# The Cauchy problem for weakly hyperbolic systems

(1)

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## The Cauchy problem for weakly hyperbolic systems

†‡

s

s

k

ε

(3)

s

s

s

t2

j,l

xj

xl

j,l

j

l

s

j,k

k

s

t

d

j=1

j

xj

t

x

j

d

j

j

d

s

d

xα

L(K)

|α|+1

s

(4)

s0

d

0

s

xα

d

xα

L( ˜K)

|α|+1

s

s

d

d

0

s

s

|t=0

0

s

s

d

d

s

(5)

−1

k,µ

k

k,µ

s

s

k,µ

k,µ

s

s

(6)

j

j

ε

ε

−γ

ε

ε

t

ε

ε

−β

ε

ε

1

−β

α

C|ξ|α

−C|ξ|α

s

t

ε

−1

ε

k,µ

ε

t

ε

(7)

1

N−2

t

x

1

−1

−1

t

1

x

−1

−1

t

N

N−1

N−2

(N−1)/N

t

d

j=1

j

xj

|t=0

(8)

t

|t=0

d

−1

−1

−M

M

2

12

d

1

L1([0,T])

t

α

d

α

s

d

1

N

N

0

N

t

|t=0

1,1

t

t

(9)

L

2

0

t

α

0

0

1

2

12

0

C1Φ(t)hξiα+tC2

12

0

C1Φ(t)hξiα+tC2

t 0

0

0

0

12

0

t

0

0

0

1

2M

γhξiα

t 0

0

0

s

−δhξi1/s

−δ|ξ|1/s

1

2M

γhξiα

−δhξi1/s

0

2

2

−δ0|ξ|1/s

s

d

d

(10)

d

−δhξi1/s

R|Imξ|

−δ|ξ|1/s

R|Imξ|

d

2

1

d

1

2M

γhξiα

(R+γ1t)|Imξ|

−δhξi1/s

1

s

0

0

s

s

s

ξ

T 0

0

d

s0

(11)

k,µ

T 0

t

1−1/(k+µ)

1/(k+µ) Ck,µ

n

k,µ

n

]

1

t

]

1−1/(k+µ)

]

L1([0,T])

1/(k+µ) Ck,µ

t

1−1/(k+µ)

]

T

0

]

j→∞

T

0

t

1−1/(k+µ)

]

1

t

]

1−1/(k+µ)

]

(12)

1

n

n

]j

1

]

]j

]

t

1−1/(k+µ)

t

]

]

t

t

]

k,µ

n

0

+

t

L1([0,T])

−1/(k+µ)

1/(k+µ) Ck,µ

t

]

1−1/(k+µ)

]

1−1/(k+µ)

]

−1/(k+µ)

0

0

(13)

2

0

2

02

0

0∗

0

0

2

2

2

2

ε

ε

0∗

0

2

ε

2

2

ε

2

ε

2

ε

2

0

ε

2

12

2

12

ε

2

12

ε

12

2

ε

k,µ

ε

k,µ

0

1

k,µ

0

t

0

0

1−1/(k+µ)

L1([0,T])

0

1/k+µ Ck,µ

(14)

t

0

t

0

1−1/2k

−1/k

ε

t

ε

1

−1/k

ε

1

0

t

1−1/2(k+µ)

−1/(k+µ)

ε

0

1

k+µ

ε

1

k+µ

k,µ

ε

1

t

ε

−1/(k+µ)

ε

ε

ε

−(k+µ)/(k+µ+1)

−1/(k+µ)

ε

−k/(k+1)

(15)

ε

−2

−2k/(k+1)

−1

t

−1/(k+µ)

1/(k+µ+1)

j

k+µ

1

N

0

0

d

j

0

0

0

s

0

0

0

d

0

0

k+µ

−1

0

m

(16)

