A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems

Nonlinear Analysis 73 (2010) 600–613

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems Zhi-Qiang Shao ∗ Department of Mathematics, Fuzhou University, Fuzhou 350002, China

article

info

Article history: Received 30 January 2010 Accepted 24 March 2010 MSC: 35L45 35L60 35Q40 Keywords: Asymptotic behavior Quasilinear hyperbolic system Linear degeneracy Global classical solution Traveling wave

abstract This paper is concerned with the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields. On the basis of the existence result for the global classical solution, we prove that when t tends to the infinity, the solution approaches a combination of C 1 traveling wave solutions, provided that the C 1 norm and the BV norm of the initial data are bounded but possibly large. In contrast to former results obtained by Liu and Zhou [J. Liu, Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Methods Appl. Sci. 30 (2007) 479–500], ours do not require their assumption that the system is rich in the sense of Serre. Applications include that to the one-dimensional Born–Infeld system arising in string theory and high energy physics. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction and main result Consider the following first-order diagonalizable quasilinear hyperbolic system:

∂ ui ∂ ui + λi (u) = 0 (i = 1, . . . , n), ∂t ∂x

(1.1)

where u = (u1 , . . . , un )T is the unknown vector-valued function of (t , x), λi (u) (i = 1, . . . , n) are assumed to be the C 2 vector-valued functions of u and are linearly degenerate, i.e.,

∂λi (u) ≡ 0 (i = 1, . . . , n) ∂ ui

(1.2)

for any given u on the domain under consideration. It is assumed that system (1.1) is strictly hyperbolic on the domain under consideration, i.e., the eigenvalues satisfy

λ1 (u) < λ2 (u) < · · · < λn (u).

(1.3)

We assume that there exists a positive constant δ such that

λi+1 (u) − λi (v) ≥ δ (i = 1, . . . , n − 1).

∗

Tel.: +86 0591 83852790. E-mail address: [email protected].

0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.03.029

(1.4)

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

601

Consider the Cauchy problem for system (1.1) with the initial data t = 0 : u(0, x) = f (x),

(1.5)

where f (·) = (f1 (·), . . . , fn (·))T ∈ C 1 with both bounded C 1 norm and bounded total variation, such that

kf kC 1 ≤ M

(1.6)

for some M > 0 bounded but possibly large, and also such that

Z

+∞

|f 0 (x)|dx ≤ N

(1.7)

−∞

for some N > 0 bounded but possibly large. The global existence of classical solutions is well-known (see [1]). Our goal in this paper is to describe the asymptotic behavior of global classical solutions to the Cauchy problem (1.1) and (1.5). Our main results can be summarized as follows: Theorem 1.1 (Asymptotic Behavior). Under the assumptions mentioned above, if 4

+∞

Z

|f (x)|dx < +∞,

N1 =

(1.8)

−∞

then there exists a unique C 1 vector-valued function φ(x) = (φ1 (x), . . . , φn (x))T such that u( t , x ) →

n X

φi (x − λi (0)t )ei as t → +∞,

(1.9)

i=1

where (i)

ei = (0, . . . , 0, 1 , 0, . . . , 0)T . Moreover, φi (x) (i = 1, . . . , n) are globally Lipschitz continuous; more precisely, there exists a positive constant κ1 depending only on δ , M and independent of N , x1 , x2 such that

|φi (x1 ) − φi (x2 )| ≤ κ1 eκ1 N (1 + N )|x1 − x2 |,

∀x1 , x2 ∈ R.

(1.10)

Furthermore, if the derivatives of the initial data, i.e., f (x), are globally ρ -Hölder continuous, where 0 < ρ ≤ 1, that is, there exists a positive constant ς such that 0

|f 0 (x1 ) − f 0 (x2 )| ≤ ς |x1 − x2 |ρ ,

∀x1 , x2 ∈ R,

(1.11)

then φ (x) is also globally ρ -Hölder continuous and satisfies that 0

|φ 0 (x1 ) − φ 0 (x2 )| ≤ eκ2 N {ς (1 + κ2 Neκ2 N )ρ |x1 − x2 |ρ + κ2 (N + 1)2 eκ2 N |x1 − x2 |},

(1.12)

where κ2 is a positive constant depending only on δ , M and independent of N , ς , x1 , x2 . Remark 1.1. The system is assumed to be linearly degenerate and strictly hyperbolic, namely

∂u1 λ1 (u) = · · · = ∂un λn (u),

λ1 (u) < λ2 (u) < · · · < λn (u).

There are richness assumptions on the system and no smallness assumptions on the initial condition. Liu and Zhou [2] recently proved the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems. Comparing with Liu and Zhou’s result, in Theorem 1.1 we remove the richness assumptions on the system. Remark 1.2. The definition of rich system is given by Serre, i.e., we define aij (u) =

∂i λj (u) . λj (u) − λi (u)

(1.13)

The system (1.1) is rich if aij (u) satisfies the following property:

∂k aij (u) = ∂i ajk (u),

(1.14)

where ∂i = ∂∂u and {i, j, k} ∈ {1, . . . , n} are different from each other. In particular, a system of conservation laws which i can be put in diagonal form must be rich. See [3].

602

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

Remark 1.3. Suppose that system (1.1) is non-strictly hyperbolic but each characteristic has a constant multiplicity, say, on the domain under consideration,

λ1 (u) < · · · < λm (u) < 0 < λm (u) < · · · < λp+1 (u) ≡ · · · ≡ λn (u) (m ≤ p ≤ n).

