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The Cauchy problem for hyperbolic systems with H¨older continuous coefficients with respect to the time variable

KUNIHIKOKAJITANI ANDYASUOYUZAWA

Abstract. We discuss the local existence and uniqueness of solutions of certain nonstrictly hyperbolic systems, with H¨older continuous coefficients with respect to time variable. We reduce the nonstrictly hyperbolic systems to the parabolic ones and by use of theTanabe-Sobolevski’s methodand the Banach scale method we construct a semi-group which gives a representation of the solution to the Cauchy problem.

Mathematics Subject Classification (2000):35L45 (primary); 35A08 (second- ary).

1.

Introduction

We consider the following Cauchy problem:













tu(t,x)= d

j=1

Aj(t,x)∂ju(t,x)

+B(t,x)u(t,x)+ f(t,x), in [0,T

R

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

R

d,

(1.1)

where each Aj and BareN × N matrix functions, f,uandu0are N component vector functions andt =∂/∂t,j =∂/∂xj. We assume that this system has weak hyperbolicity, that is,

(A.I) All eigenvalues of d

j=1

Aj(t,x)ξjare real valued in [0,T

R

dx×{

R

dξ\{0}}

and their multiplicity does not exceedν.

Many papers are devoted to the study of wellposedness in the Gevrey classes for the Cauchy problem (1.1). When all Aj are smooth enough with respect tot, then this property was proved for the order 1≤s<1+1/(ν−1)by M. D. Bronstein in [1] in Received July 10, 2006; accepted in revised form October 16, 2006.

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the higher order scalar case and by K. Kajitani in [4] in the system case, respectively.

Moreover they have shown it in the case that the coefficient also depend on x.

When each Aj has onlyµ-H¨older continuity int for some 0< µ≤1, the Cauchy problem is also wellposed in the Gevrey classes but the Gevrey order must be lower than the smooth case. The first result in the H¨older continuous case was derived by F. Colombini, E. Jannelli and S. Spagnolo in [2]. They proved that the Cauchy problem to the second order equationutt =a(t)ux x was Gevrey wellposed for the order 1≤s<1+µ/2 and, it is important, this order is optimal. T. Nishitani in [8]

extended to the second order equations with coefficients also depending onx, and then Y. Ohya and S. Tarama in [9] extended that the higher order scalar equation was Gevrey wellposed for 1≤s <1+µ/ν. The system case was investigated by Kajitani in [5], and he showed that the weakly hyperbolic systems were wellposed in the Gevrey classes for 1≤s <1+µ/(ν+1).

When the coefficients depend only ont, D’Ancona, T. Kinoshita and Spagnolo in [3] proved the Gevrey wellposedness for 1 ≤ s < 1+µ/ν to 3×3 weakly hyperbolic systems with coefficients depending ont. To prove it, they derived the energy estimates for the approximate symbols and moreover Yuzawa in [11] has treated the general systems of which coefficients depend only on time variable.

In this paper, we shall extend their result to anyN × N system whose coeffi- cients depend also on the space variables by using the other approach, semi-group method calledTanabe-Sobolevski method(cf. [6, 10]) and consequently obtain the energy estimates.

To state our results we shall introduce the Gevrey classes and their properties.

Definition 1.1. Let s ≥ 1, then we denote by γ(s)(

R

d) the set of all functions satisfying the following condition: for any compact subset K of

R

d, there exist constantsCK >0 andAK >0 such that

|∂xαu(x)| ≤CKA|α|K |α|!s

for anyxK andα

N

d and we defineγ0(s)(

R

d)=γ(s)(

R

d)C0(

R

d).

Definition 1.2. Letk be an integer and 0 < µ ≤ 1. For a Banach space Y, we denote byCk([0,T];Y)the set of functionsu(t)which arektimes differentiable inYwith respect tot and(∂/∂t)ku(t)areµ-H¨older continuous inY: there exists a constantC >0 such that

tlu(t)YC (0≤lk), ||∂tku(t)tku(t)||YC|tt|µ fort,t∈[0,T]. We writeCk,0([0,T];Y)asCk([0,T];Y)in brief.

