1 Introduction and main results
For the general first order quasilinear hyperbolic systems,
∂
u
∂
t
+
A
(
u
)
∂
u
∂
x
=
0
the global existence of classical solutions of Cauchy problem has been established for lin-early degenerate characteristics or weakly linearly degenerate characteristics with various smallness assumptions on the initial data by Bressan 1, Li 2, Li and Zhou 34, Li and Peng 56, and Zhou 7. The asymptotic behavior has been obtained by Kong and Yang 8, Dai and Kong 910. For linearly degenerate diagonalizable quasilinear hyperbolic systems with "large" initial data, asymptotic behavior of the global classical solutions has been ob-tained by Liu and Zhou 11. For the initial-boundary value problem in the first quadrant Li and Wang 12 proved the global existence of classical solutions for weakly linearly degenerate positive eigenvalues with small and decay initial and boundary data. The asymptotic behavior of the global classical solutions is studied by Zhang 13. The global existence and asymptotic behavior of classical solutions of the initial-boundary value problem of diagonal-izable quasilinear hyperbolic systems in the first quadrat was obtained in 14.
However, relatively little is known for the Goursat problem with characteristic boundaries. Global existence of the global classical solutions for the Goursat problem of reducible quasilinear hyperbolic system was obtained in 15. Under the assumptions of boundary data is small and decaying, the global existence and asymptotic behavior to classical solutions can be obtained by Liu 1617. The asymptotic behavior of classical solutions of Goursat problem for reducible quasilinear hyperbolic system was shown in 18.
In this article, we consider the following diagonalizable quasilinear hyperbolic system:
∂
u
i
∂
t
+
λ
i
(
u
)
∂
u
i
∂
x
=
0
where u = (u1, ..., un)T is unknown vector-valued function of (t, x). λi(u) is given by C2 vector-valued function of u and is linearly degenerate, i.e.,
∂
λ
i
(
u
)
∂
u
i
≡
0
The system (1) is strictly hyperbolic, i.e.,
λ
1
(
u
)
<
⋯
<
λ
m
(
u
)
<
0
<
λ
m
+
1
(
u
)
<
⋯
<
λ
n
(
u
)
Suppose that there exists a positive constant δ such that
λ
i
+
1
(
u
)
-
λ
i
(
v
)
≥
δ
,
i
=
1
,
…
,
n
-
1
Consider the Goursat problem for the strictly quasilinear hyperbolic system (1), in which the solutions to system (1) is asked to satisfy the following characteristic boundary conditions:
x
=
x
1
(
t
)
:
u
=
ϕ
(
t
)
and
x
=
x
n
(
t
)
:
u
=
ψ
(
t
)
;
where x = x1(t) and x = xn(t) are the leftmost and the rightmost characteristics passing through the origin (t, x) = (0, 0), respectively, such that
d
x
1
(
t
)
d
t
=
λ
1
ϕ
(
t
)
x
1
(
0
)
=
0
and
d
x
n
(
t
)
d
t
=
λ
n
ψ
(
t
)
x
n
(
0
)
=
0
moreover,
l
1
(
ϕ
(
t
)
)
ϕ
′
(
t
)
≡
0
and
l
n
(
ψ
(
t
)
)
ψ
′
(
t
)
≡
0
where ϕ (t) = (ϕ1(t), ..., ϕn(t))T and ψ(t) = (ψ1(t), ..., ψn(t))T are any given C1 vector functions satisfying the conditions of C1 compatibility at the origin (0, 0):
ϕ
(
0
)
=
ψ
(
0
)
and
λ
n
(
ϕ
(
0
)
)
ϕ
′
(
0
)
-
λ
1
(
ψ
(
0
)
)
ψ
′
(
0
)
+
A
(
ϕ
(
0
)
)
(
ψ
′
(
0
)
-
ϕ
′
(
0
)
)
=
0
.
li(u) be a left eigenvector corresponding to λi(u) and A(u) = diag{λ1(u), ..., λn(u)}.