(1.15)

Then the conclusion of Theorem 1.1 still holds (cf. [4,5]). The global existence of classical solutions of quasilinear hyperbolic systems has been established for linearly degenerate characteristics or weakly linearly degenerate characteristics with various smallness assumptions on the initial data by Bressan [6], Kong [5,7], Li et al. [1,8,9], Zhou [10], etc. On the other hand, for the asymptotic behavior of the classical solutions of the quasilinear hyperbolic systems, many results have also been obtained in the literature (for instance, see [4,5,11–16] and the references therein). In particular, Kong and Yang [11] were the first to study the asymptotic behavior of the classical solutions of the quasilinear hyperbolic systems with some decay initial data. However, it is well-known that the BV space is a suitable framework for one-dimensional quasilinear hyperbolic systems (see [17]); the result in [6] suggests that one may achieve global smoothness even if the C 1 norm of the initial data is large. So the following question arises naturally: can we obtain the global existence and the asymptotic behavior of the classical solutions of linearly degenerate quasilinear hyperbolic systems, provided that the BV norm of the initial data is suitably small? Here, it is important to mention that for the Cauchy problem case, this problem was solved by Bressan [6], Dai and Kong [5], Zhou [10]. The mixed initial–boundary value problem case was also studied in [15,18,19]. On the other hand, by the study of Amadori and Shen [20,21], Glimm and Lax [22], Nishida and Smoller [23], Serre [3], etc., we know that proving global existence for solutions to quasilinear hyperbolic systems with large BV data is a more difficult, still largely open problem. Therefore, it is natural to consider the following problem: can we obtain the global existence and the asymptotic behavior of the classical solutions of linearly degenerate quasilinear hyperbolic systems, provided that the BV norm of the initial data is bounded but possibly large? However, this problem is quite difficult; our work may provide a simpler approach for solving this problem, and in this connection let us also mention Li and Peng’s work [24,25] and Liu and Zhou’s work [2] on quasilinear hyperbolic systems in diagonal form. The rest of the paper is organized as follows. Section 2 is devoted to establishing some new estimates, which will play an important role in the proof of Theorem 1.1. Using these estimates, we prove Theorem 1.1 in Section 3. It is easy to see that Theorem 1.1 can be applied to the physical model discussed in Liu and Zhou [2]; therefore we do not give the details in this paper. However, of particular interest are both the one-dimensional Born–Infeld system and the system of motion of relativistic closed strings in the Minkowski space–time R1+n ; as some applications of Theorem 1.1, the asymptotic behavior of the classical solutions to the Cauchy problems with large BV data for these systems is presented in Section 4. 2. Uniform estimates In this section, we shall establish some new uniform estimates which play a key role in the proof of Theorem 1.1. For any fixed T > 0, we introduce

∂ ui ( t , x ) (i = 1, . . . , n), ∂x U∞ (T ) = sup sup |u(t , x)|, wi (t , x) =

(2.1) (2.2)

0≤t ≤T x∈R

W∞ (T ) = sup sup |w(t , x)|,

(2.3)

0≤t ≤T x∈R

U1 (T ) = sup

+∞

Z

0≤t ≤T

W1 (T ) = sup 0≤t ≤T

|u(t , x)|dx,

(2.4)

−∞

Z

+∞

|w(t , x)|dx,

(2.5)

−∞

e U1 (T ) = max max sup i=1,...,n j6=i

Z

Cj

e1 (T ) = max max sup W i=1,...,n j6=i

i=1,...,n j6=i

Z

Lj

Lj

|wi (t , x)|dt ,

(2.7)

|ui (t , x)|dt ,

(2.8)

Cj

W 1 (T ) = max max sup i=1,...,n j6=i

(2.6)

Z

Cj

U 1 (T ) = max max sup

|ui (t , x)|dt , Cj

Lj

Z

|wi (t , x)|dt , Lj

(2.9)

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

603

where | · | stands for the Euclidean norm in Rn , w = (w1 , . . . , wn )T in which wi is defined by (2.1), Cj stands for any given jth characteristic on the domain [0, T ] × R, while Lj stands for any given ray with the slope λj (0) on the domain [0, T ] × R. Lemma 2.1. Under the assumptions of Theorem 1.1, there exists a positive constant K1 depending only on δ and M such that the following estimates hold:

e1 (T ), W 1 (T ), W1 (T ) ≤ K1 N , W

(2.10)

e U1 (T ), U 1 (T ) ≤ K1 N1 e

(2.11)

W∞ (T ) ≤ Me

K1 N

,

K1 N

(2.12)

and U∞ (T ) ≤ K1 .

(2.13)

The proof can be found in [2]. Lemma 2.2. Under the assumptions of Theorem 1.1, there exists a positive constant K2 depending only on δ and M such that the following holds: U1 (T ) ≤ N1 eK2 N .

(2.14)

R +∞

Proof. To estimate −∞ |ui (t , x)|dx, we need only to estimate

Z

a

|ui (t , x)|dx

(2.15)

−a

for any given a > 0 and then let a → +∞. For i = 1, . . . , n, for any given t with 0 ≤ t ≤ T , passing through point A(t , a) (resp. B(t , −a)), we draw the ith backward characteristic which intersects the x-axis at a point D(0, xD ) (resp. C (0, xC )). It follows from (1.1) that

∂(Hi ui ) ∂(λi (u)Hi ui ) + =0 ∂t ∂x

(2.16)

with

 Z 

Hi (t , x) = exp −

Ci

  ∂λi (u) dt , ∂x

(2.17)

or equivalently, d(Hi ui (dx − λi (u)dt )) = 0.

(2.18)

We rewrite (2.18) as d(Hi |ui |(dx − λi (u)dt )) = 0.

(2.19)

By (2.19), using Stokes’s formula on the domain ABCD, we have

Z

Hi (t , x)|ui (t , x)|dx ≤ BA

Z

xD

Hi (0, x)|ui (0, x)|dx.

(2.20)

xC

Noting that system (1.1) is linearly degenerate, we obtain

X ∂λi (u) ∂λi (u) X ∂λi (u) ∂ ul = = wl . ∂x ∂ ul ∂ x ∂ ul l6=i l6=i

(2.21)

Then, noting (2.13) and (2.10) we have

Z   XZ ∂λi (u) e1 (T ) ≤ cN , dt ≤ c |wl |dt ≤ c W ∂x Ci Ci l6=i

(2.22)

where here and henceforth, c will denote a positive constant depending only on δ , M and independent of N1 , N; the meaning of c may change from line to line. Substituting (2.22) into (2.17) gives e−cN ≤ Hi (t , x) ≤ ecN .

(2.23)

604

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

Therefore, it follows from (2.20) and (2.23) that

Z

a

|ui (t , x)|dx ≤ ecN · ecN

Z

+∞

|f (x)|dx ≤ N1 ecN .

(2.24)

−∞

−a

Letting a → +∞, we immediately arrive at our conclusion. The proof of Lemma 2.2 is finished.



Combining Lemmas 2.1 and 2.2 gives: Lemma 2.3. Under the assumptions of Theorem 1.1, there exists a positive constant K3 depending only on δ and M such that U1 (∞), e U1 (∞), U 1 (∞) ≤ K3 N1 eK3 N ,

(2.25)

e1 (∞), W 1 (∞) ≤ K3 N , W1 (∞), W

(2.26)

U∞ (∞) ≤ K3 ,

(2.27)

W∞ (∞) ≤ Me

K3 N

,

(2.28)

where U1 (∞) =

Z

+∞

|u(t , x)|dx,

sup 0≤t ≤+∞

(2.29)

−∞

etc. Lemma 2.4. Under the assumptions of Theorem 1.1, there exists a positive constant K4 independent of N , t , α and β such that for any t ∈ R+ and arbitrary α, β ∈ R, it holds that

|u(t , xi (t , α)) − u(t , xi (t , β))| ≤ MeK4 N |α − β|;

(2.30)

moreover, for any given C function g (u), 1

|g (u(t , xi (t , α))) − g (u(t , xi (t , β)))| ≤ K4 MeK4 N |α − β|,

(2.31)

where x = xi (t , ·) stands for the ith characteristic passing through the point (0, ·). Proof. The proof is similar to the proof of Lemma 4.3 in the paper by Dai and Kong [5], so we skip the details and refer the reader to the above mentioned paper.  For any fixed T ≥ 0 and for arbitrary α, β ∈ R, we introduce

e Uαβ (T ) = max max i=1,...,n j6=i

eαβ (T ) = max max W i=1,...,n j6=i

T

Z

|ui (t , xj (t , α)) − ui (t , xj (t , β))|dt ,

(2.32)

0 T

Z

|wi (t , xj (t , α)) − wi (t , xj (t , β))|dt ,

(2.33)

0

where x = xj (t , ·), as before, stands for any given jth characteristic passing through the point (0, ·). Lemma 2.5. Under the assumptions of Theorem 1.1, there exists a positive constant K5 depending only on δ , M and independent of N , T , α , β such that

e Uαβ (T ) ≤ K5 NeK5 N |α − β|

(2.34)

eαβ (T ) ≤ K5 (N + 1)eK5 N |α − β|. W

(2.35)

and

Proof. The proof of (2.34) can be found in [2]. We next prove (2.35). For arbitrary α, β ∈ R and for j ∈ {1, . . . , n}, let Cj (α) and Cj (β) be the jth characteristics passing through the points P1 : (0, α) and P2 : (0, β), respectively. For the sake of simplicity, we assume that α < β . We denote by P4 : (T , xj (T , α)) (respectively P3 : (T , xj (T , β))) the point of intersection of Cj (α) (resp. Cj (β)) with the straight line t = T . Differentiating system (1.1) with respect to x, we get

∂wi ∂(λi (u)wi ) + = 0, ∂t ∂x

(2.36)

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

605

equivalently, d(wi (t , x)(dx − λi (u)dt )) = 0.

(2.37)

We rewrite (2.37) as d[ξ (t )wi (dx − λi (u)dt )] = 0,

a.e.,

(2.38)

where

ξ (t ) = sgn[(wi (t , xj (t , α)) − wi (t , xj (t , β)))(λj (u)(t , xj (t , β)) − λi (u)(t , xj (t , α)))]. By (2.38), using the Green formula on the domain P1 P2 P3 P4 bounded by the curves Cj (α), Cj (β), the x-axis and the straight line t = T , we have (cf. [5]) β

Z

ξ (0)wi (0, x)dx +

0 = α

T

Z

ξ (t )[wi (λj (u) − λi (u))](t , xj (t , β))dt 0

β

Z − α

ξ (T )wi (T , xj (T , γ ))

∂ xj (T , γ ) dγ − ∂γ

T

Z

ξ (t )[wi (λj (u) − λi (u))](t , xj (t , α))dt , 0

i.e., T

Z

|(wi (t , xj (t , α)) − wi (t , xj (t , β)))(λj (u)(t , xj (t , β)) − λi (u)(t , xj (t , α)))|dt 0 T

Z

ξ (t )wi (t , xj (t , β))[λi (u)(t , xj (t , α)) − λi (u)(t , xj (t , β))]dt

= 0

T

Z

ξ (t )wi (t , xj (t , α))[λj (u)(t , xj (t , α)) − λj (u)(t , xj (t , β))]dt

− 0

β

Z + α

  ∂ xj (T , γ ) ξ (0)wi (0, γ ) − ξ (T )wi (T , xj (T , γ )) dγ . ∂γ

(2.39)

eαβ (T ), j 6= i, we thus have from (1.4) that In the definition of W |λj (u)(t , xj (t , β)) − λi (u)(t , xj (t , α))| ≥ δ.

(2.40)

Therefore, by (71) in [2], it follows from Lemmas 2.3 and 2.4 that T

Z

|wi (t , xj (t , α)) − wi (t , xj (t , β))|dt ≤ 0

1

δ



[2e W∞ (∞) + 2K4 Me cN

≤ K5 (N + 1)eK5 N |α − β|,

K4 N

 e W1 (∞)]|α − β| (2.41)

where here and henceforth, as before, c will denote a positive constant depending only on δ , M and independent of N , T , α , β , and the meaning of c may change from line to line. This proves (2.35). The proof of Lemma 2.5 is finished.  For any fixed T ≥ 0 and for arbitrary α, β ∈ R, we introduce β

U α (T ) = max max

T

Z

i=1,...,n j6=i

|ui (t , α + λj (0)t ) − ui (t , β + λj (0)t )dt

(2.42)

0

and β

W α (T ) = max max i=1,...,n j6=i

T

Z

|wi (t , α + λj (0)t ) − wi (t , β + λj (0)t )dt .

