Lp–Lq decay estimates for dissipative linear hyperbolic systems in 1D

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Lp –Lq decay estimates for dissipative linear hyperbolic systems in 1D Corrado Mascia a , Thinh Tien Nguyen b,∗ a Dipartimento di Matematica “G. Castelnuovo”, Sapienza – Università di Roma, P.le Aldo Moro, 2 – 00185 Roma,

Italy b Department of Mathematics, Gran Sasso Science Institute – Istituto Nazionale di Fisica Nucleare, Viale Francesco

Crispi, 7 – 67100 L’Aquila, Italy Received 12 April 2017

Abstract Given A, B ∈ Rn×n , we consider the Cauchy problem for partially dissipative hyperbolic systems having the form ∂t u + A∂x u + Bu = 0, with the aim of providing a detailed description of the large-time behavior. Sharp Lp –Lq estimates, for 1 ≤ q ≤ p ≤ ∞, are established for the distance between the solution to the system and a time-asymptotic profile, where the profile is the superposition of diffusion waves and exponentially decaying waves. They show that the solution to the system decays to the diffusion waves 1/2 faster than the diffusive decay rate (1/q − 1/p)/2 while the exponential decaying waves are negligible for large time. In particular, under a symmetry property, it decays 1 faster than. The proof is based on a complex interpolation argument once Lr estimates for the fundamental solution to the system in the frequency space and Fourier multiplier estimates are accomplished. © 2017 Elsevier Inc. All rights reserved. MSC: 35L45; 35C20 Keywords: Large-time behavior; Dissipative linear hyperbolic systems; Asymptotic expansions

* Corresponding author.

E-mail addresses: [email protected] (C. Mascia), [email protected] (T.T. Nguyen). http://dx.doi.org/10.1016/j.jde.2017.07.011 0022-0396/© 2017 Elsevier Inc. All rights reserved.

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1. Introduction This work is concerned with the Cauchy problem ∂t u + Lu := ∂t u + A∂x u + Bu = 0,

u(0, x) = u0 (x).

(1.1)

Although we are going to discuss properties of the system (1.1) on its own, we primarily regard the system (1.1) as a linearization of the nonlinear hyperbolic system ∂t u + ∂x F (u) + G(u) = 0

(1.2)

at a constant stationary state u¯ satisfying G(u) ¯ = 0 (see [1,7] and descendants). In addition, linear systems fitting in the class (1.1) emerge as models for velocity jump processes such as the Goldstein–Kac model [3,5] (a generalization was introduced in [9]) and in other fields of application. Decay estimates for the system (1.1) have been accomplished for years. In [10], it is proved that the L2 -norm of the solution u to (1.1) is bounded by the sum of the two terms: the first term, with respect to the L2 -norm of the initial datum of u, decays exponentially and the second one, with respect to the Lq -norm of the initial datum of u for q ∈ [1, ∞], decays at the rate (1/q − 1/2)/2. In that work, the matrices A and B are symmetric and they satisfy the Shizuta– Kawashima condition: if z is an eigenvector of A, then z ∈ / ker B, which is required for designing a compensating matrix to capture the dissipation of the system (1.1) over the degenerate kernel space of B since the symmetric structure is not strong enough to guarantee the decay. The result is improved in [2] where (1.1) can be written in a conservative–dissipative form, where A is symmetric and B has the form diag (Om×m , D) with Om×m the m × m null matrix and D an (n − m) × (n − m) positive definite matrix. If the Shizuta–Kawashima condition holds, then by considering the parabolic equation given by applying the Chapman–Enskog expansion into the system (1.1), the Lp -norm of the difference between the solution u and the solution U to the parabolic equation decays 1/2 faster than the rate (1 − 1/p)/2 in terms of the L1 ∩ L2 -norm of the initial datum of u. Recently, [11] can be seen as a generalization of [10] for any nonsymmetric matrix B under appropriate conditions. More detailed descriptions of the asymptotic behavior have been provided for specific classes of equations by Lp –Lq estimates e.g. the Lp –Lq estimates for the Cauchy problem for the damped wave equation ∂t u + ∂tt u − u = 0.

(1.3)

These estimates show that the time-asymptotic profile of the solution u to (1.3) includes the solution to a heat equation and the solution to a wave equation, and when measuring the initial datum of u in Lq , the Lp distance between u and the profile decays ε > 0 faster than the rate α(p, q) := d(1/q − 1/p)/2, where d is the spacial dimension, for any 1 ≤ q ≤ p ≤ ∞ (see [4,8]). We are not aware of any results on such kind of estimates for the general system (1.1) with general exponents p and q. Hence, the aim of this paper is to establish the Lp –Lq estimates for the Cauchy problem (1.1) for any 1 ≤ q ≤ p ≤ ∞. We start with the following assumptions on the matrices A and B. Condition A. [Hyperbolicity] The matrix A is diagonalizable with real eigenvalues.

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Condition B. [Partial dissipativity] 0 is a semi-simple eigenvalue of B with algebraic multiplicity m ≥ 1 and the spectrum σ (B) of B can be decomposed as {0} ∪ σ0 where σ0 ⊂ {λ ∈ C : Re λ > 0}. (0)

Let P0 be the unique eigenprojection associated with the eigenvalue 0 of B, then the reduced system is given by ∂t w + C∂x w ≈ 0, (0)

(0)

(1.4)

(0)

where w := P0 u and C := P0 AP0 . One assumes Condition C. [Reduced hyperbolicity] The matrix C considered in ker(B) is diagonalizable with real eigenvalues. On the other hand, a requisite condition for the decay of the solution to the system (1.1) that is related to the Shizuta–Kawashima condition is that the eigenvalues λ(iξ ) of the matrix operator E(iξ ) := −(B + iξ A) satisfy Condition D. [Uniform dissipativity] There is θ > 0 such that Re (λ(iξ )) ≤ −

θ |ξ |2 , 1 + |ξ |2

for ξ = 0.

(1.5)

The above structure follows the ones introduced in [2,10,11]. Then, under the assumptions A, B, C and D, the time-asymptotic profile of the solution to the system (1.1) is the superposition of diffusion waves and exponentially decaying waves. The diffusion waves are constructed as follows. Let 0 be an oriented closed curve in the resolvent set ρ(B) of B enclosing 0 except for the nonzero eigenvalues of B, one sets (0)

S0 :=

1 2πi



z−1 (B − zI )−1 dz.

(1.6)

0

On the other hand, let (1)

(0)

(0)

(0)

(0)

P0 := −P0 AS0 − S0 AP0 ,

(1.7)

 (1) (1) (0) (1) (1) (0)  D := − P0 BP0 + P0 AP0 + P0 AP0 .

(1.8)

one defines

 (0)  Then, we consider the Cauchy problem with respect to U in ran Pj , that is (0)

∂t U + cj ∂x U − Pj D∂xx U = 0,

(0)

U (0, x) = Pj u0 (x),

(1.9) (0)

where cj is the j -th element of the spectrum σ (C) of C considered in ker(B), Pj is the eigenprojection associated with it for j ∈ {1, . . . , h} and h ≤ m is the cardinality of the spectrum σ (C)

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 of C considered in ker(B). Then, we consider U := hj=1 Uj where Uj is the solution to the system (1.9) for j ∈ {1, . . . , h}. (0) (0) In particular, the coefficients P0 and S0 can be computed by the formulas (0)

P0 = Pm−1 (B)

(0)

S0 = Sm−1 (B),

and

(1.10)

where the matrix-valued functions P and S are introduced in (A.1) and (A.2) in the appendix (0) section respectively. Moreover, let α > max{|λ| : λ ∈ σ (C)} and let C := C + αP0 , then C has h distinct nonzero eigenvalues denoted by cj with algebraic multiplicities mj ≥ 1 for j ∈ (0)

{1, . . . , h}. Then, Pj

can be computed by the formula (0)

Pj

= Pmj −1 (C − cj I ),

j ∈ {1, . . . , h}.

(1.11)

Noting that the shifting from C to C is requisite since we consider only the eigenvalues of C considered in ker(B). The exponentially decaying waves are constructed as follows. Due to the diagonalizable property of A, let Q ∈ Rn×n be the invertible matrix diagonalizing A. Then, one sets B¯ := Q−1 BQ,

A¯ := Q−1 AQ = diag (a1 , . . . , an ),

(1.12)

where aj for j ∈ {1, . . . , n} are the repeated eigenvalues of A. Let define a partition denoted by {Sj : j = 1, . . . , s} of {1, . . . , n} for some s ≤ n such that h, k ∈ Sj if ah = ak , it is easy to see ¯ of A. ¯ On the other hand, we also define the matrix that s is the cardinality of the spectrum σ (A) 



(0) j hk

 :=

1 if h = k ∈ Sj , 0 otherwise,

(1.13)

 (0)  for h, k = 1, . . . , n. Then, we consider the Cauchy problem with respect to V in ran j , that is (0) ¯ ∂t V + αj ∂x V + j BV = 0,

V (0, x) = j Q−1 u0 (x), (0)

where αj = ah if h ∈ Sj for j ∈ {1, . . . , s}. Then, we consider V := Q solution to the system (1.14) for j ∈ {1, . . . , s}.

s

j =1 Vj

(1.14) where Vj is the

Theorem 1.1 (Lp –Lq decay estimates). If u is the solution to the system (1.1) with the initial datum u0 ∈ Lq (R), the conditions A, B, C and D imply that, for 1 ≤ q ≤ p ≤ ∞ and t ≥ 1, there are constants C := C(p, q) > 0 and δ > 0 such that

u − U − V Lp ≤ Ct

− 12

1

1 1 q −p −2

u0 Lq ,

(1.15)

where

U Lp ≤ Ct

− 12

1

1 q −p

u0 Lq

and

V L2 ≤ Ce−δt u0 L2 .

(1.16)

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The estimates (1.15) and (1.16) show that the term V can be neglected and u has the same Lp –Lq decay estimates for U for large time. Moreover, (1.15) also shows how fast the solution u converges to U similarly to [2]. Noting that, (1.15) holds for general exponents p and q due to the fact that the explicit corrector term V is subtracted. The proof of Theorem 1.1 is based on a complex interpolation argument once the Lr estimates for the fundamental solution in the frequency space and the analysis of the Fourier multipliers are established. Going back to the Lp –Lq decay estimates in [8], if the initial condition of the Cauchy problem for (1.3) is given by (u, ∂t u)|t=0 = (u0 , u1 ), then, in one dimension, the following estimate holds     11 1  u − U − e−t/2 V  p ≤ C t − 2 q − p −1 u0 , u1  q , L L

∀t ≥ 1,

(1.17)

where respectively, U and V are the solutions to the Cauchy problems 



∂t U − ∂xx U = 0, U (x, 0) = u0 (x) + u1 (x),

and

∂tt V − ∂xx V = 0, V (x, 0) = u0 (x),

∂t V (x, 0) = 0.

Comparing (1.15) with (1.17), we recognize a difference of 1/2 in the decay rates. The better decay, which is valid for the linear damped wave equation, is a consequence of an additional property, namely the invariance with respect to the transformation x → −x. Indeed, in terms of the Goldstein–Kac system, such symmetry implies that the eigenvalue curves of E(iξ ) = −(B + iξ A) which pass through 0 can be expanded as λ(iξ ) := −d0 ξ 2 + O(|ξ |4 ) for some d0 > 0 as |ξ | → 0, and the fact that the error terms are O(|ξ |4 ) guarantees the gain of 1 instead of 1/2 in the decay rate, where 1/2 holds for general cases where the error terms are O(|ξ |3 ). Thus, we are also interested in systems fitting in the class (1.1) that have an analogous property, namely Condition S. [Symmetry] There is an invertible symmetric matrix S such that AS = −SA

BS = SB.

and

When the above assumption holds, if u := u(x, t) is a solution to (1.1), then the reflection v := v(x, t) = u(−x, t) is a solution to the same system as well. Let us consider a stronger assumption than the condition C on the reduced system, namely Condition C’. [Reduced strictly hyperbolicity] The matrix C considered in ker(B) is diagonalizable with m real distinct eigenvalues. Let U :=

m

j =1 Uj

where Uj , for j ∈ {1, . . . , m}, is the solution to the system

∂t U + cj ∂x U − dj ∂xx U = 0,

 (0) (1)  U (0, x) = Pj + Pj ∂x u0 (x), (0)

(0)

where dj is the “unique” eigenvalue of the matrix Pj DPj where

(0) Pj

is already introduced and

(1) Pj

(1.18)

considered in ker(C − cj I )|ker(B)

is defined as follows.

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For j ∈ {1, . . . , m}, let j be an oriented closed curve in the resolvent set ρ(C ) of C enclosing cj except for the other eigenvalues of C . One sets (0) Sj

1 := 2πi



z−1 (C − zI )−1 dz,

j ∈ {1, . . . , m}.

