z R n ξ β e is − / b x · ξ e A ( ξ ) dξ, that is, after the change s = σt α , D βx Z α ( t, x ) = (2 π ) − n t − α n + | β | b ∞ Z Φ α ( σ ) σ − n + | β | b dσ Z R n ξ β e it − α/ b σ − / b x · ξ e A ( ξ ) dξ. (4.7)It follows from (2.2) that (cid:12)(cid:12) e A ( ξ ) (cid:12)(cid:12) ≤ e − δ | ξ | b (4.8)(see Sect. I.4.4 in [22]). Using the fact that Φ α decays rapidly at infinity, and n + | β | b <
1, wecome to the inequality (2.12). The proof of (2.13), based on (2.8), (3.19), and (3.21), is similar.Finally, consider the most complicated case where R ≤ n + | β | = 2 b . The representation(4.7) takes the form D βx Z α ( t, x ) = (2 π ) − n t − α ∞ Z Φ α ( σ ) σ − dσ Z R n ξ β e it − α/ b σ − / b x · ξ e A ( ξ ) dξ (4.9)or, after the change ξ = σ / b η , D βx Z α ( t, x ) = (2 π ) − n t − α ∞ Z Φ α ( σ ) dσ Z R n η β e it − α/ b x · η e σA ( η ) dη. (4.10)The identity (2.6) remains valid when the matrix − A ( η ) is substituted for ζ . Indeed, wemay rewrite (2.6) in the form E α ( − ζ ) = ∞ X k =0 ( − k ζ k k ! ∞ Z Φ α ( t ) t k dt (the convergence of the series follows from the asymptotics of Φ α and Stirling’s formula). Recallthat an entire function of a matrix is defined by substituting the matrix into the power seriesexpansion. Thus we set ζ = − A ( η ), and then gather the power series into the exponential(keeping in mind the inequality (4.8)). Therefore E α ( A ( η )) = ∞ Z Φ α ( σ ) e σA ( η ) dσ. (4.11)18f we substitute (4.11) into (4.10) and change the order of integration, we obtain the repre-sentation D βx Z α ( t, x ) = (2 π ) − n t − α Z R n η β e it − α/ b x · η E α ( A ( η )) dη. (4.12)However this change of the order of integration requires a justification.It follows from (2.2) and Proposition 1 that | E α ( A ( η )) | ≤ C | η | − b , | η | ≥ , so that | η | | β | | E α ( A ( η )) | ≤ C | η | − n , | η | ≥
1. Denote X ε ( t, x, β ) = (2 π ) − n t − α ∞ Z ε Φ α ( σ ) σ − dσ Z R n ξ β e it − α/ b σ − / b x · ξ e A ( ξ ) dξ, ε > . Here we make the change of variables ξ = σ / b η , and change the order of integration (wehave moved away from the singularity!). Thus X ε ( t, x, β ) = (2 π ) − n t − α Z R n η β e it − α/ b x · η dη ∞ Z ε Φ α ( σ ) e σA ( η ) dσ. Let us show that η β ∞ Z ε Φ α ( σ ) e σA ( η ) dσ ε → −−→ η β ∞ Z Φ α ( σ ) e σA ( η ) dσ = E α ( A ( η )) η β , (4.13)as functions of η , in the topology of L ( R n ).Indeed, using the estimate (4.8) we see that Z R n | η | | β | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε Z Φ α ( σ ) e σA ( η ) dσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dη ≤ C Z R n | η | | β | ε Z e − δσ | η | b dσ dη ≤ Z R n | η | − n (cid:16) − e − δε | η | b (cid:17) dη = C ( I + I )where I = Z | η |≤ | η | − n (cid:16) − e − δε | η | b (cid:17) dη, Z | η | > | η | − n (cid:16) − e − δε | η | b (cid:17) dη. Using the inequality 1 − e − x ≤ x , x ≥
0, we find that I ≤ Cε Z | η |≤ | η | − n +4 b dη = Cε Z | η |≤ | η | | β | dη → , ε →
0. By the dominated convergence theorem, we obtain also that I →