(2.43)

0

Similarly, we can prove the following lemma. Lemma 2.6. Under the assumptions of Theorem 1.1, there exists a positive constant K6 depending only on δ , M and independent of N , T , α , β such that β

U α (T ) ≤ K6 NeK6 N |α − β|

(2.44)

and β

W α (T ) ≤ K6 (N + 1)eK6 N |α − β|.

(2.45)

606

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

Combining Lemmas 2.5 and 2.6 gives: Lemma 2.7. Under the assumptions of Theorem 1.1, there exists a positive constant K7 depending only on δ , M and independent of N , α , β such that β e Uαβ (∞), U α (∞) ≤ K7 NeK7 N |α − β|

(2.46)

and β

eαβ (∞), W α (∞) ≤ K7 (N + 1)eK7 N |α − β|. W

(2.47)

3. Asymptotic behavior of the global classical solution—proof of Theorem 1.1 This section is devoted to both the study of asymptotic behavior of the global classical solution of the Cauchy problem (1.1), (1.5) and the proof of Theorem 1.1. Let D Di t

=

∂ ∂ + λi (0) . ∂t ∂x

(3.1)

Noting (1.1) and (1.2) we have Dui Di t

= −λi (u)

∂ ui ∂ ui + λi (0) = (λi (0) − λi (u))wi = (λi (ui ei ) − λi (u))wi . ∂x ∂x

(3.2)

Using Hadamard’s formula yields Dui Di t

=

X

Λij (u)uj wi ,

(3.3)

j6=i

where Λij (u) are all C 1 functions of u, which are defined by

Λij (u) = −

1

Z 0

∂λi (τ u1 , . . . , τ ui−1 , ui , τ ui+1 , . . . , τ un ) dτ , ∂ uj

∀j 6= i.

(3.4)

For any fixed (t , x) ∈ R+ × R, define

α = x − λi (0)t .

(3.5)

It follows from (3.3) that ui (t , x) = ui (t , α + λi (0)t ) = ui (0, α) +

X Z tn j6=i

o Λij (u)uj wi (s, α + λi (0)s)ds.

(3.6)

0

By Lemma 2.3, for any fixed α ∈ R we have

X Z t n o Λij (u)uj wi (s, α + λi (0)s)ds ≤ cW∞ (∞)U 1 (∞) ≤ cN1 ecN . j6=i 0

(3.7)

This implies that the integral in the right hand side of (3.6) converges absolutely when t tends to +∞. Therefore, there exists a unique function φi (α) such that ui (t , α + λi (0)t ) → φi (α),

as t → +∞.

(3.8)

Moreover, using Lemma 2.3 we obtain that there exists a positive constant K8 depending only on δ , M and independent of N1 , N, α such that

|φi (α)| ≤ K8 (1 + N1 eK8 N ).

(3.9)

Then, we have proved the following lemma. Lemma 3.1. For any i ∈ {1, . . . , n} and any given α ∈ R, the limit lim ui (t , α + λi (0)t ) = φi (α)

t →+∞

exists and the limit function φi (α) satisfies the estimate (3.9).

(3.10)

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

607

Lemma 3.2. Suppose that the limit lim wi (t , α + λi (0)t )

t →+∞

exists; then we have dφi (α) dα

= lim wi (t , α + λi (0)t ).

(3.11)

t →+∞

Proof. The proof is similar to the proof of Lemma 5.4 in the paper by Dai and Kong [5], so we skip the details and refer the reader to the above mentioned paper.  In what follows, we shall investigate the regularity of the limit function φi (α). For any fixed (t , α + λi (0)t ), there exists a unique θi (t , α) ∈ R such that

θi (t , α) +

t

Z

λi (u(s, xi (s, θi (t , α))))ds = α + λi (0)t ,

(3.12)

0

namely,

θi (t , α) = α +

t

Z

[λi (0) − λi (u(s, xi (s, θi (t , α))))]ds,

(3.13)

0

where x = xi (s, θi (t , α)) stands for the ith characteristic passing through the point (0, θi (t , α)), which is defined by

  dxi (s, θi (t , α))

= λi (u(s, xi (s, θi (t , α)))), ds x (0, θ (t , α)) = θ (t , α). i

i

(3.14)

i

Lemma 3.3. Under the assumptions of Theorem 1.1 and for any fixed α ∈ R, there exists a unique ϑi (α) such that

θi (t , α) → ϑi (α),

t → +∞

(3.15)

and

|ϑi (α) − α| ≤ K9 N1 eK9 N .

(3.16)

The ϑi (α) defined above is globally Lipschitz continuous with respect to α , i.e.,

|ϑi (α) − ϑi (β)| ≤ [1 + K10 NeK10 N ]|α − β|,

(3.17)

where K9 is a positive constant depending only on δ , M and independent of N1 , N, α , while K10 is another positive constant depending only on δ , M and independent of N , α , β . The proof can be found in [2]. Lemma 3.4. For every i ∈ {1, . . . , n}, there exists a positive constant K11 depending only on δ , M and independent of N , α , β such that

|φi (α) − φi (β)| ≤ K11 eK11 N (1 + N )|α − β|,

∀α, β ∈ R.

(3.18)

Proof. The proof is similar to the proof of Lemma 5.3 in the paper by Dai and Kong [5], so we skip the details and refer the reader to the above mentioned paper.  For arbitrary α, β ∈ R and for any fixed i ∈ {1, . . . , n}, we introduce i Wα,β (∞) = sup |wi (t , xi (t , α)) − wi (t , xi (t , β))|,

(3.19)

t ∈R+

where x = xi (t , ·) stands for the ith characteristic passing through the point (0, ·). Lemma 3.5. Under the assumptions of Theorem 1.1, for any given i ∈ {1, . . . , n} and for any fixed α ∈ R, the limit lim wi (t , xi (t , α))

t →+∞

exists, and we denote it by ψi (α), i.e., lim wi (t , xi (t , α)) = ψi (α),

t →+∞

∀α ∈ R,

(3.20)

608

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

where x = xi (t , α) stands for the ith characteristic passing through the point (0, α). Moreover, ψi (α) is continuous with respect to α ∈ R and satisfies that there exists a positive constant K12 depending only on δ , M and independent of N, α such that

|ψi (α)| ≤ MeK12 N ,

∀α ∈ R.

