Global existence and asymptotic behavior of classical solutions to Goursat problem for diagonalizable quasilinear hyperbolic system

Boundary Value Problems20122012:36

DOI: 10.1186/1687-2770-2012-36

Received: 27 October 2011

Accepted: 3 April 2012

Published: 3 April 2012

Abstract

In this article, we investigate the global existence and asymptotic behavior of classical solutions to Goursat problem for diagonalizable quasilinear hyperbolic system. Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C1 solutions with the bounded L1 ∩ L norm of the boundary data as well as their derivatives. Based on the existence result, we can prove that when t tends to in nity, the solutions approach a combination of piece-wised C1 traveling wave solutions. As the important example, we apply the results to the chaplygin gas system.

Mathematics Subject Classi cation (2000): 35B40; 35L50; 35Q72.

Keywords

Goursat problem global classical solutions linearly degenerate asymptotic behavior traveling wave solutions.

1 Introduction and main results

For the general first order quasilinear hyperbolic systems,
u t + A ( u ) u x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equa_HTML.gif

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 [3, 4], Li and Peng [5, 6], and Zhou [7]. The asymptotic behavior has been obtained by Kong and Yang [8], Dai and Kong [9, 10]. 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 [16, 17]. 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ1_HTML.gif
(1)
where u = (u1, ..., u n ) T is unknown vector-valued function of (t, x). λ i (u) is given by C2 vector-valued function of u and is linearly degenerate, i.e.,
λ i ( u ) u i 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ2_HTML.gif
(2)
The system (1) is strictly hyperbolic, i.e.,
λ 1 ( u ) < < λ m ( u ) < 0 < λ m + 1 ( u ) < < λ n ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ3_HTML.gif
(3)
Suppose that there exists a positive constant δ such that
λ i + 1 ( u ) - λ i ( v ) δ , i = 1 , , n - 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ4_HTML.gif
(4)
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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ5_HTML.gif
(5)
and
x = x n ( t ) : u = ψ ( t ) ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ6_HTML.gif
(6)
where x = x1(t) and x = x n (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ7_HTML.gif
(7)
and
d x n ( t ) d t = λ n ψ ( t ) x n ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ8_HTML.gif
(8)
moreover,
l 1 ( ϕ ( t ) ) ϕ ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ9_HTML.gif
(9)
and
l n ( ψ ( t ) ) ψ ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ10_HTML.gif
(10)
where ϕ (t) = (ϕ1(t), ..., ϕ n (t)) T and ψ(t) = (ψ1(t), ..., ψ n (t)) T are any given C1 vector functions satisfying the conditions of C1 compatibility at the origin (0, 0):
ϕ ( 0 ) = ψ ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ11_HTML.gif
(11)
and
λ n ( ϕ ( 0 ) ) ϕ ( 0 ) - λ 1 ( ψ ( 0 ) ) ψ ( 0 ) + A ( ϕ ( 0 ) ) ( ψ ( 0 ) - ϕ ( 0 ) ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ12_HTML.gif
(12)

l i (u) be a left eigenvector corresponding to λ i (u) and A(u) = diag{λ1(u), ..., λ n (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 < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ13_HTML.gif
(13)
0 + | ϕ ( t ) | + | ψ ( t ) | d t N 1 < , 0 + | ϕ ( t ) | + | ψ ( t ) | d t N 2 < http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ14_HTML.gif
(14)

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) ≤ × ≤ x n (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), ..., Φ n (x)) T such that
u ( t , x ) i = 1 n Φ i ( x - λ i ( 0 ) t ) e i , t + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ15_HTML.gif
(15)

uniformly as t tends to infinity, where e i = (0, ..., 0, 1 i , 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 C1solutions

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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq1_HTML.gif is bounded. For any fixed T ≥ 0, we denote D T = {(t, x)| 0 ≤ t ≤ T, x1(t) ≤ x ≤ x n (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ16_HTML.gif
(16)
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ17_HTML.gif
(17)
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ18_HTML.gif
(18)
W ( T ) = sup ( t , x ) D T | w ( t , x ) , U ( T ) = sup ( t , x ) D T | u ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ19_HTML.gif
(19)

where C ̃ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq2_HTML.gif stands for any given j th characteristic d x d t = λ j ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq3_HTML.gif, L j stands for any given radial that has the slope λ j (0) on the domain D T .