0, and we haveproved (4.13).Now, by the properties of the Fourier transform, for any fixed t > x ∈ R n ,lim ε → X ε ( t, x, β ) = (2 π ) − n t − α Z R n η β e it − α/ b x · η E α ( A ( η )) dη. On the other hand, X ε ( t, x, β ) → D βx Z α ( t, x ), if x = 0, by (4.9). Thus we have proved theequality (4.12) for almost all x = 0.Denote temporarily the right-hand side of (4.12) by X ( t, x, β ). Next we prove that X ( t, x, β )is continuous in x = 0. This will establish the equality (4.12) for all x = 0; simultaneously wewill get the required estimate.By Proposition 1, X ( t, x, β ) = X + X + X where X = (2 π ) − n t − α Z | η |≤ η β e it − α/ b x · η E α ( A ( η )) dη,X = − t − α (2 π ) n Γ(1 − α ) Z | η | > η β e it − α/ b x · η [ A ( η )] − dη,X = (2 π ) − n t − α Z | η | > η β e it − α/ b x · η H ( η ) dη, | H ( η ) | ≤ C | η | − b . Since | β | − b = − n − b , X is continuous in x , and | X | ≤ Ct − α . We seealso that X is continuous in x , and | X | ≤ Ct − α .Let us write X = X + X + X where X = − t − α (2 π ) n Γ(1 − α ) Z | η | > η β e it − α/ b x · η (cid:8) [ A ( η )] − − [ A ( η ) − I ] − (cid:9) dη,X = − t − α (2 π ) n Γ(1 − α ) Z R n η β e it − α/ b x · η [ A ( η ) − I ] − dη,X = − t − α (2 π ) n Γ(1 − α ) Z | η |≤ η β e it − α/ b x · η [ A ( η ) − I ] − dη. We have (cid:12)(cid:12) [ A ( η ) − I ] − (cid:12)(cid:12) ≤
11 + δ | η | b (4.14)(see Lemma V.6.1 in [13]). By the resolvent identity[ A ( η )] − − [ A ( η ) − I ] − = [ A ( η )] − [ A ( η ) − I ] − , we get the estimate (cid:12)(cid:12) [ A ( η )] − − [ A ( η ) − I ] − (cid:12)(cid:12) ≤ δ | η | b (1 + δ | η | b ) . (4.15)20t follows from (4.14) and (4.15) that X and X are continuous in x , | X | ≤ Ct − α , | X | ≤ Ct − α .As for X , we note that X = − t − α Γ(1 − α ) D βy G ( y ) (cid:12)(cid:12) y = t − α/ b x where G is the Green matrix of the elliptic operator A ( D ) − I . This means that X iscontinuous in x = 0, thus X has the same property and coincides with D β Z α for all x = 0.Now the required inequality (2.17) is a consequence of (3.10).For n = 1, we will refine this estimate. In this case, A ( η ) = a η b whereRe h a z, z i ≤ − δ | z | for all z ∈ C N . We have D βx Z α ( t, x ) = 12 π t − α ∞ Z −∞ η β e it − α/ b xη E α ( a η b ) dη where 1 + β = 2 b . In particular, the natural number β is odd, so that D βx Z α ( t, x ) = 1 π t − α ∞ Z η β sin( t − α/ b xη ) E α ( a η b ) dη. By Proposition 1, E α ( a η b ) = − a − Γ(1 − α ) η − b + O ( η − b ) , η → ∞ . The contribution of the remainder term in the estimate of D βx Z α is clearly O ( t − α ). Thereforewe have to consider the function F ( y ) = ∞ Z ϕ ( η ) sin( yη ) dη, < y ≤ , (4.16)where ϕ is continuous on [0 , ∞ ), ϕ ( η ) ∼ η − , η → ∞ .It was shown in [42] that the integral in (4.16) exists as an improper one, and F ( y ) ∼ const · y − ϕ ( y − ) , y → F isbounded near the origin. This implies the inequality (2.16).For the function Y α with n + | β | = 2 b , we use, in a similar way, the identity (3.22) with amatrix argument ζ , which results in the representation D βx Y α ( t, x ) = (2 π ) − n t − Z R n ξ β e it − α/ b x · ξ E α,α ( A ( ξ )) dξ. ∂∂t Z α ( t, x ) are obtained just as those for Z α itself.We