The mixed initial–boundary value problem for quasilinear hyperbolic systems with linearly degenerate characteristics

Nonlinear Analysis 71 (2009) 1350–1368

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The mixed initial–boundary value problem for quasilinear hyperbolic systems with linearly degenerate characteristics Zhi-Qiang Shao ∗ Department of Mathematics, Fuzhou University, Fuzhou 350002, China

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Article history: Received 15 November 2008 Accepted 1 December 2008 MSC: 35L45 35L50 35Q72 Keywords: Quasilinear hyperbolic system Mixed initial–boundary value problem Global classical solution Linear degeneracy

In this paper, we investigate the mixed initial–boundary value problem with small BV data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t , x)|t ≥ 0, x ≥ 0}. As the result in [A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421] suggests that one may achieve global smoothness even if the C 1 norm of the initial data is large, we prove that, if the C 1 norm of the initial and boundary data is bounded but possibly large, and the BV norm of the initial and boundary data is sufficiently small, then the solution remains C 1 globally in time. Applications to quasilinear hyperbolic systems arising in physics and mechanics, particularly to the system describing the motion of the relativistic string in the Minkowski space–time R1+n , are also given. © 2008 Elsevier Ltd. All rights reserved.

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

∂u ∂u + A(u) = 0, (1.1) ∂t ∂x where u = (u1 , . . . , un )T is the unknown vector function of (t , x) and A(u) is an n × n matrix with suitably smooth elements aij (u) (i, j = 1, . . . , n). It is assumed that system (1.1) is strictly hyperbolic, i.e., for any given u on the domain under consideration, A(u) has n real distinct eigenvalues

λ1 (u) < λ2 (u) < · · · < λn (u).

(1.2)

Let li (u) = (li1 (u), . . . , lin (u)) (resp. ri (u) = (ri1 (u), . . . , rin (u)) ) be a left (resp. right) eigenvector corresponding to λi (u) (i = 1, . . . , n): T

li (u)A(u) = λi (u)li (u)

(resp. A(u)ri (u) = λi (u)ri (u)),

(1.3)

then we have det |lij (u)| 6= 0 (equivalently, det |rij (u)| 6= 0).

(1.4)

Without loss of generality, we may assume that on the domain under consideration li (u)rj (u) ≡ δij

∗

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

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

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.12.002

(1.5)

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

1351

and riT (u)ri (u) ≡ 1 (i = 1, . . . , n),

(1.6)

where δij stands for the Kronecker’s symbol. Clearly, all λi (u), lij (u) and rij (u) (i, j = 1, . . . , n) have the same regularity as aij (u) (i, j = 1, . . . , n). We assume that on the domain under consideration, the eigenvalues satisfy the non-characteristic condition

λr (u) < 0 < λs (u) (r = 1, . . . , m; s = m + 1, . . . , n).

(1.7)

We also assume that on the domain under consideration, system (1.1) is linearly degenerate, i.e., each characteristic field is linearly degenerate in the sense of Lax:

∇λi (u)ri (u) ≡ 0 (i = 1, . . . , n).

(1.8)

We are concerned with the existence and uniqueness of global classical solutions to the mixed initial–boundary value problem with small BV data for system (1.1) in the half space D = {(t , x) | t ≥ 0, x ≥ 0}

(1.9)

with the initial condition: t = 0 : u = ϕ(x)

(x ≥ 0)

(1.10)

and the nonlinear boundary condition (cf. [1–4]) x = 0 : vs = Gs (α(t ), v1 , . . . , vm ) + hs (t ),

s = m + 1, . . . , n (t ≥ 0),

(1.11)

where

vi = li (u)u (i = 1, . . . , n)

(1.12)

and

α(t ) = (α1 (t ), . . . , αk (t )). Here, Gs ∈ C 1 (s = m + 1, . . . , n), ϕ = (ϕ1 , . . . , ϕn )T , α and h(·) = (hm+1 (·), . . . , hn (·)) ∈ C 1 with bounded C 1 norm, such that

kϕ(x)kC 1 , kα(t )kC 1 , kh(t )kC 1 ≤ M ,

(1.13) 1

for some positive constant M (bounded but possibly large). Also, we assume that the conditions of C compatibility are satisfied at the point (0, 0). Without loss of generality, we assume that Gs (α(t ), 0, . . . , 0) ≡ 0 (s = m + 1, . . . , n).