Our goal in this article is to get the global existence and asymptotic behavior of the global classical solutions of the Goursat problem (1), (5), and (6) with "large" boundary data. With the assumptions that
sup
t
∈
R
+
{
|
ϕ
′
(
t
)
|
+
|
ψ
′
(
t
)
|
}
≐
M
<
∞
,
sup
t
∈
R
+
{
|
ϕ
(
t
)
|
+
|
ψ
(
t
)
|
}
≐
M
0
<
∞
∫
0
+
∞
|
ϕ
(
t
)
|
+
|
ψ
(
t
)
|
d
t
≐
N
1
<
∞
,
∫
0
+
∞
|
ϕ
′
(
t
)
|
+
|
ψ
′
(
t
)
|
d
t
≐
N
2
<
∞
we can prove the following results:
Theorem 1.1. The above assumptions and the conditions of C1 compatibility at the point (0, 0) are satisfied, the Goursat problem (1), (5), and (6) admits a unique global C1 solutions u = u(t, x) on the domain D = {(t, x)| t ≥ 0, x1(t) ≤ × ≤ xn(t)}.
If the leftmost and rightmost characteristics are convex, we can get the following result:
Theorem 1.2. Under the assumptions of Theorem 1.1, there exists a unique piece-wised C1 vector-valued function Φ(x) = (Φ1(x), ..., Φn(x))T such that
u
(
t
,
x
)
→
∑
i
=
1
n
Φ
i
(
x
-
λ
i
(
0
)
t
)
e
i
,
t
→
+
∞
uniformly as t tends to infinity, where ei = (0, ..., 0, 1i, 0, ..., 0)T.
Remark 1.1. If the system (1) is non-strictly hyperbolic but each characteristic has constant multiplicity, then the result is similar as Theorems above.
2 Global existence of C1 solutions
In this section, we will obtain some uniform a priori estimate which also play an important role in the proof of Theorem 1.1. In order to proving the global existence of classical solutions of the Goursat problem (1), (5), and (6), we will prove that |
|
u
|
|
C
1
(
D
)
is bounded. For any fixed T ≥ 0, we denote DT = {(t, x)|0 ≤ t ≤ T, x1(t) ≤ x ≤ xn(t)} and introduce
w
i
(
t
,
x
)
=
∂
u
i
(
t
,
x
)
∂
x
i
=
1
,
…
,
n
,
W
1
(
T
)
=
sup
0
≤
t
≤
T
∫
x
1
(
t
)
x
n
(
t
)
|
w
(
t
,
x
)
|
d
x
W
̃
1
(
T
)
=
max
i
≠
j
sup
C
̃
j
∫
C
̃
j
|
w
i
(
t
,
x
)
|
d
t
,
U
̃
1
(
T
)
=
max
i
≠
j
sup
C
̃
j
∫
C
̃
j
|
u
i
(
t
,
x
)
|
d
t
W
̄
1
(
T
)
=
max
i
≠
j
sup
L
j
∫
L
j
|
w
i
(
t
,
x
)
|
d
t
,
U
̄
1
(
T
)
=
max
i
≠
j
sup
L
j
∫
L
j
|
u
i
(
t
,
x
)
|
d
t
W
∞
(
T
)
=
sup
(
t
,
x
)
∈
D
T
|
w
(
t
,
x
)
,
U
∞
(
T
)
=
sup
(
t
,
x
)
∈
D
T
|
u
(
t
,
x
)
where
C
̃
j
stands for any given jth characteristic
d
x
d
t
=
λ
j
(
u
)
, Lj stands for any given radial that has the slope λj(0) on the domain DT .
Lemma 2.1. Under the assumptions of Theorem 1.1, there exists a positive constant C such that, the following estimates hold
W
̃
1
(
T
)
,
W
̄
1
(
T
)
,
W
1
(
T
)
≤
C
N
2
U
̃
1
(
T
)
,
U
̄
1
(
T
)
≤
C
N
1
e
C
N
2
W
∞
(
T
)
≤
C
M
e
C
N
2
U
∞
(
T
)
≤
C
Remark 2.1. The positive constant C is only depend on δ, M0 and independent of M, N1, N2, T. In the following, the meaning of C is similar but may change from line to line.