(1.19)

j (1)

Then, Pj

can be computed by (1)

Pj

(0)

(0)

(0)

(0)

:= Pj DSj + Sj DPj ,

j ∈ {1, . . . , m}.

(1.20)

(0)

Similarly to before, Sj can be obtained by Sj = Smj −1 (C − cj I ), (0)

j ∈ {1, . . . , m}.

(1.21)

Let V be the same as before, one has Theorem 1.2 (Increased decay rate). With the same hypotheses as in Theorem 1.1, if the condition C is substituted by the condition C’ and if the condition S holds, then, for 1 ≤ q ≤ p ≤ ∞ and t ≥ 1, there are constants C := C(p, q) > 0 and δ > 0 such that     11 1  u − U − V  p ≤ C t − 2 q − p −1 u0  q , L L

(1.22)

where

U Lp ≤ Ct

− 12

1

1 q −p

u0 Lq

and

V L2 ≤ Ce−δt u0 L2 .

(1.23)

Once relaxing the condition C’ by the condition C, the decay rate in the estimate (1.15) does not increase in general since the condition S cannot prevent the eigenvalues of E which converge to 0 from exhibiting nonzero terms (iξ )3+α for α ∈ [0, 1) in their expansions, and thus, it does not permit to have the gain of 1 in the decay rate. The paper is organized as follows. In order to study the behavior of the solution to the system (1.1), we will study the asymptotic expansion of the operator E(iξ ) = −(B + iξ A) in Section 2. Then, Section 3 and Section 4 are devoted to give the a priori estimates for the solution to the system (1.1). Moreover, the symmetry property of the system (1.1) is also discussed in Section 5. Then, we prove the main theorems in Section 6. Finally, we let the appendix section for some useful facts of the perturbation theory for linear operators in finite dimensional space together with a tool for computing the eigenprojections that we used in the previous sections. Notations and definitions Given a matrix T , we denote by ker(T ), ran(T ), ρ(T ) and σ (T ) the kernel, the range, the resolvent set and the spectrum of T respectively. On the other hand, λ is an eigenvalue of T considered in a domain D ⊆ Cn if there is u ∈ D such that u = On×1 and T u = λu. If the domain is not considered explicitly, it means D ≡ Cn .

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Moreover, the notation σ (T , D) means the set of the eigenvalues of T considered in D and σ (T , Cn ) ≡ σ (T ). For x ∈ C, assume that T (x) = T (0) + O(|x|) as |x| → 0, the set of all λ ∈ σ (T ) satisfying λ(x) → λ(0) as |x| → 0 where λ(0) ∈ σ (T (0) ) is called the λ(0) -group of T . Moreover, P := P (x) is called the total projection of a λ(0) -group of T if P is the sum of the eigenprojections associated with the eigenvalues of T belonging to that group. Let T : R → B where B is a Banach space endowed with some suitable norm | · |B . The Lp (R, B) space is the space of all T equipped with the norm

T Lp

⎛ +∞ ⎞1/p  := ⎝ |T (x)|B dx ⎠ , −∞

for 1 ≤ p < ∞ and T L∞ :=

ess sup |T (x)|B . From here, we use the notation | · | instead of

−∞
| · |B to indicate the norm associated with B. Let m be a tempered distribution, m is called a Fourier multiplier on Lp , for 1 ≤ p ≤ ∞, if

m Mp :=

sup

f Lp ≤1

F −1 (m) ∗ f Lp < +∞.

The Mp space is the space of Fourier multipliers on Lp endowed with the norm · Mp for 1 ≤ p ≤ ∞. 2. Asymptotic expansions In this section, we will study the asymptotic expansions of the operator E(iξ ) = −(B + iξ A) by dividing the frequency domain ξ ∈ R into the low frequency as |ξ | → 0, the intermediate frequency as |ξ | different from 0 and +∞ and the high frequency as |ξ | → +∞. Before going further in details, noting that if an eigenvalue λ of a matrix T is considered in a domain D ⊂ Cn , it may happen the cases where λ is considered in Cn − D as   the complement   n well. For instance, we will see that it may happen 0 ∈ σ C, ker(B) and 0 ∈ σ C, C − ker(B) . Thus, in order to avoid heavy notations, if we denote by mλ and Pλ the algebraic multiplicity and the eigenprojection associated with λ respectively, where λ ∈ σ (T , D), we consider that ran(Pλ ) ⊆ D and mλ = dim ran(Pλ ). Primarily, we consider the low frequency. Let the matrices C and D be in (1.4) and (1.8) respectively. Recall cj the j -th element of σ (C, ker(B)) together with the algebraic multiplicity mj and the eigenprojection Pj(0) associated with it for j ∈ {1, . . . , h}, where h is the cardinality (0)

of σ (C, ker(B)). Recall the algebraic multiplicity m and the eigenprojection P0 associated with the eigenvalue 0 of B. Noting that under the condition B, h ≤ m since  (0)  (0) (0)  (0)  C I − P0 = P0 AP0 I − P0 = On×1   (0)    (0)  (0)  and dim ran I − P0 = n − dim ran P0 = n − m i.e. 0 ∈ σ C, ran I − P0 with algebraic multiplicity n − m. Thus, the number of the eigenvalues of C considered in ker(B) that is h cannot exceed m.

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Proposition 2.1 (Low frequency 1). If the conditions B and C hold, the 0-group is included in σ (E) and for j ∈ {1, . . . , h}, there is an integer hj ≤ mj such that the 0-group of E is divided  into hj=1 hj distinct groups. Moreover, the elements of the j -th group are approximated by λj  (iξ ) = −icj ξ − dj  ξ 2 + O(|ξ |2 ),

|ξ | → 0,

(2.1)

  (0) (0) where dj  ∈ σ Pj DPj , ker(C − cj I )|ker(B) for j ∈ {1, . . . , h} and  ∈ {1, . . . , hj }. In particular, if the condition D holds, then Re (dj  ) ≥ θ > 0,

for j ∈ {1, . . . , h} and  ∈ {1, . . . , hj }.

(2.2)

Moreover, the total projection associated with the j -th group is approximated by Pj  (iξ ) = Pj(0)  + O(|ξ |),

|ξ | → 0,

(2.3)

(0)

where Pj  is the eigenprojection associated with dj  for j ∈ {1, . . . , h} and  ∈ {1, . . . , hj }. Proof. We can consider T (ζ ) := B + ζ A where ζ = iξ instead of E since E = −T . The proof then includes three steps of approximation and reduction steps interlacing them. Step 0. Since 0 ∈ σ (B) by the condition B, the 0-group is included in σ (T ) due to Proposition Appendix B.2 and it is obvious that by the definition of the 0-group, the approximation of the elements of the group has the form λ0 (ζ ) = O(1),

|ζ | → 0.

On the other hand, also from Proposition Appendix B.2, the total projection associated with this group is approximated by (0)

P0 (ζ ) = P0 + O(|ζ |),

|ζ | → 0.

In particular, a more accurate expansion of P0 is given by (0)

(1)

P0 (ζ ) = P0 + ζ P0 + O(|ζ |2 ),

|ζ | → 0,

(2.4)

(1)

where P0 can be computed by the formula (1.7) (see [6]). Reduction step. Moreover, P0 and I − P0 satisfy (B.1) and (B.2) due to Proposition Appendix B.2. Thus, one obtains   Cn = ran(P0 ) ⊕ Cn − ran(P0 )

and

T = T P0 + T (I − P0 ).

  The study of the 0-group of T considered in Cn is then reduced to the study of σ T P0 , ran(P0 ) . The remainder is related to the other groups of T that we will study later in Proposition 2.2. Step 1. Since 0 is semi-simple by the condition B, based on the expansion (2.4) of P0 and the fact that T P0 = P0 T P0 , one has

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 (0)   (0)  (1) (1) T (ζ )P0 (ζ ) = P0 + ζ P0 + O(|ζ |2 ) (B + ζ A) P0 + ζ P0 + O(|ζ |2 )   = ζ C − ζ D + O(|ζ |2 ) , |ζ | → 0.     It follows that λ ∈ σ T P0 , ran(P0 ) if and only if ζ −1 λ =: λ˜ ∈ σ T0 , ran(P0 ) , where T0 (ζ ) := C − ζ D + O(|ζ |2 ) as |ζ | → 0. Therefore, it returns to the eigenvalue problem of T0 considered in ran(P0 ). Hence, from the definition of cj for j ∈ {1, . . . , h} and Proposition Appendix B.2, we obtain   that λ˜ → cj as |ζ | → 0 for j ∈ {1, . . . , h}. Hence, λ ∈ σ T P0 , ran(P0) if and only if ζ −1 λ → cj as |ζ | → 0 for j ∈ {1, . . . , h}. One concludes that σ T P0 , ran(P0 ) is characterized by cj for j ∈ {1, . . . , h} and thus, it is divided into h distinct groups such that the approximation of the elements of the j -th group has the form λj (ζ ) = cj ζ + O(|ζ |),

|ζ | → 0,

j ∈ {1, . . . , h}.

On the other hand, Proposition Appendix B.2 also implies that the total projection associated with this group is approximated by (0)

Pj (ζ ) = Pj

+ O(|ζ |),

|ζ | → 0,

j ∈ {1, . . . , h}.

(2.5)

Reduction step. Then, since Pj for j ∈ {1, . . . , h} satisfy (B.1) and (B.2) by Proposition Appendix B.2, one has

ran(P0 ) =

h j =1

ran(Pj )

and

T0 =

h

(T0 Pj ). j =1

    The study of σ T0 , ran(P0 ) is then reduced to the study of σ T0 Pj , ran(Pj ) for j ∈ {1, . . . , h}. Final step. For j ∈ {1, . . . , h}, since cj is semi-simple by the condition C, based on the expansion (2.5) of Pj and the fact that T0 Pj = Pj T0 Pj , one has (T0 (ζ ) − cj I )Pj (ζ )     = Pj(0) + O(|ζ |) C − cj I − ζ D + O(|ζ |2 ) Pj(0) + O(|ζ |)   (0) (0) = ζ −Pj DPj + O(|ζ |) , |ζ | → 0.     It follows that λ ∈ σ T0 Pj , ran(Pj ) if and only if ζ −1 (λ − cj ) =: λ˜ ∈ σ Tj , ran(Pj ) , where Tj (ζ ) := −Pj(0) DPj(0) + O(|ζ |) as |ζ | → 0. Thus, it returns to the eigenvalue problem of Tj considered in ran(Pj ).   (0) (0) Let dj  be the -th element of σ Pj DPj , ker(C − cj I )|ker(B) for  ∈ {1, . . . , hj }, where   (0) (0) hj is the cardinality of σ Pj DPj , ker(C − cj I )|ker(B) . Noting that similarly to the argument for h, one has hj ≤ mj , where mj is the algebraic multiplicity of cj . Then, Proposition   Appendix B.2 implies that λ˜ → −dj  as |ζ | → 0 for  ∈ {1, . . . , hj }. Thus, λ ∈ σ T0 Pj , ran(Pj ) if and only if ζ −1 (λ − cj ) → −dj  as |ζ | → 0 for  ∈ {1, . . . , hj }. One concludes that

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  σ T0 Pj , ran(Pj ) is characterized by dj  for  ∈ {1, . . . , hj } and thus, it is divided into hj distinct groups such that the approximation of the elements of the -th group has the form λj  (ζ ) = cj ζ − dj  ζ 2 + O(|ζ |2 ),

|ζ | → 0,

 ∈ {1, . . . , hj }.

Moreover, also from Proposition Appendix B.2, the total projection associated with this group is approximated by Pj  (ζ ) = Pj(0)  + O(|ζ |),

|ζ | → 0,

(2.6)

(0)

where Pj  is the eigenprojection associated with dj  for  ∈ {1, . . . , hj }. We then deduce from the above steps of approximation for E(iξ ) = −T (iξ ) by multiplying λj  (iξ ) by −1 to obtain (2.1), and (2.3) is the same as Pj  (iξ ) for j ∈ {1, . . . , h} and  ∈ {1, . . . , hj }. Finally, we prove the estimate (2.2). For j ∈ {1, . . . , h} and  ∈ {1, . . . , hj }, since λj  in (2.1) can be seen as an eigenvalue of E and since cj is real by the condition C, if the condition D holds, then for |ξ | small, one has Re (λj  (iξ )) = −Re(dj  )|ξ |2 + Re (O(|ξ |2 )) ≤ −θ|ξ |2 (1 + |ξ |2 )−1 . Passing through the limit as |ξ | → 0, one has the desired estimate. The proof is done. Remark 1. As a consequence, for |ξ | small, for j ∈ {1, . . . , h} and  ∈ {1, . . . , hj }, in ran(Pj  ), the operator E has the representation (0)

Ej  (iξ ) = (−icj ξ − dj  ξ 2 )I − ξ 2 Nj  + O(|ξ |3 ),

(2.7)

(0)

where Nj  is the nilpotent matrix associated with dj  . Let k be the number of the nonzero distinct eigenvalues of B. Proposition 2.2 (Low frequency 2). If the condition B holds, then except for the 0-group, there are k distinct groups of the eigenvalues of E such that the approximation of the elements of the j -th group has the form ηj (iξ ) = −ej + O(1),

|ξ | → 0,

(2.8)

where ej ∈ σ (B) with Re (ej ) > 0 for j ∈ {1, . . . , k}. Moreover, the total projection associated with the j -th group is approximated by (0)

Fj (iξ ) = Fj (0)

where Fj

  + O |ξ | ,

|ξ | → 0,

is the eigenprojection associated with ej for j ∈ {1, . . . , k}.