Definition 1.3. Forρ≥0,s >1,h>0 andl

R

, we define

H(ρ)l (

R

d)=uL2x(

R

d); ξlhe(ρ)uˆ(ξ)L2ξ(

R

d),

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whereξh =

h2+ |ξ|2,(ρ)=(ρ, ξ;s,h)=ρξ1h/sanduˆ(ξ)stands for a Fourier transform ofu(x):

ˆ u(ξ)=

Rdei x·ξu(x)d x,

and forρ <0 we defineH(ρ)l (

R

d)as the dual space ofH(−ρ)l (

R

d).

Whenρ =0, H(l 0) = H0l is a usual Sobolev space and we write them as Hl in brief. H(ρ)l is a Hilbert space with inner product

(u, v)Hl

(ρ) =leρξ1/sh u,ξleρξ1/sh v)L2 and we define the norm of H(ρ)l by||u||Hl

(ρ) = ξleρξ1/sh uˆ(ξ)L2.

Definition 1.4. We definee(ρ) =eρD1/sh a pseudo differential operator of order infinity such as

e(ρ)u(x)=eρDx1/sh u(x)= 1

(2π)d Rdei x·ξ+(ρ,ξ)uˆ(ξ)dξ foru(x)inH(ρ)l .

Definition 1.5. Let ρ(t) be a positive definite function in [0,T],k an integer ≥ 0,0< µ ≤1 andl

R

. Then, we denote bye−(ρ(t))Ck([0,T];Hl)the class of functions f(t,x)for which to everyt ∈[0,T],

e(ρ(t))f(t,x)=eρ(t)Dx1/s f(t,x)Ck([0,T];Hl).

We note the relations betweenγ0(s)(

R

d)andH(ρ)l (

R

d).

Proposition 1.6 (cf. Lemma 1.2 in [4]). For any u(x)inγ0(s)(

R

d)and l

R

, there exists a constantρu >0such that u(x)in Hl

u)(

R

d).

Conversely, if u(x)belongs to H(ρ)l (

R

d)for someρ > 0, then u belongs to γ(s)(

R

d).

Now, we shall state the main theorems.

Theorem 1.7. Let1≤ s < 1+µ/ν, σ =−1)(1−1/s)and0< µ ≤1and takeδ ≥0such that s(δ+σ +1) >1+µ. Assume that(A.I)and the following condition(A.II)are valid;

(A.II) each Aj(t,x)belongs to C0([0,T];γ(s)(

R

d))for j =1, . . . ,d and B(t,x)C0([0,T];γ(s)(

R

d)).

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Then for every u0(x)in H(l T

0)(

R

d)and for every f(t,x)in e−(Tt0)C([0,T]; Hl(

R

d)), there exists a unique solution u(t,x)of the Cauchy problem(1.1), which is in e−(Tt)C([0,T];Hl(

R

d))e−(Tt)C1([0,T];Hl1(

R

d))and satis- fies

||u(t,·)||Hl+δ (T−t)C

||u0||Hl

(T0)+ t

0

||f(r,·)||Hl

(T−r+ρ0)dr

, (1.2) for any l

R

and0t T .

Considering the property of the finite propagation of the solution for the weakly hyperbolic system and Proposition 1.6, the following theorem is concluded by The- orem 1.7.

Theorem 1.8. Assume that(A.I)and(A.II). If1≤s < 1+µ/νand0< µ ≤1, then for any f(t,x)in C([0,T];γ(s)(

R

d))and u0(x)inγ(s)(

R

d), there is a unique solution u(t,x)in C1([0,T];γ(s)(

R

d))of the equation(1.1).

2.

Preliminaries

In this section we shall introduce some notation and fundamental propositions on the pseudo differential operator theory.