Proof. For any fixed point (t, x) ∈ DT , we draw the ith characteristic
C
̃
i
:
x
=
x
i
(
t
)
through this point and intersecting x1(t) or xn(t) at a point (t*, x1(t*)) or (t*, xn(t*)). Noting system (1), ui(t, x) is a constant along the ith characteristic, then we have ui(t, x) = ϕi(t*) or ui(t, x) = ψi(t*). Then
|
u
i
(
t
,
x
)
|
≤
sup
t
∈
R
+
|
ϕ
i
(
t
*
)
|
+
|
ψ
(
t
*
)
|
≤
M
0
≤
C
Then, we obtain the estimate (23).
Differentiating (1) with respect to x we can get
∂
w
i
∂
t
+
∂
λ
i
(
u
)
w
i
∂
x
=
0
We rewrite (25) as
d
(
w
i
(
t
,
x
)
(
d
x
-
λ
i
(
u
)
d
t
)
)
=
0
Multiplying the system above by sign(wi), we have
d
(
|
w
i
(
t
,
x
)
|
(
d
x
-
λ
i
(
u
)
d
t
)
)
=
0
There are only the following cases(as shown in Figure 1):
<p>Figure 1</p>Cases in estimating
W
̃
1
(
T
)
Cases in estimating W ̃1(T).
![]()
Case 1. For any fixed t0 ∈ R+, let
C
̃
j
:
x
=
x
j
(
t
)
,
j
>
1
stands for any given jth characteristic, passing through any point A(t0, x1(t0)) on the boundary x = x1(t) and intersects t = T at point P. We draw an ith characteristic C ̃i:x=xi(t) from P downward, intersecting x = x1(t) at a point B(t1, x1(t1)). Without loss of generality, we assume t0 < t1, then j > i. Integrating (26) in the region APB to get
∫
C
̃
j
(
λ
j
(
u
)
-
λ
i
(
u
)
)
|
w
i
(
t
,
x
)
|
d
t
=
∫
t
0
t
1
(
λ
i
(
u
)
-
λ
i
(
u
)
|
w
i
(
t
,
x
1
(
t
)
)
|
d
t
Along the 1th characteristic,
d
x
d
t
=
λ
1
(
u
)
, then
d
t
d
x
=
1
λ
1
(
u
)
∂
u
i
(
t
,
x
1
(
t
)
)
∂
x
=
∂
u
i
(
t
,
x
1
(
t
)
)
∂
t
∂
t
∂
x
=
ϕ
i
′
(
t
)
λ
1
(
u
)
Noting (4), (13), and (23), we can get
∫
C
̃
j
|
w
i
(
t
,
x
)
|
d
t
≤
C
∫
0
∞
|
ϕ
i
′
(
t
)
|
d
t
≤
C
N
2
Case 2. For any fixed t0 ∈ R+, passing through the point A(t0, x1(t0)), we draw
C
̃
j
:
x
=
x
j
(
t
)
and intersecting t = T at point P. We draw the ith characteristic C ̃i:x=xi(t) from P downward, intersecting x = xn(t) at B(t1, xn(t1)). Then, we integrate (26) in the region PAOB to get
∫
C
̃
j
(
λ
j
(
u
)
-
λ
i
(
u
)
)
|
w
i
(
t
,
x
)
|
d
t
=
∫
0
t
0
(
λ
1
(
u
)
-
(
λ
i
(
u
)
)
∂
u
i
(