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ20_HTML.gif
(20)
U ̃ 1 ( T ) , U ̄ 1 ( T ) C N 1 e C N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ21_HTML.gif
(21)
W ( T ) C M e C N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ22_HTML.gif
(22)
U ( T ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ23_HTML.gif
(23)

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) ∈ D T , we draw the i th characteristic C ̃ i : x = x i ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq4_HTML.gif through this point and intersecting x1(t) or x n (t) at a point (t*, x1(t*)) or (t*, x n (t*)). Noting system (1), u i (t, x) is a constant along the i th characteristic, then we have u i (t, x) = ϕ i (t*) or u i (t, x) = ψ i (t*). Then
| u i ( t , x ) | sup t R + | ϕ i ( t * ) | + | ψ ( t * ) | M 0 C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ24_HTML.gif
(24)

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ25_HTML.gif
(25)
We rewrite (25) as
d ( w i ( t , x ) ( d x - λ i ( u ) d t ) ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equb_HTML.gif
Multiplying the system above by sign(w i ), we have
d ( | w i ( t , x ) | ( d x - λ i ( u ) d t ) ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ26_HTML.gif
(26)
There are only the following cases(as shown in Figure 1):
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Fig1_HTML.jpg
Figure 1

Cases in estimating W ̃ 1 ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq5_HTML.gif.

Case 1. For any fixed t0R+, let C ̃ j : x = x j ( t ) , j > 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq6_HTML.gif stands for any given j th 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 i th characteristic C ̃ i : x = x i ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq4_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ27_HTML.gif
(27)
Along the 1th characteristic, d x d t = λ 1 ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq7_HTML.gif, then d t d x = 1 λ 1 ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq8_HTML.gif
u i ( t , x 1 ( t ) ) x = u i ( t , x 1 ( t ) ) t t x = ϕ i ( t ) λ 1 ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equc_HTML.gif
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ28_HTML.gif
(28)
Case 2. For any fixed t0R + , passing through the point A(t0, x1(t0)), we draw C ̃ j : x = x j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq9_HTML.gif and intersecting t = T at point P. We draw the i th characteristic C ̃ i : x = x i ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq4_HTML.gif from P downward, intersecting x = x n (t) at B(t1, x n (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equd_HTML.gif
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ29_HTML.gif
(29)
If the point A is on the characteristic boundary x = x n (t), using the same method as above cases we can get the desired estimates. Then, we can obtain
W ̃ 1 ( T ) C N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ30_HTML.gif
(30)
In the similar way, substituting L j for C ̃ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq2_HTML.gif in the above cases we also can get
W ̄ 1 ( T ) C N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ31_HTML.gif
(31)
In the following, we will get the estimate W1(T) ≤ CN2. Integrating Equation (26) with respect to x from x1(t) to x n (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ32_HTML.gif
(32)
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ33_HTML.gif
(33)
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ34_HTML.gif
(34)

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ35_HTML.gif
(35)
Case 1. When i = 1, ..., m, we draw the i th characteristic intersecting x = x n (t) with the point (t0, x n (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Eque_HTML.gif
Noting w i ( t 0 , x n ( t 0 ) ) = ψ i ( t 0 ) λ n ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq10_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ36_HTML.gif
(36)
Case 2. When i = m + 1, ..., n and the i th 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equf_HTML.gif
Then, we can get
| w i ( t , x i ( t ) ) | C M e C N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equg_HTML.gif
Then, we can obtain the estimate
W T C M e C N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ37_HTML.gif
(37)

In the following we estimate U ̃ 1 ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq11_HTML.gif.

Noting system (1), we denote the multiplier H i C1, (i = 1, ..., n) such that,
( H i u i ) t + λ i ( u ) H i u i x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ38_HTML.gif
(38)
Noting (1), then
H i t + λ i ( u ) H i x = - λ i ( u ) x H i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equh_HTML.gif
Let H i (t, x1(t)) = 1 and H i (t, x n (t)) = 1 we can get
H i = e - C ̃ i λ i ( u ( τ , x ) ) x d τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equi_HTML.gif
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ39_HTML.gif
(39)
Noting Equation (38), we can rewrite it as
d ( H i | u i | ( d x - λ i ( u ) d t ) ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ40_HTML.gif
(40)

There are only the following cases like we estimate W ̃ 1 ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq5_HTML.gif:

Case 1. We can integrate (40) in the region APB which is same to the case 1 of proof of W ̃ 1 ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq5_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equj_HTML.gif
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ41_HTML.gif
(41)
Case 2. Integrating the Equation (40) in the region PAOB which is same to the case 2 of proof of W ̃ 1 ( T ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq5_HTML.gif, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equk_HTML.gif
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ42_HTML.gif
(42)
Then
U ̃ 1 ( T ) C N 1 e C N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ43_HTML.gif
(43)
In the similar way, substituting L j for C ̃ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq2_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ44_HTML.gif
(44)