(2.9)

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11

Proof. Similarly to the proof of Proposition 2.1, we consider the operator T (ζ ) = B + ζ A where ζ = iξ . Let ej be the j -th element of σ (B) except for 0 for j ∈ {1, . . . , k}. Then, it is implied from Proposition Appendix B.2 that except for the 0-group of T , there are k distinct groups of the eigenvalues of T such that the approximation of the elements of the j -th group has the form ηj (ζ ) = ej + O(1),

|ζ | → 0,

and also from Proposition Appendix B.2, the total projection associated with this group is approximated by (0)

Fj (ζ ) = Fj

+ O(|ζ |),

|ζ | → 0,

(2.10)

(0)

where Fj is the eigenprojection associated with ej for j ∈ {1, . . . , k}. In particular, Re (ej ) > 0 due to the condition B. Finally, since E(iξ ) = −T (iξ ), we obtain (2.8) by multiplying ηj (iξ ) by −1, and (2.9) is the same as Fj (iξ ) for j ∈ {1, . . . , k}. Remark 2. As a consequence, for |ξ | small, in ran(Fj ), the operator E has the representation Ej (iξ ) = −ej I − Mj(0) + O(|ξ |), (0)

where Mj

(2.11)

is the nilpotent matrix associated with ej for j ∈ {1, . . . , k}.

For the high frequency, in order to analyze the eigenvalues of E(iξ ) = −(B + iξ A), one can analyze the eigenvalues of the operator ¯ ) := Q−1 E(iξ )Q = (−iξ )(A¯ + (iξ )−1 B) ¯ E(iξ where A¯ and B¯ are already introduced in (1.12). ¯ with algebraic multiplicity nj for j ∈ {1, . . . , s}, where s Let αj be the j -th element of σ (A) ¯ is the cardinality of σ (A). Proposition 2.3 (High frequency). If the  condition A holds, for j ∈ {1, . . . , s}, there is an integer ¯ is divided into sj =1 sj distinct groups and the elements of the j -th sj ≤ nj such that σ (E) group are approximated by μj  (iξ ) = −iαj ξ − βj  + O(|ξ |−1 ),

|ξ | → +∞,

(2.12)

 (0)  ¯ (0) , ker(A¯ − αj I ) and (0) that is the eigenprojection associated with where βj  ∈ σ j B j j αj is the same as (1.13) for j ∈ {1, . . . , s} and  ∈ {1, . . . , sj }. In particular, if the condition D holds, then Re (βj  ) ≥ θ > 0,

for j ∈ {1, . . . , s} and  ∈ {1, . . . , sj }.

Moreover, the total projection associated with the j -th group is then approximated by

(2.13)

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12

j  (iξ ) = j  + O(|ξ |−1 ), (0)

|ξ | → +∞,

(2.14)

(0)

where j  is the eigenprojection associated with βj  for j ∈ {1, . . . , s} and  ∈ {1, . . . , sj }. Proof. Similarly to before, we can consider T (ζ ) := A¯ + ζ B¯ where ζ = (iξ )−1 since |ζ | → 0 as |ξ | → +∞. The proof then consists of two steps of approximation and one reduction step between them. ¯ = {αj , j = 1, . . . , s}, σ (T ) is decomFirst step. From Proposition Appendix B.2, since σ (A) posed into s distinct groups such that the approximation of the elements of the j -th group of T is given by μj (ζ ) = αj + O(1),

|ζ | → 0,

and also from Proposition Appendix B.2, the total projection associated with this group is approximated by

j (ζ ) = (0) j + O(|ζ |),

|ζ | → 0,

(2.15)

(0)

where j is the eigenprojection associated with αj for j ∈ {1, . . . , s}. (0)

In particular, j Cauchy integral (0)

j = −

is exactly the same as (1.13) since it can be computed explicitly by the

1 2πi



(A¯ − zI )−1 dz = −

1 2πi

j



diag (a1 − z, . . . , an − z)−1 dz,

j

¯ enclosing αj except for the other eigenvalues of A. ¯ where j is an oriented closed curve in ρ(A) Hence, one obtains (0) ( j )hk

 1 = 0

 if h = k, ah = αj , 1 if h = k ∈ Sj , = otherwise, 0 otherwise,

for all h, k = 1, . . . , n due to the definition of Sj the j -th element of the partition {Sj : j = 1, . . . , s} of {1, . . . , n} for j ∈ {1, . . . , s}. Reduction step. By Proposition Appendix B.2, since j for j ∈ {1, . . . , s} satisfy (B.1) and (B.2), one also has C = n

s j =1

ran( j )

and

s

T= (T j ). j =1

  It implies that the study of σ (T ) is reduced to the study of σ T j , ran( j ) for j ∈ {1, . . . , s}. Final step. For j ∈ {1, . . . , s}, since αj is semi-simple by the condition A, based on the expansion (2.15) of j and the fact that T j = j T j , one has

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13

 (0)    ¯ (0) + O(|ζ |) (T (ζ ) − αj I ) j (ζ ) = j + O(|ζ |) (A¯ − αj I + ζ B) j  (0)  ¯ (0) + O(|ζ |) , = ζ j B |ζ | → 0. j     It follows that μ ∈ σ T j , ran( j ) if and only if ζ −1 (μ − αj ) =: μ˜ ∈ σ Tj , ran( j ) , where ¯ (0) Tj (ζ ) := (0) j B j + O(|ζ |) as |ζ | → 0. Therefore, it returns to the eigenvalue problem of Tj considered in ran( j ) for j ∈ {1, . . . , s}.  (0)  ¯ (0) , ker(A¯ − αj I ) for  ∈ For j ∈ {1, . . . , s}, let βj  be the -th element of σ j B j  (0)  ¯ (0) , ker(A¯ − αj I ) . Noting that similarly to {1, . . . , sj }, where sj is the cardinality of σ j B j before, one can prove sj ≤ nj , where nj is the algebraic multiplicity of αj . Then,  Proposition Ap pendix B.2 implies that μ˜ → βj  as |ζ | → 0 for  ∈ {1, . . . , sj }. Thus, μ ∈ σ T j , ran( j ) if   and only if ζ −1 (μ − αj ) → βj  as |ζ | → 0 for  ∈ {1, . . . , sj }. Hence, σ T j , ran( j ) is characterized by βj  and thus, it is divided into sj distinct groups such that the elements of the -th group are approximated by μj  (ζ ) = αj + βj  ζ + O(|ζ |),

|ζ | → 0,

 ∈ {1, . . . , sj }.

Furthermore, also from Proposition Appendix B.2, the total projection associated with this group is approximated by (0)

j  (ζ ) = j  + O(|ζ |),

|ζ | → 0,

(0)

where j  is the eigenprojection associated with βj  for  ∈ {1, . . . , sj }.   Therefore, from the above steps of approximation and by multiplying λj  (iξ )−1 by (−iξ ),    ¯ ) = (−iξ )T (iξ )−1 , and (2.14) is the same as j  (iξ )−1 ) for j ∈ we obtain (2.12) for E(iξ {1, . . . , s} and  ∈ {1, . . . , sj }. Finally, we prove the estimate (2.13). For j ∈ {1, . . . , s} and  ∈ {1, . . . , sj }, since μj  in ¯ −1 and since αj is real by the (2.12) can be seen as an eigenvalue of E¯ and thus of E = QEQ condition A, if the condition D holds, then for |ξ | large, one has Re (μj  (iξ )) = −Re(βj  ) + Re (O(1)) ≤ −θ|ξ |2 (1 + |ξ |2 )−1 . Passing through the limit as |ξ | → +∞, one obtains the desired estimate. We finish the proof. Remark 3. As a consequence, for |ξ | large, for j ∈ {1, . . . , s} and  ∈ {1, . . . , sj }, in ran( j  ), the operator E has the representation Ej  (iξ ) = (−iαj ξ − βj  )I − j  + O(|ξ |−1 ), (0)

(2.16)

(0)

where j  is the nilpotent matrix associated with βj  . For the intermediate-frequency case, since we will not use any expansion of the eigenvalues of the operator E(iξ ) = −(B + iξ A) in this domain, we only give the following remark (see [6] for more details).

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14

Remark 4 (Intermediate frequency). In the compact domain ε ≤ |ξ | ≤ R, from [6], there is only a finite number of the exceptional points at which the eigenprojections and the nilpotent parts associated with the eigenvalues of E may have poles even the eigenvalues are continuous there. On the other hand, in every simple domain excluded the exceptional points, the operator E has r (independent from ξ ) distinct holomorphic eigenvalues denoted by νj with constant algebraic multiplicity together with holomorphic eigenprojections and nilpotent parts denoted by j and j associated with them respectively for j ∈ {1, . . . , r}. Moreover, if the condition D holds, then Re (ν) < 0 for any ν ∈ σ (E). 3. Fundamental solution The aim of this section is to introduce the estimates for the fundamental solution to (1.1) in the frequency space. Let consider the fundamental system ˆ − EG ˆ = 0, ∂t G

ˆ t=0 = I, G

(3.1)

where E = E(iξ ) = −(B + iξ A) with ξ ∈ R. One sets the following kernel h,hj

ˆ K(ξ, t) :=

e(−icj ξ −dj  ξ

2 )t

(0) 2

e−Nj  ξ t Pj  , (0)

(3.2)

j,=1

and the kernel Vˆ (ξ, t) := Q

s,sj

(0)

e(−iαj ξ −βj  )t e−j  t j  Q−1 , (0)

(3.3)

j,=1

where the coefficients are introduced in the previous section. Moreover, we introduce the two useful lemmas as follows. Lemma 3.1. If X is a constant complex nilpotent matrix, then for all ε > 0, there exists C = C(ε ) > 0 such that  cX+Y  e − ecX  ≤ Ceε |c|+C|Y | |Y | and  cX+Y  e − ecX − ecX Y  ≤ Ceε |c|+C|Y | |Y |2 for every complex constant c and matrix Y . Proof. The proof is based on the existence of a basis of Cn such that |X| ≤ ε for any fixed ε > 0 once written in this basis. The existence of such a basis is based on the existence of the basis of Cn where X can be written in the Jordan normal form. One can find a detailed proof in [2]. ε

From here, if X is a constant nilpotent matrix, we can assume that |X| ≤ ε for any fixed > 0.

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15

Lemma 3.2. For a, b ≥ 0 and c, d > 0 and r ∈ [1, ∞], there exists C := C(r) > 0 such that for all t ≥ 1, one has  a b −c|·|d t  | · | t e 

1 1

Lr

a

≤ Ct − d r +b− d .

(3.4)

Proof. By changing of variables. Proposition 3.3 (Fundamental solution estimates). For 0 < ε < R < +∞, for r ∈ [1, ∞] and t ≥ 1, there exist positive constants C := C(r) and δ such that if the conditions A, B, C and D are satisfied, the following hold. 1. For |ξ | < ε, one has   G ˆ − Kˆ 

Lr

1 1

  Vˆ 

1

≤ Ct − 2 r − 2 ,

Lr

≤ Ce−δt .

(3.5)

2. For ε ≤ |ξ | ≤ R, one has       G ˆ  r , Kˆ  r , Vˆ  r ≤ Ce−δt . L L L

(3.6)

3. For |ξ | > R, one has   G ˆ − Vˆ 

Lr

≤ Ce−δt

  Kˆ 

for r > 1,

Lr

≤ Ce−δt .

(3.7)

Moreover, for |ξ | > R, we also have   −1 F (G ˆ − Vˆ )

L∞

≤ Ce−δt .

(3.8)

ˆ to the system (3.1) is given by Proof. For |ξ | < ε, by Remark 1 and Remark 2, the solution G ˆ ˆ ˆ G = G1 + G2 where h,hj

ˆ 1 (ξ, t) := G

e(−icj ξ −dj  ξ

2 )t

(0) 2 t+O (|ξ |3 )t

e−Nj  ξ



 (0) Pj  + O(|ξ |) ,

(3.9)

j,=1

and ˆ 2 (ξ, t) := G

k

(0)

e−ej t e−Mj

t+O (|ξ |)t 

(0)

Fj

 + O(|ξ |) .