t
,
x
1
(
t
)
∂
t
∂
t
∂
x
d
t
+
∫
0
t
1
(
λ
n
(
u
)
-
λ
i
(
u
)
)
∂
u
i
(
t
,
x
n
(
t
)
)
∂
t
∂
t
∂
x
d
t
Noting i = 1, ..., m i < j, Equations (4), (13) and using the above procedure, we have
∫
C
̃
j
|
w
i
(
t
,
x
)
|
d
t
≤
C
∫
0
∞
|
ϕ
i
′
(
t
)
|
d
t
+
∫
0
∞
|
ψ
i
′
(
t
)
|
d
t
≤
C
N
2
If the point A is on the characteristic boundary x = xn(t), using the same method as above cases we can get the desired estimates. Then, we can obtain
W
̃
1
(
T
)
≤
C
N
2
In the similar way, substituting Lj for C ̃j in the above cases we also can get
W
̄
1
(
T
)
≤
C
N
2
In the following, we will get the estimate W1(T) ≤ CN2. Integrating Equation (26) with respect to x from x1(t) to xn(t) for any t ∈ [0, T ] leads to
∫
x
1
(
t
)
x
n
(
t
)
∂
t
|
w
i
(
t
,
x
)
|
d
x
=
λ
i
(
u
(
t
,
x
n
(
t
)
)
)
∂
u
i
(
t
,
x
n
(
t
)
)
∂
x
-
λ
i
(
u
(
t
,
x
1
(
t
)
)
∂
u
i
(
t
,
x
1
(
t
)
)
∂
x
∂
t
∫
x
1
(
t
)
x
n
(
t
)
|
w
i
(
t
,
x
)
|
d
x
=
(
λ
n
-
λ
i
)
(
ψ
(
t
)
)
∂
u
i
(
t
,
x
n
(
t
)
)
∂
x
-
(
λ
1
-
λ
i
)
(
ψ
(
t
)
)
∂
u
i
(
t
,
x
1
(
t
)
)
∂
x
Using the same procedure as (28), (29), we can get
∫
x
1
(
t
)
x
n
(
t
)
|
w
i
(
t
,
x
)
|
d
x
≤
C
N
2
Then, we can get the desired estimate.
Rewriting the Equation (25), we get
∂
w
i
∂
t
+
λ
i
(
u
)
∂
w
i
∂
x
=
-
∂
λ
i
(
u
)
∂
x
w
i
Case 1. When i = 1, ..., m, we draw the ith characteristic intersecting x = xn(t) with the point (t0, xn(t0)). Solving the ODE system we can get
w
i
(
t
,
x
i
(
t
)
)
=
w
i
t
0
,
x
n
(
t
0
)
e
∫
C
̄
i
-
∂
λ
i
(
u
)
∂
x
d
t
Noting
w
i
(
t
0
,
x
n
(
t
0
)
)
=
ψ
i
′
(
t
0
)
λ
n
(
u
)
and the estimate (30), then
|
w
i
(
t
,
x
i
(
t
)
)
|
≤
C
M
e
C
∫
C
̃
i
∑
j
≠
i
|
w
j
|
d
t
≤
C
M
e
C
W
̃
1
(
T
)
≤
C
M
e
C
N
2
Case 2. When i = m + 1, ..., n and the ith characteristic intersecting x = x1(t) with a point (t0, x1(t0)), then
w
i
(
t
,
x
i
(
t
)
)
=
w
i
(
t
0
,
x
1
(
t
0
)
)
e
∫
C
̃
i
-
∂
λ
i
(
u
)
λ
x
d
t
Then, we can get
|
w
i
(
t
,
x
i
(
t
)
)
|
≤
C
M
e
C
N
2
Then, we can obtain the estimate
W
∞
T
≤
C
M
e
C
N
2
In the following we estimate
U
̃
1
(
T
)
.