use the representations (3.23), (3.24), and (3.26), as well as the estimate (3.25) and thematrix asymptotics given by Proposition 3. With this input, the proofs of (2.19)-(2.22) aresimilar to the ones given above. (cid:4) . In Theorem 1, the estimates are given separately for large and smallvalues of R . In order to justify the iteration procedures of the Levi method, we need unifiedestimates valid for all values of the variables. Proposition 4. If n + | β | < b , then (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α n + | β | b e − cρ ( t,x ) , c >
0; (4.17) (cid:12)(cid:12) D βx Y α ( t, x ) (cid:12)(cid:12) ≤ Ct − α − α n + | β | b e − cρ ( t,x ) . (4.18) If n + | β | > b , then (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α | x | − n +2 b −| β | e − cρ ( t,x ) . (4.19) If n + | β | = 2 b , then (cid:12)(cid:12) D βx Z α ( t, x ) (cid:12)(cid:12) ≤ Ct − α (cid:2)(cid:12)(cid:12) log (cid:0) t − α | x | b (cid:1)(cid:12)(cid:12) + 1 (cid:3) e − cρ ( t,x ) . (4.20) If n + | β | ≥ b , then (cid:12)(cid:12) D βx Y α ( t, x ) (cid:12)(cid:12) ≤ Ct − | x | − n +2 b −| β | e − cρ ( t,x ) . (4.21) The constants can depend only on the parameters listed in the formulation of Theorem 1.Proof . The estimate (4.17) coincides with (2.10), if R ≥
1, being obviously equivalent to(2.12), if R ≤ n + | β | < b .Let us consider the case where n + | β | > b . It is clear that (4.19) is equivalent to (2.14), if R ≤
1. If R ≥
1, we rewrite the right-hand side of (2.10) as follows. Let σ = c ′ + c ′′ ( c ′ , c ′′ > t − α n + | β | b e − c ( t − α/ b | x | ) b b − α = t − α | x | − n +2 b −| β | (cid:20)(cid:0) t − α/ b | x | (cid:1) n + | β |− b e − c ′ ( t − α/ b | x | ) b b − α (cid:21) e − c ′′ ( t − α/ b | x | ) b b − α ≤ Ct − α | x | − n +2 b −| β | e − cρ ( t,x ) where c = c ′′ , and we have proved (4.19).The proofs of (4.18), (4.20), and (4.21) are similar. (cid:4) Parametrix
The parametrix kernels Z (0) α ( t, x ; y ) and Y (0) α ( t, x ; y ) defined in Section 2.2 satisfy all the esti-mates of Theorem 1 and Proposition 4, with all the constants independent of y .We need also estimates of the differences Z (0) α ( t, x ; y ′ ) − Z (0) α ( t, x ; y ′′ ) and Y (0) α ( t, x ; y ′ ) − Y (0) α ( t, x ; y ′′ ). These estimates are identical to those for Z (0) α and Y (0) α themselves, with anadditional factor | y ′ − y ′′ | γ . The proofs are the same as in Theorem 1 and Proposition 4, withthe following additional tools: the difference estimates for classical parabolic systems [6]; thedifference estimate (3.13) for the Green matrices of elliptic systems; the estimate for (cid:12)(cid:12)(cid:12) e A ( y ′ ,η ) − e A ( y ′′ ,η ) (cid:12)(cid:12)(cid:12) given in Chapter 1 of [6]. We omit further details since they just repeat the above material.As in Section 2.1, we have the integral identities Z R n Z (0) α ( t, x ; y ) dx = 1 , Z R n Y (0) α ( t, x ; y ) dx = t α − Γ( α ) . (5.1)It follows from the difference estimates and the first identity in (5.1) that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R n ∂∂t Z (0) α ( t, x − ξ ; ξ ) dξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ct − αγ b . (5.2) Given the estimates of Theorem 1 and Proposition 4, the proof of Theorem 2 is carried out justas its counterpart for fractional diffusion equations [8, 7]. The integral inequalities needed forthe proof are given in sufficient generality in [7]. Therefore we drop the detailed calculationsand give only the scheme and the main estimates.We look for the functions Z (1) α ( t, x ; ξ ), Y (1) α ( t, x ; ξ ) appearing in Theorem 2 assuming thefollowing integral representations: Z (1) α ( t, x ; ξ ) = Z (0) α ( t, x − ξ ; ξ ) + t Z dλ Z R n Y (0) α ( t − λ, x − y ; y ) Q ( λ, y ; ξ ) dy ; (6.1) Y (1) α ( t, x ; ξ ) = Y (0) α ( t, x − ξ ; ξ ) + t Z dλ Z R n Y (0) α ( t − λ, x − y ; y )Φ( λ, y ; ξ ) dy. (6.2)For the functions Q, Φ we assume the integral equations Q ( t, x ; ξ ) = M ( t, x ; ξ ) + t Z dλ Z R n K ( t − λ, x ; y ) Q ( λ, y ; ξ ) dy, (6.3)23( t, x ; ξ ) = K ( t, x ; ξ ) + t Z dλ Z R n K ( t − λ, x ; y )Φ( λ, y ; ξ ) dy, (6.4)where M ( t, x ; ξ ) = [ A ( x, D x ) − A ( ξ, D x )] Z (0) α ( t, x − ξ ; ξ ) ,K ( t, x ; ξ ) = [ A ( x, D x ) − A ( ξ, D x )] Y (0) α ( t, x − ξ ; ξ )Using the estimates from Proposition 4 we find that | M ( t, x ; ξ ) | ≤ Ct − α | x − ξ | − n + γ e − cρ ( t,x − ξ ) , (6.5) | K ( t, x ; ξ ) | ≤ Ct − γ − η ) α b | x − ξ | − n + η e − cρ ( t,x − ξ ) , (6.6) c >
0, 0 < η < γ .The increments of M and K are estimated as follows. Let ∆ x M ( t, x ; ξ ) = M ( t, x ; ξ ) − M ( t, x ′ ; ξ ), ∆ x K ( t, x ; ξ ) = K ( t, x ; ξ ) − K ( t, x ′ ; ξ ). Denote by x ′′ one of the points x, x ′ , forwhich | x ′′ − ξ | = min {| x − ξ | , | x ′ − ξ |} . Then | ∆ x M ( t, x ; ξ ) | ≤ Ct − α | x − x ′ | γ − ε | x ′′ − ξ | − n + ε exp {− σρ ( t, x ′′ − ξ ) } , (6.7) | ∆ x K ( t, x ; ξ ) | ≤ Ct − | x − x ′ | γ − ε | x ′′ − ξ | − n + ε exp {− σρ ( t, x ′′ − ξ ) } , (6.8) ε > n = 2 and ν < | Q ( t, x ; ξ ) | ≤ Ct − α | x − ξ | − n + γ exp {− σρ ( t, x − ξ ) } , (6.9) | Φ( t, x ; ξ ) | ≤ Ct − | x − ξ | − n + γ exp {− σρ ( t, x − ξ ) } , (6.10) | ∆ x Q ( t, x ; ξ ) | ≤ Ct − α | x − x ′ | γ − ε | x ′′ − ξ | − n + ε exp {− σρ ( t, x ′′ − ξ ) } , (6.11) | ∆ x Φ( t, x ; ξ ) | ≤ Ct − | x − x ′ | γ − ε | x ′′ − ξ | − n + ε exp {− σρ ( t, x ′′ − ξ ) } , (6.12)Now the representation of the Green matrix stated in Theorem 2 follows from its construc-tion (6.1)-(6.2) while the estimates (2.26)-(2.31) are obtained from (6.9)-(6.12) and Lemmas1.12, 1.13 from [7]. The above estimates, together with the inequality (5.2), make it possiblealso to repeat, without significant changes, the whole reasoning from [8] or [7] regarding theheat potential and the initial condition.It also follows from (6.11), (6.12), and the difference estimates of Section 5, that the dif-ferences D βx [ V Z ( t, x ′ ; ξ ) − V Z ( t, x ′′ ; ξ )], D βx [ V Y ( t, x ′ ; ξ ) − V Y ( t, x ′′ ; ξ )], | β | ≤ b , satisfy the esti-mates similar to those for D βx V Z , D βx V Y , with the additional factor | x ′ − x ′′ | γ − ε .Let us find out when the solution u ( t, x ) of the form (2.25) is a uniform classical solutionwith the global H¨older