(3.10)

j =1

ˆ − Kˆ = G ˆ 1 − Kˆ + G ˆ 2 = I1 + I2 + J where J := G ˆ 2 and It follows that G h,hj

I1 :=

j,=1

and

e(−icj ξ −dj  ξ

2 )t



(0) 2 t+O (|ξ |3 )t

e−Nj  ξ

(0) 2 t

− e−Nj  ξ

 Pj(0) 

(3.11)

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C. Mascia, T.T. Nguyen / J. Differential Equations ••• (••••) •••–••• h,hj

I2 :=

e(−icj ξ −dj  ξ

2 )t

(0)

e

−Nj  ξ 2 t+O (|ξ |3 )t

O(|ξ |).

(3.12)

j,=1

We begin with I1 . Since cj ∈ R for all j ∈ {1, . . . , h} by the condition C, one has   h,hj

(0)   2  −N (0) ξ 2 t+O (|ξ |3 )t −N ξ 2 t  I1  ≤ C e−Re (dj  )|ξ | t e j  − e j   . j,=1

On the other hand, for j ∈ {1, . . . , h} and  ∈ {1, . . . , hj }, from Proposition 2.1 and Remark 1, 1 Re (dj  ) ≥ θ > 0 and Nj(0)  is a nilpotent matrix. Thus, by choosing ε = 4 Re (dj  ), from Lemma 3.1, we have h,hj

  1 2 2 3 I1  ≤ C e−Re (dj  )|ξ | t e 4 Re (dj  )|ξ | t+C|ξ | t |ξ |3 t j,=1 h,hj

≤C

1

e−Re (dj  )|ξ | t e 4 Re (dj  )|ξ | 2

2 t+Cε|ξ |2 t

1

|ξ |3 t ≤ Ce− 2 θ|ξ | t |ξ |3 t. 2

j,=1

Hence, by applying Lemma 3.2, we have 1 1

1

I1 Lr ≤ Ct − 2 r − 2 ,

for r ∈ [1, ∞] and t ≥ 1.

(3.13)

Similarly, we also have the estimate for I2 . Indeed, from (3.12), for |ξ | < ε, one has h,hj

|I2 | ≤ C

 (0)  2  |ξ | t+C|ξ |3 t

e−Re (dj  )|ξ | t e Nj  2

|ξ |

j,=1 h,hj

≤C

1

e−Re (dj  )|ξ | t eCε|ξ | t |ξ | ≤ Ce− 2 θ|ξ | t |ξ |, 2

2

2

j,=1

 (0)  since Nj   can be assumed to be small for all j ∈ {1, . . . , h} and  ∈ {1, . . . , hj }. Hence, by applying Lemma 3.2, we have 1 1

1

I2 Lr ≤ Ct − 2 r − 2 ,

for r ∈ [1, ∞] and t ≥ 1.

We estimate J . From (3.10), we have

|J | ≤ C

k

j =1

 

(0) 

e−Re (ej )t e Mj

t+C|ξ |t

(1 + |ξ |).

(3.14)

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17

(0)

Then, from Proposition 2.2 and Remark 2, since Re (ej ) > 0 and Mj is a nilpotent matrix for  (0)  j ∈ {1, . . . , k}, Mj  can be assumed to be small for all j ∈ {1, . . . , k} and thus, there is δ > 0 such that

J

Lr

≤C

k

  e−Re (ej )t eCεt 1 + ε ≤ Ce−δt .

(3.15)

j =1

Therefore, from (3.13), (3.14) and (3.15), one obtains for |ξ | < ε that   G ˆ − Kˆ 

1 1

Lr

1

≤ I1 Lr + I2 Lr + J Lr ≤ Ct − 2 r − 2 ,

for r ∈ [1, ∞] and t ≥ 1.

We now estimate Vˆ in (3.3) for |ξ | < ε. Since αj ∈ R for all j ∈ {1, . . . , s} by the condition A, one has s,sj  (0) 

    Vˆ (ξ, t) ≤ C e−Re (βj  )t e j  t . j,=1 (0)

Thus, by Proposition 2.3 and Remark 3, since Re (βj  ) ≥ θ > 0 and j  is a nilpotent matrix for  (0)  j ∈ {1, . . . , s} and  ∈ {1, . . . , sj }, j   can be assumed to be small for all j ∈ {1, . . . , s} and  ∈ {1, . . . , sj } and thus, one obtains   Vˆ 

1

Lr

≤ Ce− 2 θt ,

for r ∈ [1, ∞] and t ≥ 1.

From here, in order not to repeat many times, we always consider Re (dj  ) and Re (βj  ) ≥    (0)     θ > 0, Nj(0)  and j  are small enough such that they are absorbed by Re (dj  ) and Re (βj  ) respectively for all j and . For ε ≤ |ξ | ≤ R, although there are the exceptional points at which the eigenprojections and the nilpotent parts associated with the eigenvalues of E(iξ ) = −(B + iξ A) may have poles, the ˆ = eEt and number of these points is finite as introduced in Remark 4, once integrating, for G from the condition D, we have   G ˆ

Lr

   − θ|·|2 t   1+|·|2  ≤ e   

≤ Ce−δt ,

for r ∈ [1, ∞] and t ≥ 1.

Lr

We estimate Kˆ in (3.2) for ε ≤ |ξ | ≤ R. One has h,hj  

  1 2 2  (0)  2 K(ξ,  ˆ e−Re (dj  )|ξ | t e Nj  |ξ | t ≤ Ce− 2 θε t . t) ≤ C j,=1

Thus, for ε ≤ |ξ | ≤ R, one obtains   Kˆ 

1

Lr

≤ Ce− 2 θε t , 2

for r ∈ [1, ∞] and t ≥ 1.

(3.16)

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18

Similarly, for Vˆ in (3.3) for ε ≤ |ξ | ≤ R, one has s,sj  (0) 

  1   Vˆ (ξ, t) ≤ C e−Re (βj  )t e j  t ≤ Ce− 2 θt .

(3.17)

j,=1

Hence, for ε ≤ |ξ | ≤ R, we have   Vˆ 

1

≤ Ce− 2 θt ,

Lr

for r ∈ [1, ∞] and t ≥ 1.

ˆ to the system (3.1) is given Finally, we study the case |ξ | > R. By Remark 3, the solution G by ˆ G(ξ, t) :=

s,sj

e(−iαj ξ −βj  )t e

−j  t+O (|ξ |−1 )t 

 (0)

j  + O(|ξ |−1 ) .

(0)

(3.18)

j,=1

ˆ − Vˆ = I + J where Then, we have G I :=

  (0) (0) −1 (0) e(−iαj ξ −βj  )t e−j  t+O(|ξ | )t − e−j  t j 

s,sj

(3.19)

j,=1

and we have s,sj

J :=

e(−iαj ξ −βj  )t e

−j  t+O (|ξ |−1 )t (0)

O(|ξ |−1 ).

(3.20)

j,=1

We estimate I firstly and then for J . Since αj ∈ R for all j ∈ {1, . . . , s}, we have |I | ≤ C

s,sj

  (0)  −(0) t+O(|ξ |−1 )t − t  e−Re (βj  )t e j  − e j   .

j,=1

Hence, let ε = 14 Re (βj  ) and applying Lemma 3.1, for R large enough, we obtain |I | ≤ C

s,sj

−1 t

1

e−Re (βj  )t e 4 Re (βj  )t+C|ξ |

|ξ |−1 t

j,=1

≤C

s,sj

1

e−Re (βj  )t e 4 Re (βj  )t+CR

−1 t

1

|ξ |−1 t ≤ Ce− 2 θt |ξ |−1 t.

j,=1

Thus, for |ξ | > R large, one has 1

I Lr ≤ Ce− 4 θt ,

for r ∈ (1, ∞] and t ≥ 1.

(3.21)

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19

Similarly, we estimate J for |ξ | > R large. From (3.20), one has s,sj

|J | ≤

e−Re (βj  )t e

 (0)   t+C|ξ |−1 t j

|ξ |−1

j,=1 s,sj

≤

e−Re (βj  )t eCR

−1 t

1

|ξ |−1 ≤ Ce− 2 θt |ξ |−1 .

j,=1

It follows that 1

J Lr ≤ Ce− 2 θt ,

for r ∈ (1, ∞] and t ≥ 1.

(3.22)

Therefore, from (3.21) and (3.22), there is a constant δ > 0 such that   G ˆ − Vˆ 

≤ I Lr + J Lr ≤ Ce−δt ,

Lr

for r ∈ (1, ∞] and t ≥ 1.

On the other hand, we estimate Kˆ in (3.2) for |ξ | > R. We have h,hj

ˆ |K(ξ, t)| ≤ C

e−Re (dj  )|ξ | t e 2

 (0)  2 N |ξ | t j

j,=1 h,hj

≤ Ce

− 12 θR 2 t

1

e− 2 Re (dj  )|ξ | t e 2

 (0)  2 N |ξ | t j

1

1

≤ Ce− 2 θR t e− 4 θ|ξ | t . 2

2

(3.23)

j,=1

Thus, for |ξ | > R, we have ˆ Lr ≤ Ce− 2 θR 2 t t − 2 ≤ Ce− 4 θR 2 t ,

K 1

1

1

for r ∈ [1, ∞] and t ≥ 1.

ˆ − Vˆ ) = F −1 (I ) + F −1 (J ) in L∞ We now estimate the L∞ -norm of the function F −1 (G for |ξ | > R where I and J are in (3.19) and (3.20) respectively. Primarily, from (3.19) and by applying the Taylor expansion to the application X → eX , we have I = I1 + I2 + I3 where s,sj

e−iαj ξ t −βj  t −(0) t (0) I1 := t e j  M j  , e iξ

(3.24)

j,=1

where M is the coefficient in O(|ξ |−1 ) associated with (iξ )−1 , and I2 := t

s,sj

j,=1

and

e(−iαj ξ −βj  )t e

(0)

−j  t

O(|ξ |−2 ) j  , (0)

(3.25)

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20

I3 :=

s,sj

  (0) (0) (0) −1 (0) e(−iαj ξ −βj  )t e−j  t+O(|ξ | )t − e−j  t − e−j  t O(|ξ |−1 )t j  .

(3.26)

j,=1

We estimate F −1 (I1 ) =

s

−1 j =1 Fj (I1 )

Fj−1 (I1 ) := t

sj

F −1

where 

=1

e−iαj ξ t iξ



e−βj  t e

(0)

−j  t

(0)

M j  ,

for j ∈ {1, . . . , s} firstly. For j ∈ {1, . . . , s}, one has   −R +∞  sj    i(x−αj t)ξ

  −1  −Re (β )t (0) t  e j e j F (I1 )(x, t) ≤ Ct e  + dξ j   iξ   =1 −∞ R   +∞  sj  

  −Re (β )t (0) t sin((x − α t)ξ ) j j e j e  ≤ Ct dξ 2   ξ   =1 R

≤ Cte

− 12 θt

|x − αj t|.

(3.27)

Hence, if |x| ≤ Ct where C is a positive constant, then, for all j ∈ {1, . . . , s} and t ≥ 1, we have   −1 F (I1 ) j

1

L∞

1

≤ Ct 2 e− 2 θt ≤ Ce− 4 θt .

We now estimate Fj−1 (I1 ) in the case where |x| > Ct and C large enough for j ∈ {1, . . . , s}. Noting that in this case we have exαj ≤ e|x||αj | ≤ e

|x|2 −1 t |αj ||x| t

≤ eε

|x|2 t

(3.28)

where ε is small enough. One has   −R +∞  sj    i(x−αj t)ξ

  −1  −Re (β )t (0) t  e j e j . F (I1 )(x, t) ≤ Ct e  + dξ j   iξ  =1  −∞

(3.29)

R

We estimate the integral −R H := −∞

+∞ +

ei(x−αj t)ξ dξ iξ

R

−R K i(x−αj t)ξ e + dξ = H1 + H2 . = lim K→+∞ iξ −K

R

(3.30)

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Due to the fact that the integrand is holomorphic in this domain, we can estimate H2 by considering ξ = ζ +iη ∈ C and by changing the path {(ζ, 0) : ζ from R to K} to the path γ := γ1 ∪γ2 ∪γ3 in the complex plane where γ1 := {(ζ, η) : ζ = R, η from 0 to x/t} ,

(3.31)

γ2 := {(ζ, η) : ζ from R to K, η = x/t}

(3.32)

γ3 := {(ζ, η) : ζ = K, η from x/t to 0} .