Noting system (1), we denote the multiplier Hi ∈ C1, (i = 1, ..., n) such that,
∂
(
H
i
u
i
)
∂
t
+
∂
λ
i
(
u
)
H
i
u
i
∂
x
=
0
Noting (1), then
∂
H
i
∂
t
+
λ
i
(
u
)
∂
H
i
∂
x
=
-
∂
λ
i
(
u
)
∂
x
⋅
H
i
Let Hi(t, x1(t)) = 1 and Hi(t, xn(t)) = 1 we can get
H
i
=
e
-
∫
C
̃
i
∂
λ
i
(
u
(
τ
,
x
)
)
∂
x
d
τ
We note the system (1) is linearly degenerate, then
∂
λ
i
(
u
)
∂
x
=
∑
j
≠
i
n
∂
λ
i
(
u
)
∂
u
j
w
j
≤
C
∑
j
≠
i
|
w
j
|
e
-
C
∑
j
≠
i
∫
C
̃
i
|
w
j
|
d
t
≤
H
i
≤
e
C
∑
j
≠
i
∫
C
̃
i
|
w
j
|
d
t
e
-
C
W
̃
1
(
T
)
≤
H
i
≤
e
C
W
̃
1
(
T
)
e
-
C
N
2
≤
H
i
≤
e
C
N
2
Noting Equation (38), we can rewrite it as
d
(
H
i
|
u
i
|
(
d
x
-
λ
i
(
u
)
d
t
)
)
=
0
There are only the following cases like we estimate W ̃1(T):
Case 1. We can integrate (40) in the region APB which is same to the case 1 of proof of W ̃1(T) to get
∫
C
̃
j
H
i
|
u
i
|
(
t
,
x
)
d
x
≤
C
∫
t
0
t
1
H
i
|
u
i
|
(
t
,
x
1
(
t
)
)
d
t
Then
∫
C
̃
j
|
u
i
|
(
t
,
x
)
d
x
≤
C
e
C
N
2
∫
0
+
∞
|
ϕ
(
t
)
|
d
t
≤
C
N
1
e
C
N
2
Case 2. Integrating the Equation (40) in the region PAOB which is same to the case 2 of proof of W ̃1(T), we can get
∫
c
j
(
λ
i
(
u
)
-
λ
j
(
u
)
)
H
i
|
u
i
|
(
t
,
x
)
d
x
=
∫
0
t
0
(
λ
1
(
u
)
-
λ
i
(
u
)
)
H
i
|
u
i
(
t
,
x
1
(
t
)
)
|
d
t
+
∫
0
t
1
(
λ
n
(
u
)
-
λ
i
(
u
)
)
H
i
λ
i
(
u
)
|
u
i
(
t
,
x
n
(
t
)
)
|
d
t
Using the above procedures, we can get
∫
C
̃
j
|
u
i
(
t
,
x
)
|
d
x
≤
C
e
C
N
2
∫
0
+
∞
|
ϕ
(
t
)
|
d
t
+
∫
0
+
∞
|
ψ
(
t
)
|
d
t
≤
C
N
1
e
C
N
2
Then
U
̃
1
(
T
)
≤
C
N
1
e
C
N
2
In the similar way, substituting Lj for C ̃j we can get
∫
L
j
|
u
i
(
t
,
x
)
|
d
x
≤
C
N
1
e
C
N
2
U
̄
1
(
T
)
≤
C
N
1
e
C
N
2
Combining (24), (30), (31), (34), (37), (43), and (44) together we can obtain the conclusion of Lemma 2.1.
Proof of Theorem 1.1.
Noting the conclusion of Lemma 2.1, we can get
W
̃
1
(
∞
)
,
W
̄
1
(
∞
)
,
W
1
(
∞
)
≤
C
N
2
U
̃
1
(
∞
)
,
U
̄
1
(
∞
)
,
≤
C
N
1
e
C
N
2
W
∞
(
∞
)
≤
C
M
e
C
N
2
U
∞
(
∞
)
≤
C
Therefore, we can obtain that the system (1), (5), and (6) have global classical solutions on the domain D = {(t, x)| t ≥ 0, x1(t) ≤ x ≤ xn(t)}.
3 Asymptotic behavior of global classical solutions
In this section, under the assumption of the leftmost and rightmost characteristics, we will study the asymptotic behavior of the global classical solutions of system (1), (5), and (6) and give the proof of Theorem 1.2.