We denote bySρ,δ,m l (0≤δ < ρ≤1 ) a class of symbolsp(x, ξ)satisfying

|p|Smρ,δ,l = sup

(x,ξ)∈R2d,|α+β|≤l

|p(α)(β)(x, ξ)|

ξm−ρ|α|+δ|β| <∞, where p(β)(α)(x, ξ)= Dxβξαp(x, ξ). We writeS1m,0,lasSlmin brief.

We denote by(Sρ,δ,m l)N×Na class of matrix symbolsP(x,ξ)=(pi j(x,ξ))1i,jN

such that all pi jare inSρ,δ,m l and we define

|P|Sρ,δ,lm = max

1i,jN|pi j|Smρ,δ,l.

We often write(Sρ,δ,m l)N×N asSρ,δ,m l in brief andSρ,δm = ∩lSρ,δ,m l.

For p(x, ξ)Sρ,δm , we define the pseudo differential operator p(x,Dx): p(x,Dx)u(x)= 1

(2π)d ei xξp(x, ξ)uˆ(ξ)dξ.

The following proposition is well known as the boudedness inHqof pseudo differ- ential operators.

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Proposition 2.1. Let0≤δ < ρ ≤1and M0(q)=

2(|q| +l0+d+1) 2−δ

+l0, l0=

δ(d+1) ρδ

+1. Assume that lM0(q)and p(x, ξ)be in Sρ,δ,m l. Then, we have

||p(x,Dx)u||HqCq|p|Smρ,δ,M

0(q)uHq+m, (2.1) for uHq+m.

Next, we denote bysm,r,l a class of symbolsp(x, ξ)satisfying

|p|ms,r,l = sup

(x,ξ)∈R2d,|α+β|≤l,γ∈Zd+

|p(β+γ )(α) (x, ξ)|

r||γ|!sξm−|α| <∞. (2.2) For a symbol p(x, ξ)sm,r,l, we define an operator p(ρ)(x,Dx):

p(ρ)(x,Dx)=e(ρ)p(x,Dx)e−(ρ),

where(ρ)=ρDx1h/s. It follows from Kumano-go’s formula that the symbol of p(ρ)(x,Dx)give by

σ (p(ρ))(x, ξ)= 1 (2π)dOs-

R2dei yη+(ξ+η)−(ξ)p(x+y, ξ)d ydη, (2.3) where Os-

means an oscillatory integral. We denote by p(x, ξ)the symbol of p(ρ)(x,Dx)in brief.

From (2.3) and Taylor’s formula, p(x, ξ)= p(x, ξ)+

0<|γ|<N

p(γ )(x, ξ)ωγ(ρ, ξ)+rN(p)(x, ξ), (2.4) where

ωγ(ρ, ξ)= 1

γ!e−ρξκhξγ(eρξκh) and

rN(p)(x, ξ)= N (2π)d

×

|=N

Os-

R2d 1

0

(1−θ)N1ei yη+(ξ+η)−(ξ)p(γ )(xy,ξ)ωγ(ρ,ξ+η)dθd ydη.

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Proposition 2.2 ([5]). Let p(x, ξ)be insm,r,l,(ρ)=ρξ1h/sand s>1.

(i) ωγ and rN(p)satisfy the following inequalities:

|∂ξαωγ(ρ, ξ)| ≤Cαγ(|ρ| +h−κ)|ξ−(h 1−κ)|γ|−|α|, (2.5)

|∂ξαDβxrN(p)(x, ξ)| ≤CαβN(|ρ| +h−κ)NξmhN(1−κ)−|α|, (2.6) for(x, ξ)

R

2d, α, β

Z

d+,whereκ= 1s.

(ii) Ifρsatisfies

|ρ| ≤(24sdr)1/s, (2.7) and l satisfies

lsl

s−1

+l+ d

2

+1, (2.8)

then the symbol p(x, ξ)belongs to Slm. Moreover, there exists a constant Cl

such that

|p|SlmCl|p|s,r,lm .