(3.33)

and

Then, by parameterizing γ1 (s) = R + i xt s for s ∈ [0, 1], since |x| > Ct and t ≥ 1, we have   1     i(x−αj t)R+xαj s− |x|2 s    i(x−αj t)ξ    t e x  = e  lim dξ ds   K→+∞ iξ R + i xt s t     γ1

0

C ≤ R

1  0

≤

C R



 |x| |x|2 ε |x|2 s − |x|2 s + 2 e t e t ds t t

  C |x|2 1  1 + 1 − e− 2t ≤ . |x| t R

(3.34)

On the other hand, noting that   1 1 1 1 η + = −η . −η + iζ iζ ζ 2 + η2 iζ ζ 2 + η2

(3.35)

Thus, for |x| > Ct and t ≥ 1, there is c > 0 such that we also have   +∞      ixζ −iαj ζ t− |x|2 +xαj    i(x−αj t)ξ     t e e   lim  dξ  =  dζ  x K→+∞ iξ − t + iζ     γ2

R

 +∞ ⎛ ⎛ ⎞⎞    x 2   |x| 1 1 1 x t ⎠⎠ dζ  eixζ ⎝ − ⎝ + ≤ e− 2t  2 2  iζ t ζ 2 + |x| iζ ζ 2 + |x|   t2 t2 R  ⎞ ⎛ +∞ +∞     ixζ 2    |x|2 e 1 |x| |x| ≤ Ce− 2t ⎝ dζ ⎠ dζ  + + 2 iζ t t ζ2   R

≤ Ce

2 − |x| 2t

|x|2 2t



t 1 1 + + |x| |x| t

R

 ≤ Ce

2 − |x| ct

.

(3.36)

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Similarly, by considering γ3 (s) = K + i xt (1 − s) for s ∈ [0, 1], we have          i(x−αj t)ξ 1 i(x−αj t)K+xαj (1−s)− |x|2 (1−s)     t e e x   lim  dξ  =  lim ds  . x K→+∞ K→+∞ iξ K + i (1 − s) t     t γ3

(3.37)

0

On the other hand, for K > 0, since t ≥ 1, we have   1   i(x−αj t)K+xαj (1−s)− |x|2 (1−s)  1   e t x  C |x| |x|2 − |x|2 (1−s)  ds ds ≤ + e 2t  K + i xt (1 − s) t  K t t2  0 0    C  2 2 |x| 1 − |x| C 1 + e 2t e 2t − 1 ≤ . = K |x| t K

(3.38)

One deduces that 1

|x|2

ei(x−αj t)K+xαj (1−s)− t K + i xt (1 − s)

lim

K→+∞ 0

(1−s)

x ds = 0. t

(3.39)

Hence, it implies that      i(x−αj t)ξ   e  lim dξ  = 0. K→+∞ iξ  

(3.40)

γ3

Finally, one can estimate H1 similarly by substituting R and K by −R and −K respectively. Therefore, from (3.29), (3.30), (3.34), (3.36) and (3.40), one obtains   −1 F (I1 ) j

1

L∞

≤ Ce− 2 θt

for |x| > Ct where C large enough and t ≥ 1 for j ∈ {1, . . . , s}. Therefore, it implies that   −1 F (I1 )

L∞

≤C

s

  −1 F (I1 ) j

j =1

1

L∞

≤ Ce− 2 θt ,

t ≥ 1.

We estimate F −1 (I2 ) and F −1 (I3 ) where I2 and I3 are in (3.25) and (3.26) respectively. Since : L1 → L∞ , one has

F −1

 −1  F (I2,3 )

L∞

Hence, we only need to estimate I2 and I3 in L1 .

  ≤ C I2,3 L1 .

(3.41)

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From (3.25), we have s,sj

|I2 | ≤ C

e−Re (βj  )t e

 (0)   t

|ξ |−2 t.

j

j,=1

Thus, we obtain   I2 

1

L1

1

≤ Cte− 2 θt ≤ Ce− 4 θt ,

t ≥ 1.

From (3.26), we have |I3 | ≤

s,sj

e

−Re (βj  )t

  (0) (0)  −(0) t+O(|ξ |−1 )t  −j  t −j  t −1  e j  −e −e O(|ξ | )t  . 

j,=1

Then, by Lemma 3.1, we obtain |I3 | ≤

s,sj



e−Re (βj  )t eε t+C|ξ |

−1 t

|ξ |−2 t 2 ,

j,=1

where ε is small enough. Therefore, since |ξ | large enough, we have   I3 

L1

1

≤ Ce− 2 θt ,

t ≥ 1.

Thus, for t ≥ 1, we have  −1  F (I )

L∞

      1 ≤ F −1 (I1 )L∞ + F −1 (I2 )L∞ + F −1 (I3 )L∞ ≤ Ce− 2 θt .

We now estimate F −1 (J ) where J is given by (3.20). From (3.20), one has J = J1 + J2 where J1 :=

s,sj

e−iαj ξ t −βj  t −(0) t+O (|ξ |−1 )t e j M, e iξ

(3.42)

j,=1

where M is the coefficient associated with the term (iξ )−1 in O(|ξ |−1 ), and J2 :=

s,sj

(0)

−1 )t

e(−iαj ξ −βj  )t e−j  t+O(|ξ |

O(|ξ |−2 ).

j,=1

Then, we can estimate F −1 (J1 ) as the case of F −1 (I1 ) and we can estimate F −1 (J2 ) as the case of F −1 (I2 ) and F −1 (I3 ). Thus, we obtain  −1  F (J )

L∞

    1 ≤ F −1 (J1 )L∞ + F −1 (J2 )L∞ ≤ Ce− 2 θt ,

for t ≥ 1.

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24

Therefore, we conclude  −1  F (G ˆ − Vˆ )

L∞

    1 ≤ F −1 (I )L∞ + F −1 (J )L∞ ≤ Ce− 2 θt

for t ≥ 1 and the proof is done. 4. Multiplier estimates This section provides some useful Fourier multiplier estimates by recalling the Young inequality. Lemma 4.1 (Young’s inequality). For all (p, q, r) ∈ [1, ∞]3 such that q1 − p1 = 1 − 1r and (f, g) ∈ Lr × Lq , we have f ∗ g ∈ Lp and f ∗ g Lp ≤ f Lr g Lq . We thus obtain the follows. 4.1. Case |x| ≤ Ct Let χ1 and χ3 be cutoff functions defined on [−ε, ε] and (−∞, −R] ∪ [R, +∞) respectively for ε small and R large such that |χ1,3 | ≤ 1. Let χ2 := 1 − χ1 − χ3 , we introduce the multipliers ˆ − Kˆ − Vˆ ), mj := χj (G

j = 1, 2, 3.

The following holds. Proposition 4.2. For r ∈ [1, ∞], if the conditions A, B, C and D hold, then mj ∈ Mr with 1

mj Mr ≤ Ct − 2 ,

j = 1, 2, 3

and

t ≥ 1.

(4.1)

ˆ − Kˆ = I1 + I2 + J where I1 , I2 are in (3.11), Proof. We begin with m1 . For |ξ | ≤ ε, we have G ˆ (3.12) respectively and J = G2 as in (3.10). We then have F −1 (χ1 I1 )(x, t) ε



h,hj

=

χ1 (ξ )ei(x−cj t)ξ −dj  ξ

2t

  (0) (0) −N ξ 2 t+O (|ξ |3 )t −N ξ 2 t (0) − e j e j Pj  dξ.

j,=1−ε

On the other hand, for j ∈ {1, . . . , h} and  ∈ {1, . . . , hj }, let z = eiφ/2 ξ where φ = arg (dj  ) ∈ (−π/2, π/2) since Re (dj  ) > 0, one obtains



h,hj

F −1 (χ1 I1 )(x, t) =

χ1 (e−iφ/2 z)ei(x−cj t)e

−iφ/2 z−|d |z2 t j

j,=1 γ



· e

−Nj  e−iφ z2 t+O (|e−iφ/2 z|3 )t (0)

−e

−Nj  e−iφ z2 t (0)



Pj  e−iφ/2 dz, (0)

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25

where γ := {z ∈ C : z = eiφ/2 ξ, ξ ∈ [−ε, ε]}. Then, we will estimate each summand by letting  |x−c t|  η := min 2|dj j|t , 2ε . Since the integrand is holomorphic, we can change the path of the integral from γ to γ := γ1 ∪ γ2 ∪ γ3 in the complex plane with respect to z where   γ1 := −εeiφ/2 + isgn(x − cj t)ηe−iφ/2 s : s ∈ [0, 1] ,

(4.2)

  γ2 := ζ eiφ/2 + isgn(x − cj t)ηe−iφ/2 : ζ ∈ [−ε, ε]

(4.3)

  γ3 := εeiφ/2 + isgn(x − cj t)ηe−iφ/2 (1 − s) : s ∈ [0, 1] .

(4.4)

and

On the other hand, we have    i(x−cj t)e−iφ/2 z−|dj  |z2 t  e  





= e−(x−cj t) cos(φ/2)Im z−sin(φ/2)Re z e−|dj  | (Re z)

2 −(Im z)2 t

.

(0)

Moreover, by Lemma 3.1, since Nj  is a nilpotent matrix, we have   (0)  −N (0) e−iφ z2 t+O(|e−iφ/2 z|3 )t −Nj  e−iφ z2 t  3 ε |z|2 t+C|z|3 t e j  − e ,   ≤ C|z| te for any ε > 0. Thus, for z ∈ γ1 and s ∈ [0, 1], we have Re z(s) = −ε cos(φ/2) + sgn(x − cj t)η sin(φ/2)s, Im z(s) = −ε sin(φ/2) + sgn(x − cj t)η cos(φ/2)s. We then obtain from cos(φ) > 0, η ≤ ε/2, η2 s 2 ≤ ε 2 /4 for s ∈ [0, 1] and |z| ≤ 3ε small enough that |z|3 ≤ ε |z|2 /C and, by choosing ε = |dj  | cos(φ)/8, there is δ > 0 large such that, for t ≥ 1, we have     1     ≤ C e−|x−cj t|η cos(φ)s e−|dj  | cos(φ)(ε2 −η2 s 2 )t e2ε ε2 t ε 4 tds ≤ Ce−δt . (4.5)     γ1

0

For z ∈ γ2 and ζ ∈ [−ε, ε], we have Re z(ζ ) = ζ cos(φ/2) + sgn(x − cj t)η sin(φ/2), Im z(ζ ) = ζ sin(φ/2) + sgn(x − cj t)η cos(φ/2). Hence, since |z| ≤ 2(|ζ | + |η|) ≤ 3ε small enough, |z|3 ≤ ε |z|2 /C and one has

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γ2

  ε   ≤ C e−|x−cj t|η cos(φ) e−|dj  | cos(φ)(ζ 2 −η2 )t e2ε (|ζ |+|η|)2 t (|ζ | + |η|)3 tdζ.  

(4.6)

−ε

|x−c t|

If η = 2|dj j|t , then since (|ζ | + |η|)2 ≤ 2(|ζ |2 + |η|2 ), one can choose ε = |dj  | cos(φ)/8 and there is c > 0 such that      

γ2

  ε |x−cj t|2   ≤ C e− 4|dj  |t   −ε

2 cos(φ) −|d | cos(φ)ζ 2 t 4ε ζ t+ε j

e

e



· |ζ | t + 3|ζ |

≤C

e

−

|x−cj t|2 8|dj  |t

cos(φ)



k=0

≤ Ct −1 e

−

|x−cj t|2 c|dj  |t

|x − cj t|2 |x − cj t|3 − cj t| + + 3|ζ | 2|dj  | 4|dj  |2 t 8|dj  |3 t 2

2 |x

3

3

|x−cj t|2 |dj  |2 t

|x − cj t| √ t

k  ε

1

 dζ

k

e− 2 |dj  | cos(φ)ζ t |ζ |3−k t 1− 2 dζ 2

−ε

(4.7)

.