Let
D
D
i
t
=
∂
∂
t
+
λ
i
(
0
)
∂
∂
x
Obviously,
D
D
i
t
=
d
d
i
t
+
λ
i
(
0
)
-
λ
i
(
u
)
∂
∂
x
where
d
d
i
t
=
∂
∂
t
+
λ
i
(
u
)
∂
∂
x
. Thus, noting system (1)
D
u
i
D
i
t
=
λ
i
(
0
)
-
λ
i
(
u
)
∂
u
i
∂
x
Using the Hadamard's Lemma, we can obtain
D
u
i
D
i
t
=
-
∑
j
≠
i
{
Λ
i
j
(
u
)
u
j
w
i
}
where
Λ
i
j
(
u
)
=
∫
0
1
∂
λ
i
(
s
u
1
,
…
,
s
u
i
-
1
,
u
i
,
s
u
i
+
1
,
…
,
s
u
n
)
∂
u
j
d
s
.
For any fixed point (t, x) ∈ D, Passing through (t, x), we draw down the characteristic x = xr(t), which intersect with the characteristic boundary in the point
(
x
r
-
1
(
α
)
,
α
)
. Then α
=
x
-
λ
i
(
0
)
(
t
-
x
r
-
1
(
α
)
)
(where r = 1, when i = m + 1, ..., n or r = n, when i = 1, ..., m).
Then, it follows from Equation (51) that
u
i
(
t
,
x
)
=
u
i
t
,
α
+
λ
i
0
t
-
x
r
-
1
α
=
u
i
(
x
r
-
1
(
α
)
,
α
)
-
∫
x
r
-
1
(
α
)
t
∑
j
≠
i
Λ
i
j
(
u
)
u
j
w
i
(
s
,
α
+
λ
i
(
0
)
(
s
-
x
r
-
1
(
α
)
)
)
d
s
Noting (46) and (47), we can get
∫
x
r
-
1
(
α
)
t
∑
j
≠
i
Λ
i
j
(
u
)
u
j
w
i
d
s
≤
C
W
∞
(
∞
)
U
̄
1
(
∞
)
≤
C
M
N
1
e
C
N
2
This implies that the integral
∫
x
r
-
1
(
α
)
t
∑
j
≠
i
Λ
i
j
(
u
)
u
j
w
i
(
s
,
α
+
λ
i
(
0
)
(
s
-
x
r
-
1
(
α
)
)
)
d
s
converges uniformly for α ∈ R. Then, there exists a unique function Φi(α) such that,
u
i
t
,
x
→
Φ
i
α
,
t
→
+
∞
Moreover, noting (13) and Equations (52), (53), we have
|
Φ
i
α
|
≤
C
(
1
+
M
N
1
)
e
C
N
2
In what follows, we will study the regularity of Φ(α). Noting Equation (52), we can get Φ(α) ∈ C0 (R). From any fixed point A
(
t
,
x
)
=
(
t
,
a
+
λ
i
(
0
)
(
t
-
x
r
-
1
(
α
)
)
)
, we draw a characteristic C ̃j intersecting the boundary x = x1(t) or x = xn(t) at
(
x
r
-
1
(
θ
i
(
t
,
α
)
)
,
θ
i
(
t
,
α
)
)
.
Then, integrating it along the ith characteristic, we obtain
α
+
λ
i
(
0
)
t
-
x
r
-
1
(
α
)
=
θ
(
t
,
α
)
+
∫
x
r
-
1
(
θ
i
(
t
,
α
)
)
t
λ
i
u
(
τ
,
x
i
(
τ
,
θ
i
(
t
,
α
)
)
)
d
τ
Then, we can get the following Lemma
Lemma 3.1 Under the assumptions of Theorem 1.1, for the θi(t, α) defined above, there exists a unique ϑi(α), such that
lim
t
→
+
∞
θ
i
(
t
,
α
)
=
ϑ
i
(
α
)
.