The following proposition is the fundamental property on the hyperbolic polyno- mial:

Proposition 2.3 ([1]). Let p(t,x, λ, ξ)be a hyperbolic polynomial of order N , that is, p can be factorized by real roots:

p(t,x, λ, ξ)=

|α|+j=N

aj(t,x)λjξα= N j=1

λj(t,x, ξ)),

where allλj(t,x, ξ)are real valued for(t,x, ξ)∈[0,T

R

d×

R

d. Assume that the multiplicity ofλj(t,x, ξ)is at mostν(≤ N)and all the coefficients aj(t,x) and Dβxaj(t,x)(|β| ≤ν) are bounded in[0,T

R

d, then p(t,x, λ, ξ)satisfies the following estimates:

p(t,x, λ−1iξκh, ξ)C(|λ| + ξh)Nξ−κνh , (2.9)

|p(α)(β)(t,x, λiξκh, ξ)|

p(t,x, λiξκh, ξ)Cξ−κ|α|+(h 1−κ)|β| (|α+β| ≤ν), (2.10) forλ

C

such thatImλ <0and for(x, ξ)in[0,T

R

d.

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3.

The Cauchy problem in Sobolev spaces

In this section we assume that 1≤s<1+µ/νand putκ=1/s.

We reintroduce the equation (1.1):

P(t,x,D)u(t,x)= −i f(t,x), (t,x)∈[0,T

R

d,

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

R

d. (1.1)

where

P(t,x,D)= Dtd

j=1

Aj(t,x)Dj +i B(t,x),

andD=(Dt,Dx)=(Dt,D1,D2, . . . ,Dd),Dt = −i∂t andDi = −i∂i.

For some non negative and continuously differentiable function ρ(t), we set v(t,x) = e(ρ(t))u(t,x) = eρ(t)Dxκhu(t,x). Then we can reduce the problem

(1.1) to

P(t,x,Dt,Dx)v(t,x)=g(t,x),

v(0,x)=v0(x), (3.1)

where

Dt, =Dt,(ρ(t))= Dt +(t)Dxκh, P(t,x,Dt,Dx)= Dt,I

d j=1

Aj,(t,x,Dx)Dj +i B(t,x,Dx), for (t,x)

R

1+d,v0(x) = e(ρ(0))u0(x), g(t,x) = −i e(ρ(t))f(t,x), and each

Aj,(t,x,Dx)andBare pseudo differential operators such as

Aj,(t,x,Dx)= Aj,(ρ(t))(t,x,Dx)=eρ(t)DxκhAj(t,x)e−ρ(t)Dxκh, B(t,x,Dx)= B(ρ(t))(t,x,Dx)=eρ(t)DxκhB(t,x)e−ρ(t)Dxκh.

We shall solve the equation (3.1) in Sobolev space Hl by using the semi group for (t)Dxκh)IA(t)).

4.

Construction of

((λ+(τ)Dxκh))IA(τ))1

Let τ be fixed in [0,T]. In this section, we shall construct the inverse of((λ+ (τ)Dxκh)IA(τ))for a complex numberλwith Imλ≤ −h0. We consider the following resolvent equation :

((λ+(τ)Dxκh)IA(τ))v=g, (4.1)

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forgHq, where

A(τ)= A(τ,x,Dx)= d

j=1

Aj(τ,x)Dj, (4.2) and

A(τ)= A(ρ(τ))(t,x,Dx)= d

j=1

Aj,(ρ(τ))(τ,x,Dx)Dj. (4.3) Let us define several symbols as follows:

A(t,x, ξ)=σ (A)(t,x, ξ)= d

j=1

Aj(t,x)ξj,

A(t,x, ξ)=σ (A(ρ(t)))(t,x, ξ)= d

j=1

Aj,(ρ(t))(t,x, ξ)ξj,

P(t,x, λ, ξ)=σ (P)(t,x, λ, ξ)=λId

j=1

Aj(t,x, ξ)ξj+i B(t,x), P(t,x, λ, ξ)=σ (P)(t,x, λ, ξ)

=λId

j=1

Aj,(ρ(t))(t,x, ξ)ξj +i B(ρ(t))(t,x, ξ), H(t,x, λ, ξ)=coIA(t,x, ξ)),

p(t,x, λ, ξ)=det(λIA(t,x, ξ)), pL(t,x, λ, ξ)=detIA(t,x, ξ)),

M(t,x, λ, ξ)= 1

p(t,x, λ, ξ)I, λ=λ+(t)ξκh.