If η = ε/2, then since |x − cj t| ≥ ε|dj  |t by the definition of η and since |z| < ε, by choosing ε = |dj  | cos(φ)/16, we have      

γ2

  ε   ≤ C e−|x−cj t|η cos(φ) e−|dj  | cos(φ)(ζ 2 −η2 )t e2ε ε2 t ε 3 tdζ   −ε

1 2 |dj  | cos(φ)t

≤ Ce− 8 ε

ε

e−|dj  | cos(φ)ζ t tdζ

(4.8)

≤ Ce−δt ,

(4.9)

2

−ε 1 2

≤ Ct e

− 18 ε 2 |dj  | cos(φ)t

for some δ > 0 large and t ≥ 1. For z ∈ γ3 and s ∈ [0, 1], we have Re z(s) = ε cos(φ/2) + sgn(x − cj t)η sin(φ/2)(1 − s), Im z(s) = ε sin(φ/2) + sgn(x − cj t)η cos(φ/2)(1 − s). Thus, similarly to γ1 , by choosing ε = |dj  | cos(φ)/8, there is δ > 0 large such that, for t ≥ 1, we have      

γ3

  1   ≤ C e−|x−cj t|η cos(φ)(1−s) e−|dj  | cos(φ)(ε2 −η2 (1−s)2 )t e2ε ε2 t ε 4 tds ≤ Ce−δt .   0

(4.10)

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Therefore, from (4.5), (4.7), (4.9), (4.10) and δ > 0 large, one has e

−δt

≤ Ct

−1 −

e

|x−cj t|2 c|dj  |t

for some c > 0 large since |x| ≤ Ct and t ≥ 1. Thus, for t ≥ 1, there is c > 0 such that |x−cj t|2     −1 −1 − c|dj  |t . F (χ1 I1 )(x, t) ≤ Ct e

By the same way for F −1 (χ1 I2 ) and F −1 (χ1 J ), for t ≥ 1, there is c > 0 such that we also have |x−cj t|2      −1   −1  −1 − c|dj  |t . F (χ1 I2 )(x, t) , F (χ1 J )(x, t) ≤ Ct e

Hence, for r ∈ [1, ∞], by the Young inequality and since m1 = χ1 (I1 + I2 + J ), it follows that

m1 Mr =

    1 sup F −1 (m1 ) ∗ f Lr ≤ F −1 (m1 )L1 ≤ Ct − 2 ,

t ≥ 1.

f Lr ≤1

We consider m2 . Since |χ2 (ξ )|, |eixξ | ≤ 1 for ξ ∈ R, we have    −1   F (m2 )(x, t) ≤ ˆ ˆ |G(ξ, t)| + |K(ξ, t)| + |Vˆ (ξ, t)| dξ ≤ Ce−δt

(4.11)

ε≤|ξ |≤R

ˆ for some δ > 0 large due to (3.16), (3.17) and the fact that |G(ξ, t)| ≤ e for θ > 0 under the integral sign. Thus, there is c > 0 such that one has |x−cj t|2   −   −1 F (m2 )(x, t) ≤ Ct −1 e c|dj  |t ,

−

θ|ξ |2 t 1+|ξ |2

on ε ≤ |ξ | ≤ R

t ≥ 1.

Hence, for r ∈ [1, ∞], by the Young inequality, we have

m2 Mr =

    1 sup F −1 (m2 ) ∗ f Lr ≤ F −1 (m2 )L1 ≤ Ct − 2 ,

f Lr ≤1

t ≥ 1.

ˆ − Vˆ = I + J where I is defined as Finally, we consider m3 . Based on the decomposition G I1 in (3.24) and J is the remainder, from (3.27) and a same treatment for J , there is c > 0 such that, for |x| ≤ Ct and t ≥ 1, we have s |x−cj t|2  

 −1  −δt −δt −1 − c|dj  |t ˆ ˆ te |x − αj t| + Ce ≤ Ct e , F (χ3 (G − V ))(x, t) ≤ C j =1

for some δ > 0 large.

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Similarly, from (3.23), for t ≥ 1, we have     −1  − 12 θR 2 t ˆ F (χ3 K)(x, t) ≤ e

e

− 14 θ|ξ |2 t

dξ ≤ Ce

−δt

≤ Ct

−1 −

e

|x−cj t|2 c|dj  |t

.

|ξ |≥R

ˆ − Vˆ − K), ˆ one obtains Hence, for r ∈ [1, ∞], by the Young inequality and m3 = χ3 (G

m3 Mr =

    1 sup F −1 (m3 ) ∗ f Lr ≤ F −1 (m3 )L1 ≤ Ct − 2 ,

f Lr ≤1

t ≥ 1.

We finish the proof. 4.2. Case |x| > Ct We consider the multipliers ˆ − Vˆ m1 := G

and

ˆ m2 := K,

ˆ is the solution to the system (3.1), Kˆ is in (3.2) and Vˆ is in (3.3). The following holds. where G Proposition 4.3. For r ∈ [1, ∞], if the conditions A, B, C and D hold, then mj ∈ Mr with 1

mj Mr ≤ Ct − 2 ,

j = 1, 2

and

t ≥ 1.

(4.12)

Proof. We estimate the L1 -norm of F −1 (m1 ). We have

F

−1

R (m )(x, t) = lim 1

R→+∞ −R

  ˆ t) − Vˆ (ξ, t) dξ. eixξ G(ξ,

(4.13)

ˆ to (3.1) is written as G(ξ, ˆ ˆ On the other hand, noting that the solution G t) = eE(iξ )t and thus G is holomorphic on the complex plane since E(iξ ) = −(B + iξ A). Moreover, due to the formula of Vˆ in (3.3), Vˆ is also holomorphic on the complex plane. Thus, by considering ξ = ζ + iη ∈ C, one can change the path of the integral in (4.13) from {(ζ, 0) : ζ from − R to R} to the path γ := γ1 ∪ γ2 ∪ γ3 in the complex plane where γ1 := {(ζ, η) : ζ = −R, η from 0 to x/t} , γ2 := {(ζ, η) : ζ from − R to R, η = x/t} and γ3 := {(ζ, η) : ζ = R, η from x/t to 0} .

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29

ˆ has the representation Furthermore, since R and |x|/t large, along these curves, the solution G of the high frequency case (3.18). Therefore, by the same computation as in (3.34)–(3.40) and letting R → +∞, we obtain   −1 1 |x|2 |x|2 F (m )(x, t) ≤ Ce− ct ≤ Ct −1 e− 2ct |x|2

(4.14)

C2

for some c, C > 0 since e− 2ct ≤ e− 2c t ≤ t −1 due to the fact that |x| > Ct and C large enough. Hence, we obtain  −1 1  F (m )

1

L1

≤ Ct − 2 .

(4.15)

Thus, by the Young inequality, for r ∈ [1, ∞], we have  1 m 

Mr

=

    1 sup F −1 (m1 ) ∗ f Lr ≤ F −1 (m1 )L1 ≤ Ct − 2 .

f Lr ≤1

(4.16)

The estimate for m2 is similar and the proof is done. 5. Symmetry In this section, we will discuss about the conditions C’ and S in order to increase the decay rate of the solution to the system (1.1). Recall the matrices C and D as in (1.4) and (1.8) respectively. Lemma 5.1. If the conditions B and C’ hold, the operator E(iξ ) = −(B + iξ A) has m distinct analytic eigenvalues converging to 0 as |ξ | → 0, where m = dim ker(B). The expansion of the j -th eigenvalue has the form λj (iξ ) = −icj ξ − dj ξ 2 + O(|ξ |3 ),

|ξ | → 0,

(5.1)

    (0) (0) (0) where cj ∈ σ C, ker(B) and dj ∈ σ Pj DPj , ker(C − cj I )|ker(B) with Pj the eigenprojection associated with cj for j ∈ {1, . . . , m}. Proof. This is a consequence of Proposition 2.1. Indeed, from (2.1), the approximation of the eigenvalues of E converging to 0 as |ξ | → 0 is λj  (iξ ) = −icj ξ − dj  ξ 2 + O(|ξ |2 ),

|ξ | → 0,

    (0) (0) (0) where cj ∈ σ C, ker(B) and dj  ∈ σ Pj DPj , ker(C − cj I )|ker(B) with Pj the eigenprojection associated with cj for j ∈ {1, . . . , h} and  ∈ {1, . . . , hj }. Noting that hj is the cardinality     of σ Pj(0) DPj(0) , ker(C − cj I )|ker(B) for j ∈ {1, . . . , h} and h is the cardinality of σ C, ker(B) . In particular, h ≤ m and hj ≤ mj where mj = dim ker(C − cj I )|ker(B) for j ∈ {1, . . . , h} and m = dim ker(B). On the other hand, since the condition C’ holds, h = m. Moreover, also by the condition C’, one deduces that cj is simple for all j ∈ {1, . . . , m}. Thus, dim ker(C − cj I )|ker(B) = 1 and therefore hj = 1 for j ∈ {1, . . . , m}. It implies that there is only one dj := dj 1 ∈

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  (0) (0) σ Pj DPj , ker(C − cj I )|ker(B) for each j ∈ {1, . . . , m}. Moreover, dj is also simple, and thus, one can continue the reduction process as in the proof of Proposition 2.1. Furthermore, due to the simplicity of the coefficients in the expansion of λj provided cj is simple and the reduction process, there is no splitting in the expansion of the eigenvalues λj i.e. the eigenvalues λj can be expanded analytically for j ∈ {1, . . . , m} and the proof is done. Let p(λ, κ) := det(E(κ) − λI ) be the dispersion polynomial associated with E(κ) = −(B + κA), where λ, κ ∈ C. Lemma 5.2. If the condition S holds, p(λ, −κ) = p(λ, κ) for any λ, κ ∈ C. Proof. For q(λ, κ) := p(λ, −κ), there holds p(λ, κ) = 0 if and only if q(λ, κ) = 0. Indeed, if the couple (λ, κ) is such that p(λ, κ) = 0 if and only if there exist a nonzero vector u such that 

 λ I + κA + B u = 0.

In such a case, setting v = S −1 u, there also holds     0 = S −1 λ I + κA + B Sv = S −1 λ S + κAS + BS v     = S −1 λ S − κSA + SB v = λ I − κA + B v. Hence, q(λ, κ) = 0. The other implication can be proved in the same way. For fixed κ, the polynomials p and q have both degree n in λ with principal term λn . Hence, p p q q p,q there exist λ1 , . . . , λn and λ1 , . . . , λn with λi = λi (κ) such that p(λ, κ) =

n  

 p λ − λk (κ)

and q(λ, κ) =

k=1

n  

 q λ − λk (κ)

k=1 q

p

Since p and q have the same zero-set, for any k ∈ {1, . . . , n} there exists j such that λk = λj . As a consequence, p ≡ q. Corollary 5.3. If the conditions B, C’ and S hold, the approximation (5.1) is refined by λj (iξ ) = −icj ξ − dj ξ 2 + O(|ξ |4 ),

|ξ | → 0,

j ∈ {1, . . . , m}.

(5.2)

Proof. From the proof of Lemma 5.1, since dj is simple for all j ∈ {1, . . . , m}, one can continue the reduction process in the proof of Proposition 2.1 and thus the formula (5.1) is given by λj (iξ ) = −icj ξ − dj ξ 2 − ej (iξ )3 + O(|ξ |4 ), 

 (0)  (0) where ej ∈ σ Mj , ker Pj DPj − dj I 

ker(C−cj I )|ker(B)

|ξ | → 0,

 for some suitable matrix Mj for j ∈

{1, . . . , m}. By recalling the proof of Proposition 2.1 and by Lemma 5.1 one more time, substituting (−iξ ) into iξ , there are m distinct analytic eigenvalues of E(−iξ ) converging to 0 as |ξ | → 0 such that

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λj (−iξ ) = −i(−cj )ξ − dj ξ 2 − (−ej )(iξ )3 + O(|ξ |4 ),

|ξ | → 0,

31

(5.3)

where cj , dj and ej are already introduced as before. that σ (Mj ) On the other hand, since σ (E(iξ )) ≡ σ (E(−iξ )) due to Lemma 5.2, one deduces  (0)  (0) contains both ej and −ej . Moreover, since the dimension of ker Pj DPj − dj I 

ker(C−cj I )|ker(B)

is one, one concludes that ej = −ej = 0. The proof is done.

Remark 5. The nilpotent parts associated with λj for j ∈ {1, . . . , m} are zero since these eigenvalues are distinct and simple. Moreover, the total projection associated with λj is itself the eigenprojection associated with λj and has the expansion (2.5) with ζ = iξ i.e. we have (0)

Pj (iξ ) = Pj

(1)

+ iξ Pj

+ O(|ξ |2 ),

|ξ | → 0,

(5.4)

(1)

where Pj can be computed by the formula (1.20) (see [6]) for j ∈ {1, . . . , m}. This is based on the fact that there is no splitting after the second step of the reduction process in the proof of Proposition 2.1. One sets the kernel Kˆ ∗ (ξ, t) :=

m

e(−icj ξ −dj ξ

2 )t



 Pj(0) + iξ Pj(1) .