Proof. Using the Hardarmad's formula, we can rewrite (55) as following
θ
i
(
t
,
α
)
=
α
+
∫
x
r
-
1
(
θ
i
(
t
,
α
)
)
t
Λ
i
j
u
j
(
τ
,
x
i
(
τ
,
θ
i
(
t
,
α
)
)
)
d
τ
+
λ
i
(
0
)
(
x
r
-
1
(
θ
i
(
t
,
α
)
)
-
x
r
-
1
(
α
)
)
where
Λ
i
j
=
-
∫
0
1
∂
λ
i
(
s
u
1
,
…
,
s
u
i
-
1
,
u
i
,
s
u
i
+
1
,
…
,
s
u
n
)
∂
u
j
d
s
.
are C1 functions. Therefore
θ
i
(
t
,
α
)
-
λ
i
(
0
)
x
r
-
1
(
θ
i
(
t
,
α
)
)
=
α
+
∫
x
r
-
1
θ
i
(
t
,
α
)
t
Λ
i
j
u
j
τ
,
x
i
(
τ
,
θ
i
(
t
,
α
)
)
d
τ
-
λ
i
(
0
)
x
r
-
1
(
α
)
where r is either 1 or n. Then, we know that when t tends to ∞, the right hand of (58) convergence absolutely. For any given α, the right hand of (57) convergence to some function with respect to α. That implies that there exists a unique function ϑ(α), such that
lim
t
→
+
∞
θ
i
(
t
,
α
)
=
ϑ
i
(
α
)
Lemma 3.2 Under the assumptions of Theorem 1.1, for any given point (xr-1(α),α) on the boundary, there exists a unique function Ψi(ϑ(α)) ∈ C0, such that
lim
t
→
+
∞
w
i
t
,
α
+
λ
i
(
0
)
(
t
-
x
r
-
1
(
α
)
)
=
Ψ
i
(
ϑ
i
(
α
)
)
.
uniformly for any α ∈ R.
Proof. Noting
w
i
(
t
,
α
+
λ
i
0
(
t
-
x
r
-
1
(
α
)
)
)
=
w
i
(
t
,
x
i
(
t
,
θ
i
(
t
,
α
)
)
)
.
In the following, we prove that there exists a unique Ψi(ϑ(α)) ∈ C0, such that
lim
t
→
+
∞
w
i
(
t
,
x
i
(
t
,
θ
i
(
t
,
α
)
)
)
)
=
Ψ
i
(
>
ϑ
i
(
α
)
)
.
Integrating (35) along the ith characteristic C ̃j gives
w
i
(
t
,
x
i
(
t
,
α
)
)
)
=
w
i
x
r
-
1
(
θ
i
(
t
,
α
)
,
θ
i
(
t
,
α
)
)
+
∫
x
r
-
1
(
θ
i
(
t
,
α
)
)
t
∑
j
≠
i
γ
i
j
(
u
)
w
j
w
i
(
τ
,
x
i
(
τ
,
θ
i
(
t
,
α
)
)
)
d
τ
where
γ
i
j
(
u
)
=
∂
λ
i
(
u
)
∂
u
j
. Noting (45) and (47), we can get
∫
x
r
-
1
θ
i
(
t
,
α
)
t
∑
j
≠
i
γ
i
j
(
u
)
w
j
w
i
(
τ
,
x
i
(
τ
,
θ
i
(
t
,
α
)
)
)
d
τ
≤
C
M
N
2
e
C
N
2
Then, as t tends to +∞, the integrals in the right-hand side of (63) convergence, i.e., there exists a unique function Ψi(ϑi(α)), such that
lim
t
→
∞
w
i
(
t
,
x
i
(
t
,
α
)
)
)
=
Ψ
i
(
ϑ
i
(
α
)
)
.
Lemma 3.3
d
Φ
i
(
α
)
d
α
=
1
-
λ
i
(
0
)
λ
i
(
x
1
-
1
(
α
)
,
α
)
Ψ
i
(
ϑ
i
(
α
)
)
d
Φ
i
(
α
)
d
α
=
1
-
λ
i
(
0
)
λ
n
(
x
n
-
1
(
α
)
,
α
)
Ψ
i
(
ϑ
i
(
α
)
)
Proof. For any fixed α ∈ R, we calculate
d
Φ
i
α
d
α
=
lim
Δ
α
→
∞
Φ
i
α
+
Δ
α
-
Φ
i
α
Δ
α
=
lim
Δ
α
→
0
lim
t
→
+
∞
u
i
t
,
α
+
Δ
α
+
λ
i
0
t
-
x
r
-
1
α
+
Δ
α
-
u
i
t
,
α
+
λ
i
0
t
-
x
r
-
1
α
Δ
α
=
lim
t
→
+
∞
lim
Δ
α
→
0
u
i
t
,
α
+
Δ
α
+
λ
i
0
t
-
x
r
-
1
α
+
Δ
α
-
u
i
t
,
α
+
λ
i
0
t
-
x
r
-
1
α
Δ
α
=
lim
t
→
+
∞
lim
Δ
α
→
0
u
i
t
,
α
+
λ
i
0
t
-
x
r
-
1
α
+
Δ
α
-
λ
i
0
x
r
-
1
α
+
Δ
α
-
x
r
-
1
x
Δ
α
.