We shall consider an operatorP˜(t,x, λ,Dx)=(λIA(t,x,Dx))◦H(t,x, λ,Dx), where H(t,x, λ,Dx)is the operator with the symbolH(t,x, λ, ξ)and◦means an operator product. By Proposition 2.2, ifρ(t)satisfies

0≤ρ(t)(24sdr)−κ in [0,T], (4.4) then A(t,x, ξ)is in Sl1, soH(t, λ)is inSlN1, wherel satisfies (2.8). Since we can takelin (2.8) arbitarly,H(t, λ)∈ ∩l>0SlN1= SN1that is,

H(β)(α)(t,x, λ, ξ)C(|λ| + ξh)N1−|α|, (4.5) for(t,x, λ, ξ)∈[0,T

R

d×

C

×

R

d,α, β

Z

d+.

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Denote by S˜ρ,δm a set of symbols a(t,x, λ, ξ) which are holomorphic inλ ∈ {λ∈

C

: Imλ0}and satisfying

|∂ξαDβxa(t,x, λ, ξ)| ≤Cαβξmh−ρ|α|+δ|β|,x, ξ

R

d, λ

C

,Imλ0, α, β

Z

d+. Next we shall compare pL(t,x, λ, ξ)and

p(t,x, λ, ξ)=σ(ep(t,x, λ+iρ(t)Dxκh),Dx)e)(t,x, λ, ξ). Since p(t,x, λ, ξ)=

π∈SN

sgnπ ·a1π(1)a2π(2)· · ·aNπ(N),

where ai j = ai j(t,x, λ, ξ)is the (i, j)-component of (λIA(t,x, ξ)) and SN stands for the set of permutations of{1,2,· · · ,N}, we can write

p(t,x, λ,Dx)=

π∈SN

sgnπ

×a1π(1)(x,λ,Dx)a2π(2)(x,λ,Dx)◦ · · · ◦aNπ(N)(x,λ,Dx)+q1 and consequently

p(t,x,λ,Dx)=

π∈SN

sgnπa1π(1),(x,Dx)a2π(2),(x,Dx)◦ · · ·

· · · ◦aNπ(N),(x,Dx)+q1,

whereaj k,(x,D)=eaj keandq1(t,x, λ, ξ)∈ ˜S1N,01, because σ (a1π(1),(x,Dx)a2π(2),(x,Dx)◦ · · · ◦aNπ(N),(x,Dx))(x, ξ)

=a1π(1),(x, ξ)a2π(2),(x, ξ)· · ·aNπ(N),(x, ξ)+qπ(x, ξ) for anyπSN, whereqπ(t,x, λ, ξ)∈ ˜S1N,01. Hence we have

p(t,x, λ, ξ)= pL(t,x, λ, ξ)+q2(t,x, λ, ξ), (4.6) whereq2 ∈ ˜S1N,01that is,

q2(α)(β)(t,x, λ, ξ)Cαβ(|λ| + ξ)N1−|α|, (4.7) for (t,x, λ, ξ) ∈ [0,T

R

d ×

C

×

R

d,λ ≤ 0 andα, β

N

d. Therefore it follows from (4.5), (4.6) and (4.7) that we have

σ(P˜)(t,x, λ, ξ)= p(t,x, λ, ξ)I +Q1(t,x, λ, ξ), (4.8) where Q1satisfies

|Q1(α)(β)(t,x, λ, ξ)| ≤Cαβ(|λ| + ξκh)N1−|α|, (4.9) for(t,x, λ, ξ)∈[0,T

R

d×

C

×

R

d,Imλ0 andα, β

Z

d+.