(3.21)

Also, there exists a positive constant K13 depending only on δ , M and independent of N , α , β such that i Wα,β (∞) ≤ eK13 N {|wi (0, α) − wi (0, β)| + K13 (N + 1)|α − β|},

∀α, β ∈ R.

(3.22)

In particular, if the initial data satisfy (1.11), then

|ψi (α) − ψi (β)| ≤ eK13 N {ς |α − β|ρ + K13 (N + 1)|α − β|},

∀α, β ∈ R,

(3.23)

where 0 < ρ ≤ 1. Proof. It follows from (2.36) that

X ∂λi (u) ∂wi ∂wi wl wi . + λ i ( u) =− ∂t ∂x ∂ ul l6=i

(3.24)

Then, integrating (3.24) along the ith characteristic x = xi (t , α) yields

Z t  X   ∂λi (u) wi (t , xi (t , α)) = wi (0, α) exp − wl (s, xi (s, α))ds . ∂ ul 0 l6=i

(3.25)

Noting (2.26)–(2.27), we observe that the integral

Z  Z tX t X ∂λi (u)  e1 (t ) ≤ cN . |wl (s, xi (s, α))|ds ≤ c W − wl (s, xi (s, α))ds ≤ c 0 ∂ ul 0 l6=i l6=i

(3.26)

This implies that the integral in the right hand side of (3.25) converges absolutely when t tends to +∞. Thus, there exists a unique function ψi (α) such that

wi (t , xi (t , α)) → ψi (α),

as t → +∞.

(3.27)

Moreover, we obtain from (3.25) and (3.26) that

 e1 (∞) ≤ MecN . |wi (t , xi (t , α))| ≤ |f 0 (α)| exp c W

(3.28)

(3.21) follows from (3.28) directly. In the following, we calculate wi (t , xi (t , α)) − wi (t , xi (t , β)):

 XZ t    ∂λi (u) wi (t , xi (t , α)) − wi (t , xi (t , β)) = [wi (0, α) − wi (0, β)] exp − wl (s, xi (s, α))ds ∂ ul 0 l6=i   X Z t    X Z t   ∂λi (u) ∂λi (u) + wi (0, β) exp − wl (s, xi (s, α))ds − exp − wl (s, xi (s, β))ds . ∂ ul ∂ ul 0 0 l6=i l6=i

(3.29)

Noting (3.26) and Lemmas 2.3–2.5, using Taylor’s formula, we obtain

   X Z t   X Z t  ∂λi (u)  ∂λi (u) wl (s, xi (s, α))ds − exp − wl (s, xi (s, β))ds exp − ∂ u ∂ u l l 0 0 l6=i l6=i   X Z t ∂λ (u) ∂λi (u) i ≤ ecN wl (s, xi (s, β)) − wl (s, xi (s, α)) ds l6=i 0 ∂ ul ∂ ul Z   t X ∂λi (u) ∂λi (u) cN ≤e · ∂ u (s, xi (s, β)) − ∂ u (s, xi (s, α)) wl (s, xi (s, β)) ds l l 0 l6=i  Z t ∂λi (u) ds + ( s , x ( s , α))[w ( s , x ( s , β)) − w ( s , x ( s , α))] i l i l i ∂u l 0 e1 (t ) + c W eαβ (t )} ≤ ecN {K4 MeK4 N |α − β|W ≤ ecN {K4 MeK4 N |α − β| × K3 N + cK5 (N + 1)eK5 N |α − β|}.

(3.30)

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

609

Then, we get from (3.29) that

|wi (t , xi (t , α)) − wi (t , xi (t , β))| ≤ ecN |wi (0, α) − wi (0, β)| + MecN {K4 MeK4 N |α − β| × K3 N + cK5 (N + 1)eK5 N |α − β|}.

(3.31)

Thus, (3.22) follows from (3.31) directly. Because wi (0, x) is continuous, it follows from (3.22) that ψi (α) ∈ C (R ). If (1.11) holds, we see that wi (0, x) is globally ρ -Hölder continuous. (3.23) follows from (3.22) easily. The proof of Lemma 3.5 is finished.  0

+

Lemma 3.6. For every i ∈ {1, . . . , n}, the limit limt →+∞ wi (t , α + λi (0)t ) exists and lim wi (t , α + λi (0)t ) = ψi (ϑi (α)) ∈ C 0 (R).

(3.32)

t →+∞

Moreover, if (1.11) are satisfied, then the following estimate holds:

|ψi (ϑi (α)) − ψi (ϑi (β))| ≤ eK14 N {ς (1 + K14 NeK14 N )ρ |α − β|ρ + K14 (N + 1)2 eK14 N |α − β|},

∀α, β ∈ R,

(3.33)

where K14 is a positive constant depending only on δ , M and independent of N, ς , α , β . Proof. The proof is similar to the proof of Lemma 5.5 in [5], so instead of giving all the details we refer the reader to the above mentioned paper.  Combining Lemmas 3.2 and 3.6 gives: Lemma 3.7. For every i ∈ {1, . . . , n}, it holds that dφi (α) dα

= ψi (ϑi (α)) ∈ C 0 (R).