(5.5)

j =1

Then, the first estimate in (3.5) of Proposition 3.3 can be modified by Proposition 5.4. If the conditions B, C’, D and S hold, for r ∈ [1, ∞], one has   G ˆ − Kˆ ∗ 

Lr

1 1

≤ Ct − 2 r −1 ,

(5.6)

for |ξ | < ε small and t ≥ 1. Proof. For |ξ | < ε, by Remark 1, Remark 2 and Corollary 5.3, the solution to the system (3.1) ˆ 2 where G ˆ 2 is in (3.10) and ˆ =G ˆ1+G is given by G ˆ 1 (ξ, t) = G

m

e(−icj ξ −dj ξ

2 )t+O (|ξ |4 )t



(0)

Pj

(1)

+ iξ Pj

 + O(|ξ |2 ) .

j =1

ˆ − Kˆ ∗ = I1 + I2 + J where Thus, similarly to the proof of the first estimate in (3.5), we have G ˆ 2 and J =G I1 :=

m

  4 (0) (1)  eO(|ξ | )t − 1 Pj + iξ Pj ,

e(−icj ξ −dj ξ

2 )t

e(−icj ξ −dj ξ

2 )t+O (|ξ |4 )t

(5.7)

j =1

I2 :=

m

j =1

O(|ξ |2 ).

(5.8)

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Hence, similarly to before, there is a constant c > 0 such that |I1 | ≤ Ce−c|ξ | t |ξ |4 t 2

|I2 | ≤ Ce−c|ξ | t |ξ |2 . 2

and

Thus, together with (3.15), it implies that   G ˆ − Kˆ ∗ 

Lr

      1 1 ≤ I1 Lr + I2 Lr + J Lr ≤ Ct − 2 r −1 ,

for |ξ | < ε, t ≥ 1 and r ∈ [1, ∞]. We finish the proof. Similarly, by recalling the multipliers mj for j = 1, 2, 3 and mj for j = 1, 2 with Kˆ is substituted by Kˆ ∗ , we can also refine Proposition 4.2 for |x| ≤ Ct and Proposition 4.3 for |x| > Ct as follows. Proposition 5.5 (|x| ≤ Ct). For r ∈ [1, ∞], if the conditions A, B, C’, D and S hold, then mj ∈ Mr with

mj Mr ≤ Ct −1 ,

j = 1, 2, 3

and

t ≥ 1.

(5.9)

Proof. Similarly to the proof of Proposition 4.2, we only need to consider F −1 (χ1 I1 ) and F −1 (χ1 I2 ) on γ2 where I1 , I2 are now given by (5.7), (5.8) respectively and γ2 is the same |x−c t| as (4.3) with the case η = 2|dj j|t . The others are bounded by e−δt for some δ > 0 large and thus 3

since |x| ≤ Ct, they are dominated by t − 2 e Hence, noting that since

−

|x−cj t|2 c|dj |t

for some c > 0 large and t ≥ 1.

  2   O(|e−iφ/2 z|4 )t − 1 ≤ C(|Re z| + |Im z|)4 te3ε(|Re z|+|Im z|) t e for |z| ≤ 3ε and φ = arg(dj ) ∈ (−π/2, π/2) for j ∈ {1, . . . , m}, on γ2 , we have m ε  

2 2 2   −1 e−|x−cj t|η cos(φ) e−|dj | cos(φ)(ζ −η )t eε(ζ +|η|) t (|ζ | + |η|)4 tdζ F (χ1 I1 ) ≤ C j =1−ε

≤C

4 m

e

−

|x−cj t|2 8|dj |t

cos(φ)

j =1 k=0

ε · −ε

e

− 12 |dj | cos(φ)ζ 2 t

|ζ |



|x − cj t| √ t

4−k 1− k2

t

k

dζ ≤ C

m

t

− 32 −

e

|x−cj t|2 c|dj |t

j =1

for some c > 0 and t ≥ 1. The estimate for F −1 (χ1 I2 ) is similar. Therefore, taking the L1 -norm in x variable and using the Young inequality, we finish the proof.

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Proposition 5.6 (|x| > Ct ). For r ∈ [1, ∞], if the conditions A, B, C’, D and S hold, then mj ∈ Mr with

mj Mr ≤ Ct −1 ,

j = 1, 2

t ≥ 1.

and

Proof. The proof is the same as in Proposition 4.3 and the fact that e− 2

t

− 32 − |x| 2t

e

(5.10) |x|2 t

is dominated by

since |x| > Ct and t ≥ 1. The proof is done.

6. Proof of main results Recall the well-known inequality Lemma 6.1 (Interpolation inequality). Let (pj , qj )j ∈{0,1} be two elements of [1, ∞]2 . Consider a linear operator T which continuously maps Lpj into Lqj for j ∈ {0, 1}. For any θ ∈ [0, 1], if 

1 1 , pθ q θ





1 1 , := (1 − θ ) p0 q 0



 +θ

 1 1 , , p1 q 1

then T continuous maps Lpθ into Lqθ and θ

T L(Lpθ ;Lqθ ) ≤ T 1−θ L(Lp0 ;Lq0 ) T L(Lp1 ;Lq1 ) .

We then introduce detailed proofs for Theorem 1.1 and Theorem 1.2.  Proof of Theorem 1.1. Let u be the solution to (1.1), recalling that U = hj=1 Uj where Uj is s the solution to (1.9) for j ∈ {1, . . . , h} and V = Q j =1 Vj where Vj is the solution to (1.14) for j ∈ {1, . . . , s}. Then, we have   ˆ − Kˆ − Vˆ ∗ u0 . u − U − V = F −1 G On the other hand, let χ be the characteristic function, we have       ˆ − Kˆ − Vˆ = F −1 G ˆ − Kˆ − Vˆ χ[0,ε) + χ[ε,R] + χ(R,∞) (|ξ |) F −1 G     ˆ − Kˆ − Vˆ χ[0,ε) + χ[ε,R] (|ξ |) = F −1 G      ˆ − Vˆ χ(R,∞) (|ξ |) − F −1 Kχ ˆ (R,∞) (|ξ |) . + F −1 G Thus, since F −1 : L1 → L∞ , we have  −1   F ˆ − Kˆ − Vˆ  G

L∞

   ˆ − Kˆ − Vˆ χ[0,ε) + χ[ε,R]  1 ≤C  G L    −1     ˆ − Vˆ χ(R,∞) (|ξ |)  ˆ (R,∞)  1 + F G +Kχ L

L∞

.

Hence, by the estimates (3.5), (3.6), (3.7) and (3.8) in Proposition 3.3, from the Young inequality, we obtain

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     u − U − V  ∞ ≤ F −1 G ˆ − Kˆ − Vˆ  ∞ u0 L1 ≤ Ct −1 u0 L1 , L L

t ≥ 1.

Furthermore, from Proposition 4.2 and Proposition 4.3, for all r ∈ [1, ∞], we also have   u − U − V  r ≤ Ct − 12 u0 Lr , L

t ≥ 1.

Therefore, by the interpolation inequality, we obtained the desired results. The estimate for U in (1.16) is similar and the estimate for V in (1.16) is obtained by the Parseval identity. We finish the proof. Proof of Theorem 1.2. Similarly to before and from Propositions 5.4, 5.5 and 5.6. The proof is done. Appendix A. Eigenprojection computation We introduce a useful tool for computing the eigenprojection associated with a semi-simple eigenvalue of a given matrix in Rn×n based on the computation of the determinants. We have the following definitions. – a set of indices I is a set I = {k1 < · · · < k } with kp ∈ {1, . . . , n} for any p; – an index-transformation χ is an injective map from a set of indices I to {1, . . . , n}. Then, we introduce some additional notations. – given two matrices A, B ∈ Rn×n , a set of indices I = {i} and an index-transformation χ , we denote by (A, B; I, χ) the matrix obtained by substituting the i-th column of A by the χ(i)-th column of B. – given sets of indices I, K, L satisfying K ⊆ I and |K| = |L|, where |K| and |L| are the cardinalities of K and L respectively. The map χK→L is the injective map from I to {1, . . . , n} defined by χK→L (k) = k if k ∈ / K; and there is a unique  ∈ L such that χK→L (k) =  for each k ∈ K. We then set [A]k ∈ Rn×n with components defined by [A]kij :=

det (A, I ; I, χi→j ),

i, j ∈ {1, . . . , n},

where the sum is made on sets of indices I containing i and with cardinality |I| = k + 1. If k = 0, then [A]0 = adj (A). Hence, the above notation can be seen as an extended version of the adjunct of the matrix A. One sets the matrix-valued functions Pk (A) :=

(k + 1)[A]k , Tr [A]k

(A.1)

Sk (A) :=

(k + 1)(k + 2)[A]k+1 Tr [A]k − (k + 1)2 [A]k Tr [A]k+1  . (k + 2) Tr [A]k )2

(A.2)

Without loss of generality, we consider the eigenvalue 0 of A. Let be an oriented closed curve in ρ(A) enclosing 0 except for the other eigenvalues of A, one defines   1 1 P := − (A − zI )−1 dz and S := z−1 (A − zI )−1 dz. (A.3) 2πi 2πi



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35

The matrices P and S are called the eigenprojection and the reduced resolvent coefficient associated with 0 respectively. Proposition Appendix A.1. If 0 is semi-simple with algebraic multiplicity m ≥ 1, then respectively, P and S can be computed by the formulas P = Pm−1 (A)

S = Sm−1 (A).

and

(A.4)

Before giving a proof for Proposition Appendix A.1, one has the follows. Lemma Appendix A.2. Let f (x) := det (A + xB, C; I, χK→L ) for x ∈ C. By setting   J := h ∈ / I : ∃j ∈ χK→L (I), cj = bh ∈ col(B) where col(B) is the column space of B, for any nonnegative integer m satisfying m ≤ n − |I|, the following holds f (m) (x) = m!

det ((A + xB, C; I, χK→L ), B; M, χM→M )

where the sum is made on the set of indices M with |M| = m and M ∩ (I ∪ J ) = ∅. Proof. By definition, we have f (x) =

n 

 fh (x)

where

fh (x) :=

h=1

(ah + bh x) if h ∈ / I, if h ∈ I. cχK→L (h)

Hence, the derivative of order m ∈ N of f satisfies f

(m)

(x) =

n

 h=1

 (s ) fh h (x)

where

(s ) fh h (x) :=

(ah + bh x)(sh ) (sh ) cχK→L (h)

if h ∈ / I, if h ∈ I

and sh := sh1 + · · · + shm ∈ N with sh ∈ {0, 1} for all  ∈ {1, . . . , m} such that s1 + · · · + sn = m. Thus, if we denote by S ∈ {1, 0}n×m the n ×m matrix whose components are defined by Sh := sh for h ∈ {1, . . . , n} and  ∈ {1, . . . , m}, the sum is made on the set S := {S : s1 + · · · + sn = m} . It means that f (m) (x) is the sum of a finite number of determinants, where the determinants are  (s ) generated by the elements S of S and they are given by DS := nh=1 fh h (x). / I, then (ah + bh x)(sh ) = On×1 and if sh ≥ 1 for Moreover, for any S ∈ S, if sh ≥ 2 for h ∈ (sh ) h ∈ I, cχK→L (h) = On×1 ; and thus, the determinants related to these cases are zero. Thus, due to the condition s1 + · · · + sn = m where m ≤ n − |I|, we can introduce a partition for S such that its elements denoted by SM are associated with index-sets M := {h1 , . . . , hm } ⊂ {1, . . . , n}\I and they are given by SM := {S : sh = δM (h) for all h = 1, . . . , n}

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where δM (h) :=

1 0

if h ∈ M, if h ∈ / M.

In particular, for any M, if S and S belong to SM , one has DS = DS since sh = δM (h) = sh for all h ∈ {1, . . . , n} where sh and sh are the sum of the elements of the h-th rows of the matrices S and S respectively. On the other hand, we have ⎧ ⎪ if h ∈ M, ⎨bh DS = fh(sh ) (x) where fh(sh ) (x) := (ah + bh x) if h ∈ / M ∪ I, ⎪ ⎩ h=1 if h ∈ I cχK→L (h)     = det   A + xB, C; I, χK→L , B; M, χM→M . n 

Moreover, let σ be a permutation of the set {e1 , . . . , em } where e is the -th row of the identity matrix I ∈ Rm×m . Then, by definition, the rows of any S ∈ SM for M = {h1 , . . . , hm } must be in the form of    σ (e ) if  ∈ {1, . . . , m}, Sh 1 . . . Sh m = O1×m if  ∈ {m + 1, . . . , n}. Therefore, since DS = DS for S, S ∈ SM and since the number of such σ is m!, we obtain f (m) (x) =



DS

M S∈SM

= m!

    det   A + xB, C; I, χK→L , B; M, χM→M .

M

Assume that there is M to satisfy M ∩ J = ∅, then there exists h ∈ M such that h ∈ / I and bh = cj for some j ∈ χK→L (I). Hence, the determinant DS generated by S ∈ SM has (sh )

fh

(x) = bh = cj .