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We shall construct the inverse of((λ+(t)Dxκh)IA(τ))of form ((λ+iρ(t)Dxκh)IA(τ))1=H(τ, λ+iρ(t)Dxκh)◦(P˜)1(τ, λ+iρ(t)Dxκh), where(P˜)1is the inverse operator of P. Let us show that˜ (P˜)1exists. Define a operatorS(t, λ)=S(t,x, λ,Dx)as

S(t,x, λ,Dx)= I − ˜P(t,x, λ,Dx)M(t,x, λ,Dx). (4.10) where M(t,x, λ, ξ) = p(t,x, λ+(t)Dxκh, ξ)1.Noting that p(τ,x, λ, ξ)is elliptic for|λ| ≥ Mξh, we can prove the following proposition by use of Propo- sition 2.3.

Proposition 4.1. Assume that1<s< ν/(ν−1)and q satisfies M0(q) <l, where M0(q)is given in Proposition 2.1 with ρ = κ, δ = 1−κ. Then, there exists a positive constant h>0such that for anyτ ∈[0,T]andλ

C

(Imλ≤0).

|∂ξαDxβS(t,x, λ, ξ)| ≤Cαβ(|λ| + ξh)ν(1−κ)−1−κ|α|+(1−κ)|β|. (4.11) Proof. If|λ|≥Mξh, thenp(t,x,λ+iρ(t)ξκh, ξ)is elliptic for|λ|≥Mξh(M 1), so we have

|p(t,x, λ+(t)ξκh, ξ)| ≥c0(|λ| + ξh)m,

for|λ| ≥ Mξh, which implies (4.11) evidently for|λ| ≥ Mξh. From (4.8), the symbol of P˜(t,x, λ,Dx)M(t,x, λ,Dx)is

σ

P˜(t, λ)M(t, λ) (x, ξ)

= I +

0<|γ|<n

1

γ!(p(t,x, λ, ξ))(γ )

1

p(t,x, λ+(t)ξκh, ξ)

(γ )

I

+

0≤|γ|<n

1

γ!Q(γ )(t,x, λ, ξ)

1

p(t,x, λ+(t)ξκh, ξ)

(γ )

I +Rn(t,x, λ, ξ)

where Rn is in S˜κ,(11−κ)ν−κ−κ n. Moreover, by virtue of Proposition 2.2 and Proposi- tion 2.3

p(t,x,λ+(t)ξκh, ξ)(γ )

1

p(t,x,λ+iρ(t)ξκh, ξ)

(γ )

∈ ˜Sκ,(11−κ2κ)|γ|⊂ ˜Sκ,−δ11−κ for 1≤ |γ| ≤n, and

Q(γ )1 (t,x, λ, ξ) 1

p

(γ )(t,x, λ+(t)ξκh, ξ)I ∈ ˜Sκ,(11−κ)ν−−κ 1−κ|γ| ⊂ ˜Sκ,−δ11−κ for 0 ≤ |γ| ≤ n, whereδ1 = min{2κ−1,1−ν(1−κ)} = 1−ν(1−κ) > 0.

Therefore we can seeS(t,x, λ, ξ)is inS˜κ,−δ11−κ. This implies (4.11).

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From Proposition 4.1 it follows that if we take the parameterh>0 sufficiently large, there exists a(IS(τ, λ))1as Neumann series

(IS(τ, λ))1=

j=0

S(τ, λ)j

=

P˜(τ,x, λ,Dx)M(τ,x, λ,Dx)1 .

(4.12)

Therefore we obtain the inverse operator of+(τ)Dxκh)IA(ρ(τ))(τ))as ((λ+(τ)Dxκh)IA(ρ(τ))(τ,x,Dx))1

= H(τ,x, λ,Dx)M(τ,x, λ,Dx)

j=0

S(τ,x, λ,Dx)j

= H(τ)M(τ)+H(τ)M(τ)

j=1

S(τ)j.