(3.34)

Moreover, if (1.11) are satisfied, then the following estimate holds:

dφi dφi K15 N {ς (1 + K15 NeK15 N )ρ |α − β|ρ + K15 (N + 1)2 eK15 N |α − β|}, dα (α) − dα (β) ≤ e

∀ α, β ∈ R,

(3.35)

where K15 is a positive constant depending only on δ , M and independent of N, ς , α , β . Proof of Theorem 1.1. The conclusion of Theorem 1.1 follows from Lemmas 3.1, 3.4 and 3.7 given above. The proof of Theorem 1.1 is finished.  4. Applications 4.1. The Born–Infeld system The Born–Infeld model is a nonlinear version of Maxwell’s theory; it was introduced by Born and Infeld [26] in the 1930’s to cut off (in a nonlinear fashion) the singularities created by point particles in classical electrodynamics. Recently, the Born–Infeld system has attracted considerable attention because of its new applications in string theory and high energy physics. We refer the reader to [27,28,3,29] for mathematical analysis of the BI system and to [30] for its impact in modern high energy physics and string theory. The one-dimensional Born–Infeld system reads (cf. [28])

   B3 + D2 P1 − D1 P2  ∂t D2 + ∂x = 0,   h       −B2 + D3 P1 − D1 P3   ∂t D3 + ∂x = 0,    h    −D3 + B2 P1 − B1 P2 ∂t B2 + ∂x = 0,   h        ∂ B + ∂ D2 + B3 P1 − B1 P3 = 0,  t 3 x   h   p P (u) = D × B, h(u) = 1 + |B|2 + |D|2 + |D × B|2 ,

(4.1)

where u = (D2 , D3 , B2 , B3 )T are the unknown variables, B1 , D1 are real constants and B = (B1 , B2 , B3 )T ,

D = (D1 , D2 , D3 )T ,

P = (P1 , P2 , P3 )T .

From [25], the Riemann invariants and the eigenvalues of system (4.1) are

wi = h−1 (u)li (u)u (1 ≤ i ≤ 4)

(4.2)

610

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

and P1 − a

λ1 (u) = λ2 (u) =

h

,

λ3 (u) = λ4 (u) =

P1 + a

(4.3)

h

respectively, where li (u) (1 ≤ i ≤ 4) are linearly independent constant vectors, which are given by l1 (u) = (a, β1 , 0, −β3 ),

l2 (u) = (−β1 , a, β3 , 0),

l3 (u) = (0, −β2 , a, β1 ),

l4 (u) = (β2 , 0, −β1 , a),

(4.4)

in which

β1 = B1 D1 ,

β2 = 1 + B21 ,

β3 = 1 + D21 and a =

q

1 + B21 + D21 > 0.

(4.5)

Hence, system (4.1) can be written in the form (1.1). They found that it enjoys many interesting properties like nonstrict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, existence of entropy– entropy flux pairs, etc. Consider the Cauchy problem for system (4.1) with the initial data t = 0 : u = (D2 , D3 , B2 , B3 )T = (D02 (x), D03 (x), B02 (x), B03 (x))T ,

(4.6)

where (D02 (x), D03 (x), B02 (x), B03 (x))T ∈ C 1 with both bounded C 1 norm and bounded total variation, such that

kD02 (x)kC 1 , kD03 (x)kC 1 , kB02 (x)kC 1 , kB03 (x)kC 1 ≤ M

(4.7)

for some M > 0 bounded but possibly large, and also such that

Z

+∞

0

|D02 (x)|dx,

Z

−∞

+∞

0

|D03 (x)|dx,

Z

−∞

+∞

0

|B02 (x)|dx, −∞

+∞

Z

0

|B03 (x)|dx ≤ N

(4.8)

−∞

for some N > 0 bounded but possibly large. The global existence of classical solutions is well-known (see [1]). However, there are still no asymptotic behavior results on global classical solutions of the Cauchy problem (4.1) and (4.6). By Theorem 1.1 and Remark 1.3, we get: Theorem 4.1 (Asymptotic Behavior). Under the assumptions mentioned above, for the Cauchy problem (4.1) and (4.6), if

Z

+∞

|D02 (x)|dx, −∞

Z

+∞

|D03 (x)|dx, −∞

+∞

Z

|B02 (x)|dx, −∞

Z

+∞

|B03 (x)|dx < +∞,

(4.9)

−∞

then there exists a unique C 1 vector-valued function φ(x) = (φ1 (x), . . . , φ4 (x))T such that Di (t , x) → φi−1 (x − t ),

t → +∞, i = 2, 3,

(4.10)

Bi (t , x) → φi+1 (x + t ),

t → +∞, i = 2, 3.

(4.11)

Moreover, φi (x) (i = 1, . . . , 4) are globally Lipschitz continuous; more precisely, there exists a positive constant κ1 depending only on δ , M and independent of N , x1 , x2 such that

|φi (x1 ) − φi (x2 )| ≤ κ1 eκ1 N (1 + N )|x1 − x2 |,

∀x1 , x2 ∈ R. 0 D02

0 D03

Furthermore, if the derivatives of the initial data, i.e., (x), 0 < ρ ≤ 1, that is, there exists a positive constant ς such that

(x),

(4.12) 0 B02

(x) and

0 B02

(x), are globally ρ -Hölder continuous, where

|D02 (x1 ) − D02 (x2 )| + |D03 (x1 ) − D03 (x2 )| + |B02 (x1 ) − B02 (x2 )| + |B03 (x1 ) − B03 (x2 )| ≤ ς |x1 − x2 |ρ , 0

0

0

0

0

0

0

0

∀1 , x2 ∈ R,

(4.13)

then φ 0 (x) is also globally ρ -Hölder continuous and satisfies that

|φ 0 (x1 ) − φ 0 (x2 )| ≤ eκ2 N {ς (1 + κ2 Neκ2 N )ρ |x1 − x2 |ρ + κ2 (N + 1)2 eκ2 N |x1 − x2 |}, where κ2 is a positive constant depending only on δ , M and independent of N , ς , x1 , x2 .

(4.14)

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

611

4.2. The system of motion of the relativistic string in Minkowski space–time Consider the following Cauchy problem for the system of motion of relativistic closed strings in the Minkowski space R1+n (cf. [31,32]):

  

|v|2 u − hu, viv



hu, vi2 − (|u|2 − 1)|v|2 vt − uθ = 0, p

t = 0 : u = u0 (θ ),

hu, viu − (|u|2 − 1)v

 − t

p



hu, vi2 − (|u|2 − 1)|v|2

θ

= 0,

v =e v0 + v0 (θ ),

(4.16)

where u = (u1 , . . . , un )T , v = (v1 , . . . , vn )T , e v0 = (e v10 , · · · ,e vn0 )T is a constant vector with |e v0 | = u0 (·) = u01 (·), . . . , u0n (·)

(4.15)


T

and v0 (·) = v10 (·), . . . , vn0 (·)


T

q

(e v10 )2 + · · · + (e vn0 )2 > 0,

∈ C 1 with bounded C 1 norm, such that

ku0 (θ )kC 0 , kv0 (θ )kC 0 , ku00 (θ )kC 0 , kv00 (θ )kC 0 ≤ M ,

(4.17)

for some positive constant M (bounded but possibly large). The global existence of classical solutions is well-known (see [31]). Our goal in this paper is to describe the asymptotic behavior of global classical solutions to the Cauchy problem (4.15) and (4.16). Let

  U =

u

v

.