0

1

m

k,µ

0

0

−1

0

t

0

0

0

0

j

t

j

1/(k+µ+1)

j

j

j

L1

j

j

1/(k+µ+1)

j

−1

j

j

j

j

−2

j

t

j

j

1/(k+µ+1)

j

0

m

j

m

j=1

j

j

l6=j

l

0

1

1

m

m

−1

−2m

(17)

t

1/(k+µ+1)

t

j

1/(k+µ+1)

j

j

t

1/(k+µ+1)

j

j

1/(k+1)

t

t

t

t

1/(k+µ+1)

ε

0

isA

−isA

−sε

0

isM

−isM

2i

j

j

ε

0

j

j

−isA

−itA

−1

2

ε

2

(18)

s

t

−isA(t)

t

t

ε

−1

t

ε

ε

ε

ε

ε

ε

ε

ε

ε

ε

ε

−1

n

∗k

j

ε

ε

k,l

∗l

l

(19)

m−1

k=0 m−1

l=0

k,l

m−1−k

m−1−l

m−1

k=0 m−1

l=0

k,l

∗m−1−k

m−1−l

k,l

l,k

2i1

k,l

k,l

k−1,l

k,l−1

k,l

k,l

∗k

l+1

∗k+1

l

k,l−1

k−1,l

∗k

l

A

A

A

(20)

A

A

A

A

A

12

M

M

M

M

M

M

M

M

j

A

A

N

N

k=1

k

N−k

N

j=1

j

M

2i

M

A

N

j=1

j

(21)

M

M

M

M

j,k

j

k

N

N(N−1)/2

j<k

j

k

2

2

A

A

ε

M

M

M

M

ε

M

ε

N

M

ε

M

ε

N(N−1)/2

ε

ε

j<k

j

k

2

2

ε

1

m

j,k

1

M

ε

j,k

M

1

ε

ε

ε

M

1

M

ε

ε

M

1

M

1

M

M

ε

(22)

j,k

j,k

j,k

j,k

j,k

ε

j,k

ε

ε

ε

ε

ε

ε

ε

ε

ε

M

A

0

0

0

A

0

k

k

k

k

k

k

A

−1

ε

ε

ε

(23)

k,µ

d−1

ε

−1

ε

ε

ε

ε

ε

−1

N(N−1)

ε

ε,j,k

ε

k,µ

t

ε,j,s

j,k

ε,j,s

1−1/(k+µ)

j,k

L1([0,T])

t

ε,j,s

j,k

1−1/(k+µ)

j,k

−N(N−1)/(k+µ)

j,k

t

ε

−N(N−1)/(k+µ)

2

−N(N−1)/(k+µ)

ε

(24)

ε

−k/(k+µ+N(N−1))

−N(N−1)/(k+µ)

ε

### (t, ω)A(t, ξ). Their final con- tribution is O(|ξ|

(N(N−1)/(k+µ+N(N−1))

0

0

0

−1

1

m

j

j

j

t

j

αj

j

j

j

L1

j

j

αj

j

−1

j

j

j

j

j

j

j

j

j

−Nj(Nj−1)

j

t

j

j

αj

j

j

m

j=1

j

j

l6=j

l

1

1

m

m

(25)

j

j

Nj

−1

−M

j

j

t

α

1

j

j

α

t

ε

12

t

(26)

j

k,µ

d−1

[

N1

[

N(N−1)

[

[

[

[

[

[

N(N−1)

[

2

2j

j

[

j<l

j

l

2

j

l

2

1

2j

1

s

(27)

ε

t

ε

ε

[

α

j,k

n

t

α

N(N−1)−n

N(N−1)−n

α

t

α

α

1−1/(k+µ)

t

j,k

n

N(N−1)−n−1/(k+µ)

n

[

N(N−1)−n−1/(k+µ)

1

k,µ

t

j,k

−1/(k+µ)

t

ε

−1/(k+µ)

ε

−(k+µ)/(k+µ+1)

t

[

t

(28)

2

(29)

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