(4.13)

Proposition 2.2 and Proposition 2.3 yield,

σ (H(τ, λ)M(τ, λ))(x, ξ)= H(τ, λ, ξ)M(τ, λ, ξ)+Q2(τ,x, λ, ξ), where Q2 ∈ ˜Sκ,ν(11−κ−κ)−1−κ. Moreover, noting thatσ(HMS)∈ ˜S2κ.ν(1−κ1−κ)−2, we get

σ (((λ+(τ)Dxκh)IA(ρ(τ))(τ,x,Dx))1)(τ,x, λ, ξ)

= H(τ, λ, ξ)M(τ, λ, ξ)+Q3(τ,x, λ, ξ), (4.14) where Q3 ∈ ˜Sκ,−δ1−κ, where δ = min{1+ κν(1− κ),2(1− ν(1−κ))} = 2(1−ν(1−κ)) >0.

5.

The equation with time-independent coefficients

In this section, we shall solve the following equation: for fixedτ ∈[0,T),

P(ρ(τ))(τ,x,Dt,(ρ(τ)),Dx)v(t,x),=g(t,x), Tt > τ,x

R

d

v(τ,x)=v0(x). (5.1)

We define

V0(t, τ)=V0(t, τ,x,Dx)

= 1

2πi Imλ=0eiλ(t−τ)((λ+(τ)Dxκh)IA(ρ(τ))(τ,x,Dx))1 (5.2) for 0≤τtT.

(12)

Proposition 5.1. Let1<s <1+ µν, µ≤1,κ = 1/s, andσ = −1)(1−κ). Then there isκ > 0such thatκ < κ < 1,1−κ < κ and V0(t, τ)is a pseudo differential operator of which symbol is in Sκσ,1−κ. Moreover there is a constant C >0such that

V0(t, τ)uHq

(ρ)CuHq+σ

(ρ), Tt > τ, (5.3)

for uH(ρ)q .

Proof. It is sufficient to prove that there isκ>0 such that 1> κ >1−κand

|∂ξDxαV0(t, τ,x, ξ)| ≤Cαβξσh−κ|α|+(1−κ)|β|, Ttτ,x, ξ

R

n. (5.4) forα, β

N

d. Since it is known from Proposition 3.3 in [6] thatV0(t, τ)(x, ξ)is inSκ,ν(11−κ−κ), it suffices to

|V0(t, τ)(x, ξ)| ≤Cξσh, Ttτ,x, ξ

R

n. (5.5) In fact, we can chooseκ such thatκ > κandκ >1−κbecause ofκ >1−κ. V0(t, τ)(x, ξ)is inSκ,ν(11−κ−κ)implies

|∂ξDxαV0(t, τ,x, ξ)| ≤Cαβξσh−κ|α|+(1−κ)|β|, Ttτ,x, ξ

R

n. (5.6) for |α+β| ≥ κ−κ1−κ. Using the interpolation theorem we can get (5.4) from (5.5) and (5.6). Now we shall show (5.5). (5.2) yields

V0(t, τ)(x, ξ)= 1

2πi Imλ=0eiλ(t−τ)((λ+iρ(τ)Dxκh)IA(ρ(τ))1(τ,x, ξ))dλ

= 1 2πi lim

R→∞ Imλ=0,|Reλ|≤R

eiλ(t−τ)((λ+iρ(τ)Dxκh)IA(ρ(τ))1(τ,x, ξ))dλ.

Besides,

Imλ=0,|Reλ|≤R

eiλ(t−τ)((λ+(τ)Dxκh)IA(ρ(τ))1(τ,x, ξ))dλ

=

CR

eiλ(t−τ)((λ+(τ)Dxκh)IA(ρ(τ)))1(τ,x, ξ))dλ

γR

eiλ(t−τ)((λ+(τ)Dxκh)IA(ρ(τ))))1(τ,x, ξ))dλ,

where CR = {λ ∈

C

;Imλ = 0,|Reλ| ≤ R} ∪γR, γR = {|λ| = R,Imλ 0}. Noting

lim

R→∞ γR

eiλ(t−τ)((λ+(τ)Dxκh)IA(ρ(τ)))1(τ,x, ξ))dλ=0,

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