Δ
α
Δ
α
-
u
i
t
,
α
+
λ
i
0
t
-
x
r
-
1
α
Δ
α
Thus, when i = 1, ..., s, i.e., r = 1, we can get
d
Φ
i
α
d
α
=
1
-
λ
i
0
λ
1
x
1
-
1
α
,
α
lim
t
→
+
∞
w
i
t
,
α
+
λ
i
0
t
-
x
1
-
1
α
=
1
-
λ
i
0
λ
1
x
1
-
1
α
,
α
Ψ
i
ϑ
i
α
.
Using the similar procedure, when α > 0, i.e., i = s + 1, ..., n and r = n, we can get
d
Φ
i
α
d
α
=
1
-
λ
i
0
λ
n
x
n
-
1
α
,
α
Ψ
i
ϑ
i
α
.
Noting the above lemmas, we get the conclusion of Theorem 1.2.
4 Applications
Recent years observations of the luminosity of type Ia distant supernovae point towards an accelerated expansion of the universe, which implies that the pressure p and the energy density ρ of the universe should violate the strong energy condition, i.e., ρ + 3p < 0. Here, we consider a recently proposed class of simple cosmological models based on the use of peculiar perfect fluids 19. In the simplest case, we study the model of a universe filled with the so called Chaplygin gas, which is a perfect fluid characterized by the following equation of state p
=
-
A
ρ
=
-
A
τ
, where A is a positive constant.
In Lagrange coordinate, the 1D gas dynamics equations in isentropic case can be written as
τ
t
-
u
x
=
0
u
t
+
p
(
τ
)
x
=
0
Noting (69), in isentropic case we can get the system of one dimensional Chaplygin gas model
τ
t
-
u
x
=
0
u
t
-
A
τ
x
=
0
Nothing systems (70) is linear systems, it is easy to get the eigenvalues
λ
+
=
A
,
λ
-
=
-
A
.
and left eigenvectors
l
+
=
A
,
1
,
l
-
=
-
A
,
1
Introduce the Riemann invariants
v
1
=
u
+
A
τ
,
v
2
=
u
-
A
τ
we can rewrite the system as following
∂
v
1
∂
t
-
A
∂
v
1
∂
x
=
0
∂
v
2
∂
t
-
A
∂
v
2
∂
x
=
0
Consider the Goursat problem for system (69) with following characteristic boundary conditions:
x
=
-
A
t
:
τ
=
τ
-
t
,
u
=
u
-
t
;
x
=
A
t
:
τ
=
τ
+
t
,
u
=
u
+
t
Then, the above system satisfies the assumptions of Theorems 1.1 and 1.2. More precisely, we can get the following theorems:
Theorem 4.1. The Goursat problem (69) and (75) admits a unique global C1 solution (τ, u) (t, x) on the domain D
=
t
,
x
|
t
≥
0
,
-
A
t
≤
x
≤
A
t
.
Theorem 4.2. There exists a unique piece-wised C1 vector-value function Φ(x) = (Φ1 (x), ...,Φn(x))T such that
τ
,
u
t
,
x
→
∑
i
=
1
n
Φ
i
x
-
A
t
e
i
,
t
→
+
∞
uniformly as t tends to infinity, where ei = (0, ..., 0, 1i, 0, ..., 0)T.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.