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ45_HTML.gif
(45)
U ̃ 1 ( ) , U ̄ 1 ( ) , C N 1 e C N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ46_HTML.gif
(46)
W ( ) C M e C N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ47_HTML.gif
(47)
U ( ) C http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ48_HTML.gif
(48)

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 ≤ x n (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ49_HTML.gif
(49)
Obviously,
D D i t = d d i t + λ i ( 0 ) - λ i ( u ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equl_HTML.gif
where d d i t = t + λ i ( u ) x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq12_HTML.gif. Thus, noting system (1)
D u i D i t = λ i ( 0 ) - λ i ( u ) u i x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ50_HTML.gif
(50)
Using the Hadamard's Lemma, we can obtain
D u i D i t = - j i { Λ i j ( u ) u j w i } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ51_HTML.gif
(51)

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq13_HTML.gif.

For any fixed point (t, x) ∈ D, Passing through (t, x), we draw down the characteristic x = x r (t), which intersect with the characteristic boundary in the point ( x r - 1 ( α ) , α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq14_HTML.gif. Then α = x - λ i ( 0 ) ( t - x r - 1 ( α ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq15_HTML.gif (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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ52_HTML.gif
(52)
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ53_HTML.gif
(53)
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq16_HTML.gif converges uniformly for αR. Then, there exists a unique function Φ i (α) such that,
u i t , x Φ i α , t + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ54_HTML.gif
(54)
Moreover, noting (13) and Equations (52), (53), we have
| Φ i α | C ( 1 + M N 1 ) e C N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equm_HTML.gif

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 ) ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq17_HTML.gif, we draw a characteristic C ̃ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq2_HTML.gif intersecting the boundary x = x1(t) or x = x n (t) at ( x r - 1 ( θ i ( t , α ) ) , θ i ( t , α ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq18_HTML.gif.

Then, integrating it along the i th 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ55_HTML.gif
(55)

Then, we can get the following Lemma

Lemma 3.1 Under the assumptions of Theorem 1.1, for the θ i (t, α) defined above, there exists a unique ϑ i (α), such that
lim t + θ i ( t , α ) = ϑ i ( α ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ56_HTML.gif
(56)
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 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ57_HTML.gif
(57)
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equn_HTML.gif
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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ58_HTML.gif
(58)
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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ59_HTML.gif
(59)
Lemma 3.2 Under the assumptions of Theorem 1.1, for any given point ( x r - 1 ( α ) , α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq14_HTML.gif on the boundary, there exists a unique function Ψ i (ϑ(α)) ∈ C0, such that
lim t + w i t , α + λ i ( 0 ) ( t - x r - 1 ( α ) ) = Ψ i ( ϑ i ( α ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ60_HTML.gif
(60)

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 ) ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ61_HTML.gif
(61)
In the following, we prove that there exists a unique Ψ i (ϑ(α)) ∈ C0, such that
lim t + w i ( t , x i ( t , θ i ( t , α ) ) ) ) = Ψ i ( > ϑ i ( α ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ62_HTML.gif
(62)
Integrating (35) along the i th characteristic C ̃ j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq2_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ63_HTML.gif
(63)
where γ i j ( u ) = λ i ( u ) u j http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq19_HTML.gif. 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equo_HTML.gif
Then, as t tends to +, the integrals in the right-hand side of (63) convergence, i.e., there exists a unique function Ψ i (ϑ i (α)), such that
lim t w i ( t , x i ( t , α ) ) ) = Ψ i ( ϑ i ( α ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equp_HTML.gif
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 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ64_HTML.gif
(64)
d Φ i ( α ) d α = 1 - λ i ( 0 ) λ n ( x n - 1 ( α ) , α ) Ψ i ( ϑ i ( α ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ65_HTML.gif
(65)
Proof. For any fixed αR, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ66_HTML.gif
(66)
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ67_HTML.gif
(67)
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ68_HTML.gif
(68)

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq20_HTML.gif, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ69_HTML.gif
(69)
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ70_HTML.gif
(70)
Nothing systems (70) is linear systems, it is easy to get the eigenvalues
λ + = A , λ - = - A . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ71_HTML.gif
(71)
and left eigenvectors
l + = A , 1 , l - = - A , 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ72_HTML.gif
(72)
Introduce the Riemann invariants
v 1 = u + A τ , v 2 = u - A τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ73_HTML.gif
(73)
we can rewrite the system as following
v 1 t - A v 1 x = 0 v 2 t - A v 2 x = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ74_HTML.gif
(74)
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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ75_HTML.gif
(75)

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_IEq21_HTML.gif.