properties. We consider the more complicated second term w ( t, x ) = t Z dλ Z R n Y (1) α ( t − λ, x ; y ) f ( λ, y ) dy = w ( t, x ) + w ( t, x )24here w ( t, x ) = t Z dλ Z R n Y (0) α ( t − λ, x − y ; y ) f ( λ, y ) dy,w ( t, x ) = t Z dλ Z R n V Y ( t − λ, x ; y ) f ( λ, y ) dy (the first term in (2.25) can be considered similarly). The remainder kernel V Y is less singularthan Y (0) α , so that the uniform convergence of derivatives of w is verified in a straightforwardway. The same can be said about lower order derivatives of w .For the leading derivatives, we have, just as in [8] or [7], the expressions D βx w ( t, x ) = t Z dλ Z R n D βx Y (0) α ( t − λ, x − y ; y )[ f ( λ, y ) − f ( λ, x )] dy + t Z f ( λ, x ) dλ Z R n D βx Y (0) α ( t − λ, x − y ; y ) dy, | β | = 2 b, (6.13) (cid:16) D ( α ) t w (cid:17) ( t, x ) = f ( t, x ) + t Z dλ Z R n ∂Z (0) α ( t − λ, x − y ; y ) ∂t [ f ( λ, y ) − f ( λ, x )] dy + t Z f ( λ, x ) dλ Z R n ∂Z (0) α ( t − λ, x − y ; y ) ∂t dy. (6.14)The global H¨older property of the derivatives (6.13) follows from the difference estimates of D βx Y (0) α and our assumptions regarding the function f .The representation (6.14) is obtained as follows (see [8] or [7]). First of all, if v ( t, x ) = (cid:0) I − α w (cid:1) ( t, x ), then D ( α ) t w = ∂v∂t , and v ( t, x ) = t Z dλ Z R n Z (0) α ( t − λ, x − y ; y ) f ( λ, y ) dy. For a small positive number h , set v h ( t, x ) = t − h Z dλ Z R n Z (0) α ( t − λ, x − y ; y ) f ( λ, y ) dy. Then ∂v h ∂t = v (1) h + v (2) h where v (1) h ( t, x ) = Z R n Z (0) α ( h, x − y ; y ) f ( t − h, y ) dy, (2) h ( t, x ) = t − h Z dλ Z R n ∂Z (0) α ( t − λ, x − y ; y ) ∂t f ( λ, y ) dy. We have v (1) h ( t, x ) = Z R n (cid:2) Z (0) α ( h, x − y ; y ) − Z (0) α ( h, x − y ; x ) (cid:3) f ( t − h, y ) dy + Z R n Z (0) α ( h, x − y ; x )[ f ( t − h, y ) − f ( t − h, x )] dy + f ( t − h, x ) . It follows from the estimates of the parametrix kernel and its differences, and from theglobal H¨older property of f that both integrals in the last formula tend to zero, as h → x ∈ R n , t ∈ [0 , T ].Similarly, we prove the convergence of v (2) h ( t, x ) to the sum of the two integrals appearing in(6.14), uniformly with respect to x, t . This proves the uniform property of our solution. (cid:4) First we consider the model system (2.1) with constant coefficients. A uniform classical solutionof the system (2.1) can be interpreted as a classical solution of the operator-differential equation (cid:16) D ( α ) t w (cid:17) ( t ) = B w ( t ) (7.1)in the Banach space C b ( R n ) N of bounded continuous vector-functions with the supremum norm.Here B is the closed operator on C b ( R n ) N defined by A ( D ) with the domain (cid:8) v ∈ C b ( R n ) N : A ( D ) v ∈ C b ( R n ) N (cid:9) ( A ( D ) v is understood in the sense of tempered distributions). Let q be such a natural numberthat q > ( N + n ) · b . By Theorem 4.1 of the paper [16], under the condition (2.2) (in fact, thePetrowsky parabolicity condition would suffice), we have (cid:13)(cid:13) ( λI − B ) − ( I − ∆) − q/ (cid:13)(cid:13) ≤ p ( | λ | ) , Re