Furthermore, since j ∈ χK→L (I), there is i ∈ I such that χI →J (i) = j . Thus, the determinant DS also has (si )

fi

(x) = cχK→L (i) = cj

as well. Since h ∈ / I and i ∈ I, it implies that there are at least two columns are the same and the determinant DS is accordingly zero. Therefore, we can choose M ∩ J = ∅ for all M. The proof is done. Corollary Appendix A.3. Let h ∈ {0, . . . , n − 1} and A ∈ Rn×n , for x ∈ C, the following hold  [A]h = (h!)−1 (adj(A + xI ))(h) x=0 ,

 Tr [A]h = (h!)−1 (det(A + xI ))(h+1) x=0 .

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Proof. By the definition of adj(A + xI ), the elements of the matrix are the minors of the matrix A + xI . On the other hand, one defines Mj i (x) := det (A + xI, I ; {i}, χi→j ),

i, j ∈ {1, . . . , n}.

Thus, by the definition of the minors and the definition of , we have (adj(A + xI ))ij = Mj i (x) for all i, j ∈ {1, . . . , n}. Moreover, by Lemma Appendix A.2, the following holds (h)

Mj i (0) = h!

det ((A, I ; {i}, χi→j ), I ; H, χH→H )

Hi,j /

= h!

det (A, I ; H ∪ {i}, χi→j ) = h![A]hij

H∪{i}

where H has the cardinality |H| = h for h ∈ {0, . . . , n − 1} for i, j ∈ {1, . . . , n}. We finished proving the first equality in the statement. By the definition of χM→N : I → {1, . . . , n} for any set of indices I containing M, it follows that χi→i ≡ χI →I for any I containing i. Thus, by the definition of [A]h for h ∈ {0, . . . , n − 1}, we have Tr [A] = h

n

det (A, I ; I, χi→i )

i=1 I i

=

det (A, I ; I, χI →I ) + · · · +

I 1

det (A, I ; I, χI →I )

I n

where I has the cardinality |I| = h + 1. Moreover, for any fixed set of indices I satisfying |I| = h + 1, I must be considered in h + 1 terms in the right hand side of the formula of Tr [A]h . In fact, each of i ∈ I belongs to {1, . . . , n}. Thus, for any fixed I, we can collect h + 1 quantities that are the same and we have Tr [A]h = (h + 1)

det (A, I ; I, χI →I )

I

where I has the cardinality |I| = h + 1. Furthermore, by Lemma Appendix A.2, one has

 (det(A + xI ))(h+1) x=0 = (h + 1)! det (A, I ; I, χI →I ). I

The proof is done. We can now give a proof for Proposition Appendix A.1. Proof of Proposition Appendix A.1. By definition, the resolvent of the matrix A is given by R(z) := (A − zI )−1 =

adj(A − zI ) . det(A − zI )

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For z small, the resolvent can be expanded as  n−1 h −1 (h)  zh 1 h=0 (−1) (h!) (adj(A + xI ))  x=0 R(z) = m n . h −1 (h)  z zh−m h=m (−1) (h!) (det(A + xI )) x=0 Thus, Corollary Appendix A.3 implies that n−1 h [A]h zh 1 h=0 (−1)   R(z) = m n . h −1 Tr [A]h−1 zh−m z h=m (−1) h On the other hand, by using the Laurent expansion of R(z) (see [6]), we also have R(z) = −

+∞

z−h−1 (N )h − z−1 P +

h=−1

+∞

zh (S)h+1 ,

h=0

where P , S are in (A.3) and N = AP is the nilpotent matrix associated with the eigenvalue 0 of A. Then equating two sides, we obtain the formulas. We finish the proof. Appendix B. Perturbation theory for linear operators In this part, we introduce some results from the perturbation theory for linear operators in finite dimensional space that we will use for this paper. Moreover, we will sketch the proofs of them. For whom are interested in, see [6] for further details. Let P be a projection onto a domain D ⊆ Cn such that P u = u if u ∈ D and P u = On×1 otherwise, then (Pj ) for j ∈ {1, . . . , k} is called a sequence of subprojections of P if one has Pj Pj = δjj Pj ,

P=

k

j =1

Pj

and ran(P ) =

k

ran(Pj ),

(B.1)

j =1

where δjj denotes the Kronecker delta function for j, j ∈ {1, . . . , k}. Assume that T is a matrix operator considered in ran(P ). Lemma Appendix B.1. If T commutes with Pj for all j ∈ {1, . . . , k}, then, for all j ∈ {1, . . . , k}, one has T Pj = Pj T = Pj T Pj

and

TP = PT = PTP =

k

(T Pj ).

(B.2)

j =1

Proof. For j ∈ {1, . . . , k}, since Pj2 = Pj by (B.1), one has Pj T Pj = T Pj2 = T Pj = Pj T  if T commutes with Pj . Also from (B.1), one has P = kj =1 Pj and thus, we have T P = k k j =1 (T Pj ) = j =1 Pj T = P T . On the other hand, since P is a projection, we have P T P =  2 P T = P T = T P = kj =1 (T Pj ). We finish the proof.

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For x ∈ C, assume T (x) = T (0) + O(|x|) and P (x) = P (0) + O(|x|) as |x| → 0. Assume also (0) that there are k ≤ n distinct eigenvalues λj of T (0) considered in ran(P (0) ) for j ∈ {1, . . . , k}. We have the following result. Proposition Appendix B.2. The eigenvalues of T considered in ran(P ) are divided into k distinct (0) groups, where the j -th group is the λj -group for j ∈ {1, . . . , k}. On the other hand, there is a unique sequence (Pj ) satisfying (B.1) and (B.2) such that Pj (x) = Pj(0) + O(|x|), where Pj(0) is the eigenprojection associated with λ(0) j for j ∈ {1, . . . , k}. (0)

In particular, Pj is the total projection associated with the λj -group. Before going to the proof of Proposition Appendix B.2, we have the following lemma where a proof for it can be found in [6]. Let the resolvent of T where T depends on x ∈ C be R(x, z) := (T (x) − zI )−1 ,

z ∈ ρ(T ).

(B.3)

(0) Lemma Appendix B.3. The resolvent of  = T + O(|x|) is holomorphic in any neigh T(0)(x) 2 . Moreover, if is a compact subset of ρ(T (0) ), borhood of (0, y) ∈ C such that y ∈ ρ T then R(x, y) is a uniformly convergent series for sufficiently small x and y ∈ and one has the expansion

R(x, y) = R (0) (y) + O(|x|),

|x| → 0,

(B.4)

where R (0) (y) := (T (0) − yI )−1 . As a consequence, there is no eigenvalue of T included in . We are now going back to the proof of Proposition Appendix B.2. Proof of Proposition Appendix B.2. Primarily, we have the follows. For λ ∈ σ (T ), λ is a zero of the dispersion polynomial p := det(T − λI ). Moreover, it is known that λ is continuous and converges to an eigenvalue of T (0) as |x| → 0 since T (x) = T (0) + O(|x|) as |x| → 0. Thus, one can write λ(x) = λ(0) + O(1),

|x| → 0,

(B.5)

  where λ(0) ∈ σ T (0) is the limit of λ as |x| → 0. In particular, the eigenvector u ∈ Cn associated with λ satisfies u(x) = u(0) + O(1),

|x| → 0,

(B.6)

where u(0) ∈ Cn is an eigenvector associated with λ(0) .  It follows that u ∈ ran(P ) if and only if u(0) ∈ ran P (0) . Indeed, one has    P u = P (0) + O(|x|) u(0) + O(1) = P (0) u(0) + O(1) and thus P u = u if and only if P (0) u(0) = u(0) . Hence, λ ∈ σ (T , ran(P )) if and only if λ(0) ∈    (0) σ T (0) , ran P (0) . Therefore, since λj for j ∈ {1, . . . , k} are the k distinct eigenvalues of T (0)

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  considered in ran P (0) , the above argument and the expansion (B.5) show that for any λ ∈ σ (T , ran(P )), λ → λ(0) j as |x| → 0 for some j ∈ {1, . . . , k}. Thus, we can consider the formal formula σ (T ) considered in ran(P ) =

k % 

 (0) λj -group ,

(B.7)

j =1

  (0) (0) where λj -group = λ ∈ σ (T ) : λ → λj as |x| → 0 . We are going to prove the unique existence of a sequence (Pj ) satisfying (B.1) and (B.2) where j ∈ {1, . . . , k}. First of all, we consider the domain Cn i.e. P = I the identity matrix and since P (x) = P (0) + O(|x|) as |x| → 0, P (0) = I as well. Hence, the eigenvalues λ of T and λ(0) j of T (0) are considered in Cn in this case. Let λ ∈ σ (T ) and let λ be an oriented closed curve in ρ(T ) enclosing λ except for the other eigenvalues of T , since λ is a singularity of the resolvent R(z) = (T − zI )−1 of T , the Cauchy integral Pλ (x) := −

1 2πi

 (B.8)

R(x, z) dz

λ

is known as the eigenprojection associated with λ. The matrix operator Nλ := (T − λI )Pλ is then the nilpotent part associated with λ. Moreover, T Pλ = λPλ + Nλ = Pλ T . In particular, the resolvent R(x,z) is expanded analytically in x for |x| small and z belonging to any compact set contained in ρ T (0) provided Lemma Appendix B.3.   (0) Based on that, for j ∈ {1, . . . , k}, let j be an oriented closed curve in ρ T (0) enclosing λj except for the other eigenvalues of T (0) . Then, by Lemma Appendix B.3, there is no eigenvalue of T to belong to j and therefore, for |x| small enough, the interior domain bounded by j (0) encloses the λj -group except for the other groups of T . Hence, for every λ ∈ σ (T ) included in the λ(0) j -group, one can choose λ such that they are strictly contained in the interior domain bounded by j and one has  1 Pj (x) := − R(x, y) dy 2πi

j

=

(0) λ∈λj -group

⎛ ⎜ 1 ⎝− 2πi



⎞ ⎟ R(x, z) dz⎠ =

Pλ .

(0) λ∈λj -group

λ (0)

Hence, Pj is the total projection associated with the λj -group of T . The sequence of the total projections Pj for j ∈ {1, . . . , k} is a sequence of subprojections of P = I i.e. it satisfies (B.1). Indeed, since, for λ ∈ λ(0) j -group, Pλ is an eigenprojection, one has Pj2 =



(0) λ∈λj -group

Pλ

2

=

(0) λ,λ ∈λj -group

Pλ Pλ =

(0) λ∈λj -group

Pλ = Pj ,

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and for j = j , one has Pj Pj =





(0) λ∈λj -group

 Pλ =

Pλ

(0) λ ∈λj -group

(0)

Pλ Pλ = O

(0) λ∈λj -group (0) λ ∈λj -group

(

(0)

since these two groups are distinct by λj = λj . Moreover, we have Cn =  and I = λ∈σ (T ) Pλ , and thus from (B.7), one deduces I=

k



Pλ =

j =1 λ∈λ(0) -group

k

Pj ,

Cn =

j =1

j

k



ran(Pλ ) =

j =1 λ∈λ(0) -group j

k

λ∈σ (T ) ran(Pλ )

ran(Pj ).

j =1

Moreover, the sequence satisfies also (B.2). In fact, (B.2) holds if one proves that T commutes with Pj for all j ∈ {1, . . . , k} due to Lemma Appendix B.1. For all λ ∈ σ (T ), since T Pλ = Pλ T , one obtain that

T Pj = (T Pλ ) = (Pλ T ) = Pj T , j ∈ {1, . . . , k}. (0)

λ∈λj -group

(0)

λ∈λj -group

In particular, from (B.4), one has 1 Pj (x) := − 2πi



1 R(x, y) dy = − 2πi

j

 R (0) (y) dy + O(|x|),

|x| → 0,

j

  −1  where R (0) (y) = T (0) − yI for y ∈ ρ T (0) . Then, it is easy to see that R (0) is the resolvent of the matrix T (0) and thus by the definition of j , it implies that (0)

Pj (x) = Pj

+ O(|x|),

|x| → 0,

where Pj(0) is the eigenprojection associated with λ(0) since it is known that Pj(0) = j 1 ) − R (0) (y) dy. 2πi j We thus already constructed the desired sequence of Pj for j ∈ {1, . . . , k} if ran(P ) = Cn . For the case where we consider the domain ran(P ) ⊂ Cn , it is enough to define the projection P˜j onto ran(Pj ) ∩ ran(P ) by P˜j u := u for u ∈ ran(Pj ) ∩ ran(P ), where Pj is constructed in Cn as before for j ∈ {1, . . . , k}. One denote P˜j again by Pj for j ∈ {1, . . . , k}. The proof is done. References [1] S. Benzoni-Gavage, D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations: First-Order Systems and Applications, Oxford University Press on Demand, 2007. [2] S. Bianchini, B. Hanouzet, R. Natalini, Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math. 60 (11) (2007) 1559–1622.

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