Theorem 4.2. There exists a unique piece-wised C1 vector-value function Φ(x) = (Φ1 (x), ...,Φ n (x)) T such that
τ , u t , x i = 1 n Φ i x - A t e i , t + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-36/MediaObjects/13661_2011_Article_148_Equ76_HTML.gif
(76)

uniformly as t tends to infinity, where e i = (0, ..., 0, 1 i , 0, ..., 0) T .

Declarations

Acknowledgements

The first author was supported by NSFC-Tianyuan Special Foundation (No. 11126058), Excellent Young Teachers Program of Shanghai and the Shanghai Leading Academic Discipline Project (No. J50101). The second author was supported by China Postdoctoral Science Foundation (No. 2011M501295) and the Fundamental Research Funds for the Central Universities (No. 2011QNZT102). The authors would like to thank Professor Zhou Yi for his guidance and encouragements.

Authors’ Affiliations

(1)
Department of Mathematics, Shanghai University
(2)
Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, School of Geosciences and Info-Physics, Central South University
(3)
School of Mathematics and Statistics, Central South University

References

  1. Bressan A: Contractive metrics for nonlinear hyperbolic system. Nat Genet 1988, 37: 409-.MathSciNet
  2. Li TT: Global classical solutions for qusilinear hyperbolic systems. In Research in Appl Math. Volume 32. Wiley/Masson, New York; 1994.
  3. Li TT, Zhou Y, Kong DX: Weak linear degeneracy and global classical solutions for general qusilinear hyperbolic systems. Comm Partial Diff Equ 1994, 19: 1263-1317.MATHMathSciNetView Article
  4. Li TT, Zhou Y, Kong DX: Global classical solutions for general qusilinear hyperbolic systems with decay initial data. Nonl Anal 1997, 28: 1299-1322.MATHMathSciNetView Article
  5. Li TT, Peng YJ: Global C1solution to the initial-boundary value problem for dioagal hyperbolic systems with linearly degenerate characteristics. J Partial Diff Equ 2003, 16: 8-17.MATHMathSciNet
  6. Li TT, Peng YJ: The mixed initial-boundary value problem for reducible qusilinear hyperbolic systems with linearly degenerate characteristics. Nonlinear Anal 2003, 52: 573-583.MATHMathSciNetView Article
  7. Zhou Y: Global classical solutions to qusilinear hyperbolic systems with weak linear degeneracy. Chin Ann Math Ser B 2004, 25: 37-56.View Article
  8. Kong DX, Yang T: Asymptotic behavior of global classical solutions of qusilinear hy-perbolic systems. Comm Part Diff Equ 2003, 28: 1203-1220.MATHMathSciNetView Article
  9. Dai WR, Kong DX: Asymptotic behavior of global classical solutions of general qusilinear hyperbolic systems with weakly linearly degeneracy. Chinese Ann Math Ser B 2006, 27(3):263-286.MATHMathSciNetView Article
  10. Dai WR, Kong DX: Global existence and asymptotic behavior of classical solutions of qusilinear hyperbolic systems with linearly degenerate characteristic fields. J Diff Equ 2007, 251(3):127-165.MathSciNetView Article
  11. Liu JL, Zhou Y: Asymptotic behavior of global classical solutions of diagonalizable qusilinear hyperbolic systems. Math Meth Appl Sci 2007, 30: 479-500.MATHView Article
  12. Li TT, Wang LB: Global classical solutions to a kind of mixed initial-boundary value problem for qusilinear hyperbolic systems. Discrete Cont Dyn Syst 2005, 12: 59-78.MATHView Article
  13. Zhang JG: Asymptotic behavior of global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. J Partial Diff Equ 2007, 20: 114-130.MATH
  14. Liu JL: The mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant. Nonlinear Anal 2010, 72(1):533-545.MATHMathSciNetView Article
  15. Zhou Y: The Goursat problem for reducible quasilinear hyperbolic systems. Chinese Ann Math Ser A 1992, 13: 437-441.MATHMathSciNet
  16. Liu CM, Liu JL: Goursat problem for the first order quasilinear hyperbolic systems (Sumitted to Acta Mathematica Sinica, English Series).
  17. Liu JL, Pan KJ: Asymptotic behavior of global classical solutions to Goursat problem of quasilinear hyperbolic systems. (Accepted by J Math Anal Appl).
  18. Duan YZ: Asymptotic behavior of global classical solutions of reducible qusilinear hy-perbolic systems with characteristic boundaries. J Math Anal Appl 2009, 351: 186-205.MATHMathSciNetView Article
  19. Gorini V, Kamenshchik A, Moschella U, Pasquier V: Can the Chaplygin gas be a plausible model for dark energy? Phys. Rev D 2003, 67: 063509.View Article

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© Liu and Pan; licensee Springer. 2012

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