λ > , (7.2)where p is a certain polynomial.Note that the operator ( I − ∆) − q/ is bounded on C b ( R n ) N ; that follows from the integrabilityof its integral kernel [43]. Under the assumption (7.2), the equation (7.1) has only a trivialsolution w ∈ C b ( R n ) N with w (0) = 0. That is proved exactly as the uniqueness theorem from[19] where it was assumed that lim sup λ →∞ λ − /α log k ( λI − B ) − k = 0. One should just repeatthe whole reasoning from [19] for the function ( I − ∆) − q/ w , instead of w , and notice that theoperator ( I − ∆) − q/ is injective.Thus, we have proved Theorem 3 for the system (2.1). Turning to the general case, werewrite the system (1.1) with f = 0 in the form (cid:16) D ( α ) t u (cid:17) ( t, x ) − A ( y, D x ) u ( t, x ) = [ A ( x, D x ) − A ( y, D x )] u ( t, x ) + A ( x, D x ) u ( t, x ) . (7.3)26ere y ∈ R n is an arbitrary fixed point. As before, we assume that u (0 , x ) = 0.By Theorem 2, we can write down an integral representation of a uniform classical solutionof the Cauchy problem for the equation (7.3). By the above uniqueness result for model systemswith constant coefficients, we obtain the equality u ( t, x ) = t Z dτ Z R n Y (0) α ( t − τ, x − ξ ; y )[ A ( ξ, D ξ ) − A ( y, D ξ )] u ( t, ξ ) dξ + t Z dτ Z R n Y (0) α ( t − τ, x − ξ ; y ) A ( ξ, D ξ ) u ( t, ξ ) dξ. (7.4)Denote v ( t ) = X | m |≤ b sup x ∈ R n | D mx u ( t, x ) | . Differentiating both sides of (7.4) in x and taking into account the boundedness of the deriva-tives of u we come to the inequality | D mx u ( t, x ) | ≤ C t Z v ( τ ) dτ Z R n (cid:12)(cid:12) D mx Y (0) α ( t − τ, x − ξ ; y ) (cid:12)(cid:12) dξ, | m | < b. Using the estimates (4.18) and (4.21) we find that | D mx u ( t, x ) | ≤ C t Z v ( τ )( t − τ ) − b −| m | ) α b dτ, | m | < b. (7.5)For | m | = 2 b , the derivatives of a heat potential are regularized by subtraction, that is D mx u ( t, x ) = t Z dτ Z R n D mx Y (0) α ( t − τ, x − ξ ; y ) { [ A ( ξ, D ξ ) − A ( y, D ξ )] u ( t, ξ ) − [ A ( x, D x ) − A ( y, D x )] u ( t, x ) } dξ + t Z dτ Z R n D mx Y (0) α ( t − τ, x − ξ ; y )[ A ( ξ, D ξ ) u ( t, ξ ) − A ( x, D x ) u ( t, x )] dξ. Until now, y ∈ R n was an arbitrary parameter. Set y = x . Then one of the terms in the firstintegral disappears. For the remaining terms, we use the inequalities | [ A ( ξ, D ξ ) − A ( x, D ξ )] u ( t, ξ ) | ≤ C | x − ξ | γ v ( t ) , | A ( ξ, D ξ ) u ( t, ξ ) − A ( x, D x ) u ( t, x ) | ≤ C | x − ξ | sup x ∈ R n | β |≤ deg A +1 (cid:12)(cid:12) D βx u ( t, x ) (cid:12)(cid:12) ≤ C | x − ξ | v ( t )(since deg A + 1 ≤ b ). 27t follows from (4.21) that | D mx u ( t, x ) | ≤ C t Z v ( τ )( t − τ ) − dτ × Z R n ( | x − ξ | γ + | x − ξ | ) | x − ξ | − n exp n − c (cid:2) ( t − τ ) − α | x − ξ | b (cid:3) b − α o dξ ≤ C t Z v ( τ )( t − τ ) − αγ b dτ, | m | = 2 b . Adding this inequality to (7.5) we find that, for any t ∈ [0 , T ], v ( t ) ≤ C t Z v ( τ )( t − τ ) − ε dτ, ε > . (7.6)By a kind of the Bellman-Gronwall inequality proved by Henry ([15], Lemma 7.1.1), it followsfrom (7.6) that v ( t ) ≡
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