Open Access Research

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

Jianli Liu1 and Kejia Pan23*

Author Affiliations

1 Department of Mathematics, Shanghai University, Shanghai 200444, China

2 Key Laboratory of Metallogenic Prediction of Nonferrous Metals, Ministry of Education, School of Geosciences and Info-Physics, Central South University, Changsha 410083, China

3 School of Mathematics and Statistics, Central South University, Changsha 410075, China

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Boundary Value Problems 2012, 2012:36  doi:10.1186/1687-2770-2012-36


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/36


Received:27 October 2011
Accepted:3 April 2012
Published:3 April 2012

© 2012 Liu and Pan; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 ∩ Lnorm 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,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M1">View MathML</a>

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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M2">View MathML</a>

(1)

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.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M3">View MathML</a>

(2)

The system (1) is strictly hyperbolic, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M4">View MathML</a>

(3)

Suppose that there exists a positive constant δ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M5">View MathML</a>

(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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M6">View MathML</a>

(5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M7">View MathML</a>

(6)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M8">View MathML</a>

(7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M9">View MathML</a>

(8)

moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M10">View MathML</a>

(9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M11">View MathML</a>

(10)

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):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M12">View MathML</a>

(11)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M13">View MathML</a>

(12)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M14">View MathML</a>

(13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M15">View MathML</a>

(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) ≤ × ≤ 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M16">View MathML</a>

(15)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M17">View MathML</a> is bounded. For any fixed T ≥ 0, we denote DT = {(t, x)|0 ≤ t ≤ T, x1(t) ≤ x ≤ xn(t)} and introduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M18">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M19">View MathML</a>

(17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M20">View MathML</a>

(18)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M21">View MathML</a>

(19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M22">View MathML</a> stands for any given jth characteristic <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M23">View MathML</a>, 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M24">View MathML</a>

(20)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M25">View MathML</a>

(21)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M26">View MathML</a>

(22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M27">View MathML</a>

(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) ∈ DT , we draw the ith characteristic <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M28">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M29">View MathML</a>

(24)

Then, we obtain the estimate (23).

Differentiating (1) with respect to x we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M30">View MathML</a>

(25)

We rewrite (25) as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M31">View MathML</a>

Multiplying the system above by sign(wi), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M32">View MathML</a>

(26)

There are only the following cases(as shown in Figure 1):

thumbnailFigure 1. Cases in estimating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M60">View MathML</a>.

Case 1. For any fixed t0 R+, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M33">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M28">View MathML</a> 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M34">View MathML</a>

(27)

Along the 1th characteristic, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M35">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M36">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M37">View MathML</a>

Noting (4), (13), and (23), we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M38">View MathML</a>

(28)

Case 2. For any fixed t0 R+, passing through the point A(t0, x1(t0)), we draw <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M39">View MathML</a> and intersecting t = T at point P. We draw the ith characteristic <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M28">View MathML</a> from P downward, intersecting x = xn(t) at B(t1, xn(t1)). Then, we integrate (26) in the region PAOB to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M40">View MathML</a>

Noting i = 1, ..., m i < j, Equations (4), (13) and using the above procedure, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M41">View MathML</a>

(29)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M42">View MathML</a>

(30)

In the similar way, substituting Lj for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M22">View MathML</a> in the above cases we also can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M43">View MathML</a>

(31)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M44">View MathML</a>

(32)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M45">View MathML</a>

(33)

Using the same procedure as (28), (29), we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M46">View MathML</a>

(34)

Then, we can get the desired estimate.

Rewriting the Equation (25), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M47">View MathML</a>

(35)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M48">View MathML</a>

Noting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M49">View MathML</a> and the estimate (30), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M50">View MathML</a>

(36)

Case 2. When i = m + 1, ..., n and the ith characteristic intersecting x = x1(t) with a point (t0, x1(t0)), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M51">View MathML</a>

Then, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M52">View MathML</a>

Then, we can obtain the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M53">View MathML</a>

(37)

In the following we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M54">View MathML</a>.