(4.18)

Then, we can rewrite system (4.15) as Ut + A(U )Uθ = 0,

(4.19)

where 2hu, vi − In×n A(U ) =  |v|2 −In×n



 |u|2 − 1 I n×n  . |v|2

(4.20)

0

  0

It is easy to see that in a neighborhood of U0 = e v0 , (4.15) is a hyperbolic system with the following real eigenvalues:

λ1 (U ) ≡ · · · ≡ λn (U ) = λ− ,

λn+1 (U ) ≡ · · · ≡ λ2n (U ) = λ+ ,

(4.21)

where

λ± =

−hu, vi ±

p

hu, vi2 − (|u|2 − 1)|v|2 . |v|2

(4.22)

When n = 1, (4.15) is a strictly hyperbolic system; while, when n ≥ 2, (4.15) is a non-strictly hyperbolic system with characteristics with constant multiplicity. By Kong et al. [31], the characteristic fields λ± are linearly degenerate. Let



Ri = ui + λ+ vi Si = ui + λ− vi

(i = 1, . . . , n), (i = 1, . . . , n).

(4.23)

Since 4

∆(u, v) = hu, vi2 − (|u|2 − 1)|v|2 > 0,

(4.24)

by a direct computation, (4.15) implies that

 ∂λ ∂λ+ +  + λ− = 0,   ∂t ∂θ      ∂ Ri + λ ∂ Ri = 0 (i = 1, . . . , n),  − ∂t ∂θ ∂λ− ∂λ  −  + λ+ = 0,    ∂t ∂θ    ∂ Si ∂ Si  + λ+ = 0 (i = 1, . . . , n). ∂t ∂θ

(4.25)

612

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

The initial condition (4.16) can then be rewritten as t = 0 : λ+ (0, θ ) = Λ+ (θ ),

λ− (0, θ ) = Λ− (θ ),

Ri (0, θ ) = R0i (θ ),

Si (0, θ ) = Si0 (θ )

(i = 1, . . . , n),

(4.26)

where

Λ± (θ ) =

1



|e v0 + v0 (θ )|

2

−hu0 (θ ),e v0 + v0 (θ )i ±

 p hu0 (θ ),e v0 + v0 (θ )i2 − (|u0 (θ )|2 − 1)|e v0 + v0 (θ )|2 ,

(4.27)

R0i (θ ) = u0i (θ ) + Λ+ (θ )(e vi0 + vi0 (θ ))

(i = 1, . . . , n)

(4.28)

Si0 (θ ) = u0i (θ ) + Λ− (θ )(e vi0 + vi0 (θ ))

(i = 1, . . . , n).

(4.29)

and

By Theorem 1.1 and Remark 1.3, we get: Theorem 4.2 (Asymptotic Behavior). Under the assumptions mentioned above, for the Cauchy problem (4.15) and (4.16), if (4.17) holds together with

Z

+∞

|Λ± (θ )|dθ , −∞

Z

+∞

Z

| (θ )|dθ , R0i

−∞

+∞

|Λ0± (θ )|dθ , −∞

Z

+∞

−∞

Z

+∞

|Si0 (θ )|dθ < +∞,

(4.30)

−∞

0 Z +∞ 0 dRi (θ ) dSi (θ ) dθ , dθ dθ dθ ≤ N −∞

(4.31)

for some N > 0 bounded but possibly large, then there exists a unique C 1 vector-valued function φ(θ ) = (e φ(θ ), φe1 (θ ), . . . , φen (θ), φ(θ ), φ1 (θ ), . . . , φn (θ ))T such that

  t λ+ (t , θ ) → e φ θ+ , t → +∞, |e v0 |   ei θ + t , t → +∞, i = 1, . . . , n, Ri (t , θ ) → φ |e v0 |   t λ− (t , θ ) → φ θ − , t → +∞, |e v0 |   t , t → +∞, i = 1, . . . , n. Si (t , θ ) → φi θ − |e v0 |

(4.32)

(4.33)

(4.34)

(4.35)

Hence, we have ui ( t , θ ) =

e φ(θ + λ+ Si (t , θ ) − λ− Ri (t , θ ) → λ+ (t , θ ) − λ− (t , θ )

t

|e v0 |

)φi (θ − e φ(θ +

t

) − φ(θ −

t

) − φ(θ −

|e v0 | |e v0 |

t

ei (θ + )φ

t

)

|e v0 | |e v0 |

t

|e v0 |

)

,

t → +∞, i = 1, . . . , n,

ei (θ + φ Ri (t , θ ) − Si (t , θ ) vi ( t , θ ) = → e λ+ (t , θ ) − λ− (t , θ ) φ(θ +

(4.36) t

) − φi (θ −

t

) − φ(θ −

|e v0 | |e v0 |

t

)

t

)

|e v0 | |e v0 |

,

t → +∞, i = 1, . . . , n.

(4.37)

Moreover, φ(θ ) is globally Lipschitz continuous; more precisely, there exists a positive constant κ1 depending only on δ , M and independent of N , θ1 , θ2 such that

|φ(θ1 ) − φ(θ2 )| ≤ κ1 eκ1 N (1 + N )|θ1 − θ2 |,

∀θ1 , θ2 ∈ R.