Noting system (1), we denote the multiplier Hi C1, (i = 1, ..., n) such that,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M55">View MathML</a>

(38)

Noting (1), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M56">View MathML</a>

Let Hi(t, x1(t)) = 1 and Hi(t, xn(t)) = 1 we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M57">View MathML</a>

We note the system (1) is linearly degenerate, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M58">View MathML</a>

(39)

Noting Equation (38), we can rewrite it as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M59">View MathML</a>

(40)

There are only the following cases like we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M60">View MathML</a>:

Case 1. We can integrate (40) in the region APB which is same to the case 1 of proof of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M60">View MathML</a> to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M61">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M62">View MathML</a>

(41)

Case 2. Integrating the Equation (40) in the region PAOB which is same to the case 2 of proof of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M60">View MathML</a>, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M63">View MathML</a>

Using the above procedures, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M64">View MathML</a>

(42)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M65">View MathML</a>

(43)

In the similar way, substituting Lj for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M22">View MathML</a> we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M66">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M67">View MathML</a>

(45)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M68">View MathML</a>

(46)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M69">View MathML</a>

(47)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M70">View MathML</a>

(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 ≤ 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M71">View MathML</a>

(49)

Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M72">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M73">View MathML</a>. Thus, noting system (1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M74">View MathML</a>

(50)

Using the Hadamard's Lemma, we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M75">View MathML</a>

(51)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M76">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M77">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M78">View MathML</a> (where r = 1, when i = m + 1, ..., n or r = n, when i = 1, ..., m).

Then, it follows from Equation (51) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M79">View MathML</a>

(52)

Noting (46) and (47), we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M80">View MathML</a>

(53)

This implies that the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M81">View MathML</a> converges uniformly for α R. Then, there exists a unique function Φi(α) such that,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M82">View MathML</a>

(54)

Moreover, noting (13) and Equations (52), (53), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M83">View MathML</a>

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 onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M84">View MathML</a>, we draw a characteristic <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M22">View MathML</a> intersecting the boundary x = x1(t) or x = xn(t) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M85">View MathML</a>.

Then, integrating it along the ith characteristic, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M86">View MathML</a>

(55)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M87">View MathML</a>

(56)

Proof. Using the Hardarmad's formula, we can rewrite (55) as following

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M88">View MathML</a>

(57)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M89">View MathML</a>

are C1 functions. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M90">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M91">View MathML</a>

(59)

Lemma 3.2 Under the assumptions of Theorem 1.1, for any given point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M77">View MathML</a> on the boundary, there exists a unique function Ψi(ϑ(α)) ∈ C0, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M92">View MathML</a>

(60)

uniformly for any α ∈ R.

Proof. Noting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M93">View MathML</a>

(61)

In the following, we prove that there exists a unique Ψi(ϑ(α)) ∈ C0, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M94">View MathML</a>

(62)

Integrating (35) along the ith characteristic <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M22">View MathML</a> gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M95">View MathML</a>

(63)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M96">View MathML</a>. Noting (45) and (47), we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M97">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M98">View MathML</a>

Lemma 3.3

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M99">View MathML</a>

(64)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M100">View MathML</a>

(65)

Proof. For any fixed α R, we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M101">View MathML</a>

(66)

Thus, when i = 1, ..., s, i.e., r = 1, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M102">View MathML</a>

(67)

Using the similar procedure, when α > 0, i.e., i = s + 1, ..., n and r = n, we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M103">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M104">View MathML</a>, where A is a positive constant.

In Lagrange coordinate, the 1D gas dynamics equations in isentropic case can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M105">View MathML</a>

(69)

Noting (69), in isentropic case we can get the system of one dimensional Chaplygin gas model

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M106">View MathML</a>

(70)

Nothing systems (70) is linear systems, it is easy to get the eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M107">View MathML</a>

(71)

and left eigenvectors

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M108">View MathML</a>

(72)

Introduce the Riemann invariants

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M109">View MathML</a>

(73)

we can rewrite the system as following

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M110">View MathML</a>

(74)

Consider the Goursat problem for system (69) with following characteristic boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M111">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M112">View MathML</a>.

Theorem 4.2. There exists a unique piece-wised C1 vector-value function Φ(x) = (Φ1 (x), ...,Φn(x))T such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/36/mathml/M113">View MathML</a>

(76)

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.

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.

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