(4.38)

Furthermore, if the derivatives of the initial data, i.e., u0 (θ ) and v0 (θ ), are globally ρ -Hölder continuous, where 0 < ρ ≤ 1, that is, there exists a positive constant ς such that 0

0

|u00 (θ1 ) − u00 (θ2 )| + |v00 (θ1 ) − v00 (θ2 )| ≤ ς |θ1 − θ2 |ρ ,

∀θ1 , θ2 ∈ R,

(4.39)

then φ (θ ) is also globally ρ -Hölder continuous and satisfies that 0

|φ 0 (θ1 ) − φ 0 (θ2 )| ≤ eκ2 N {ς (1 + κ2 Neκ2 N )ρ |θ1 − θ2 |ρ + κ2 (N + 1)2 eκ2 N |θ1 − θ2 |}, where κ2 is a positive constant depending only on δ , M and independent of N , ς , θ1 , θ2 .

(4.40)

Z.-Q. Shao / Nonlinear Analysis 73 (2010) 600–613

613

Acknowledgements The author gratefully acknowledges the support of K. C. Wong Education Foundation, Hong Kong. The author would like to express his gratitude to the referees for their valuable comments and careful reading. The author would also like to thank Prof. Ta-tsien Li very much for his valuable lectures and guidance in Fuzhou in December 2003. The author also would like to thank Prof. Jiaxing Hong very much for his guidance and constant encouragement. The author was supported by the National Natural Science Foundation of China (Grant No. 70371025), the Scientific Research Foundation of the Ministry of Education of China (Grant No. 02JA790014), the Natural Science Foundation of Fujian Province of China (Grant No. 2009J01006), and the Science and Technology Development Foundation of Fuzhou University (Grant No. 2007-XQ-17). Appendix. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.na.2010.03.029. References [1] T.T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, in: Research in Applied Mathematics, vol. 32, Wiley/Masson, New York, 1994. [2] J. Liu, Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Methods Appl. Sci. 30 (2007) 479–500. [3] D. Serre, Systems of Conservation Laws I, II, Cambridge University Press, 2000. [4] W.-R. Dai, Asymptotic behavior of global classical solutions of quasilinear non-strictly hyperbolic systems with weakly linear degeneracy, Chin. Ann. Math. Ser. B 27 (2006) 263–286. [5] W.-R. Dai, D.-X. Kong, Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields, J. Differential Equations 235 (2007) 127–165. [6] A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421. [7] D.-X. Kong, Cauchy Problem for Quasilinear Hyperbolic Systems, in: MSJ Memoirs, Mathematical Society of Japan, Tokyo, 2000. [8] T.T. Li, Y. Zhou, D.X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Differential Equations 19 (1994) 1263–1317. [9] T.T. Li, Y. Zhou, D.X. Kong, Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Anal. TMA 28 (1997) 1299–1332. [10] Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chin. Ann. Math. Ser. B 25 (2004) 37–56. [11] D.-X. Kong, T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems, Comm. Partial Differential Equations 28 (2003) 1203–1220. [12] T.P. Liu, Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977) 767–796. [13] T.P. Liu, Large-time behavior of solutions of initial and initial–boundary value problems of a general system of hyperbolic conservation laws, Comm. Math. Phys. 55 (1977) 163–177. [14] T.P. Liu, Asymptotic behavior of solutions of general system of nonlinear hyperbolic conservation laws, Indiana Univ. Math. J. 27 (1978) 211–253. [15] Z.-Q. Shao, Asymptotic behavior of global classical solutions to the mixed initial–boundary value problem for quasilinear hyperbolic systems with small BV Data, J. Elasticity 98 (2010) 25–64. [16] Y.Z. Duan, Asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries, J. Math. Anal. Appl. 351 (2009) 186–205. [17] A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford University Press, 2000. [18] Z.-Q. Shao, The mixed initial–boundary value problem for quasilinear hyperbolic systems with linearly degenerate characteristics, Nonlinear Anal. 71 (2009) 1350–1368. [19] Z.-Q. Shao, Global existence of classical solutions to the mixed initial–boundary value problem for quasilinear hyperbolic systems of diagonal form with large BV data, J. Math. Anal. Appl. 360 (2009) 398–411. [20] D. Amadori, W. Shen, The slow erosion limit in a model of granular flow. Preprint available at http://www.math.ntnu.no/conservation/2009/003.html. [21] D. Amadori, W. Shen, Global existence of large BV solutions in a model of granular flow. Preprint available at http://www.math.ntnu.no/conservation/ 2008/023.html. [22] J. Glimm, P.D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Mem. Amer. Math. Soc. 101 (1970). [23] T. Nishida, J. Smoller, Solutions in the large for some nonlinear conservation laws, Comm. Pure Appl. Math. 26 (1973) 183–200. [24] T.T. Li, Y.J. Peng, Cauchy problem for weakly linearly degenerate hyperbolic systems in diagonal form, Nonlinear Anal. 55 (2003) 937–949. [25] T.T. Li, Y.J. Peng, J. Ruiz, Entropy solutions for linearly degenerate hyperbolic systems of rich type, J. Math. Pures Appl. 91 (2009) 553–568. [26] M. Born, L. Infeld, Foundations of the new field theory, Proc. R. Soc. Lond. A 144 (1934) 425–451. [27] G. Bolliat, C.M. Dafermos, P.D. Lax, T.P. Liu, Recent Mathematical Methods in Nonlinear Wave Propagation, in: Lecture Notes in Mathematics, vol. 1640, Springer-Verlag, Berlin, 1996. [28] Y. Brenier, Hydrodynamic structure of the augmented Born–Infeld equations, Arch. Ration Mech. Anal. 172 (2004) 65–91. [29] D. Serre, Hyperbolicity of the nonlinear models of Maxwell’s equations, Arch. Ration Mech. Anal. 172 (2004) 309–331. [30] G.W. Gibbons, Aspects of Born–Infeld theory and string/M-theory, in: Proceedings of the 4th Mexican School on Gravitation and Mathematical Physics, Membrane, 2000. Or http://arxiv.org/abs/hep-th/0106059. [31] D.-X. Kong, Q. Zhang, Q. Zhou, The dynamics of relativistic strings moving in the Minkowski space R1+n , Comm. Math. Phys. 269 (2007) 153–174. [32] Z.-Q. Shao, Global weakly discontinuous solutions to the mixed initial-boundary value problem for quasilinear hyperbolic systems, Math. Models Methods Appl. Sci. 19 (2009) 1099–1138.