(1.14)

Without loss of generality, we also assume that

ϕ(0) = 0.

(1.15)

In fact, by the following transformation

e u = u − ϕ(0),

(1.16)

we can always realize the above assumption. For the Cauchy problem of quasilinear hyperbolic systems, such kinds of problems have been extensively studied (for instance, see [5–17] and references therein). In particular, Li et al. proved that for any given initial data ϕ(x) satisfying the following small and decaying property: sup(1 + |x|)1+µ (|ϕ(x)| + |ϕ 0 (x)|)  1,

(1.17)

x∈R

where µ > 0 is a constant, Cauchy problem for system (1.1) admits a unique global C 1 solution with small C 1 norm, provided that system (1.1) is weakly linearly degenerate (see [11–13]). In their works, the condition µ > 0 is essential. If µ = 0, a counterexample was constructed by Kong [9] showing that the C 1 solution may blow up in a finite time, even when system (1.1) is weakly linearly degenerate. On the other hand, for the mixed initial–boundary value problem of quasilinear hyperbolic systems, many results on the global existence of C 1 solutions have also been obtained in the literature (for instance, see [18–22,1,23,15,2] and the references therein), and some method have been established. In particular, Li and Wang [1] have proved the existence and uniqueness of global C 1 solution in the half space {(t , x)|t ≥ 0, x ≥ 0}, provided that each characteristic with positive velocity is weakly linearly degenerate and the initial and boundary data are suitably small in the C 1 norm. However, it is well known that the BV space is a suitable framework for one-dimensional Cauchy problem for quasilinear hyperbolic systems (see Bressan [6]), the result in Bressan [5] 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 existence and uniqueness of global C 1 solutions to the mixed initial–boundary value problem (1.1), (1.10) and (1.11), provided that the BV

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Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

norm of the initial and boundary data is suitably small? Here, it is important to mention that for the Cauchy problem case, this problem was solved by Bressan [5], Zhou [17], Dai and Kong [7]. However, due to the presence of a boundary, any waves with negative speed are expected to be reflected at the boundary, some additional difficulties appear. Therefore new proofs are required to overcome them. This makes our new analysis more complicated than that for the Cauchy problem case. The present paper can be viewed as a development of [5,7,17]. Our main results can be summarized as follows: Theorem 1.1. Suppose that system (1.1) is strictly hyperbolic and linearly degenerate. Suppose furthermore that in a neighborhood of u = 0, A(u) ∈ C 2 and (1.7) holds. Suppose finally that ϕ, α, Gs , hs (s = m + 1, . . . , n) are all C 1 functions with respect to their arguments satisfying the conditions of C 1 compatibility at the point (0, 0). For any constant M > 0, there exists ε > 0 small enough such that, if (1.13)–(1.15) hold together with +∞

Z

|ϕ 0 (x)|dx,

+∞

Z

|α 0 (t )|dt , 0

0

+∞

Z

|h0 (t )|dt ≤ ε,

(1.18)

0

then the mixed initial–boundary value problem (1.1), (1.10) and (1.11) admits a unique global C 1 solution u = u(t , x) in the half space {(t , x)|t ≥ 0, x ≥ 0}. Remark 1.1. 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).

(1.19)

Then the conclusion of Theorem 1.1 still holds (cf. [7,11]). This paper is organized as follows. For the sake of completeness, in Section 2, we briefly recall John’s formula on the decomposition of waves with some supplements. Theorem 1.1 is proved in Section 3. It is easy to see that Theorem 1.1 can be applied to all physical models discussed in Li and Wang [1] on the mixed initial–boundary value problem for the system of the planar motion of an elastic string, provided that the BV norm of the initial and boundary data is suitably small, therefore we do not give the details in this paper. However, of particular interest is the system of the motion of the relativistic string in the Minkowski space–time R1+n , as an application of Theorem 1.1, global existence of classical solutions to the mixed initial–boundary value problem with small BV data for this system is presented in Section 4. 2. Preliminaries Suppose that on the domain under consideration, system (1.1) is strictly hyperbolic and (1.5) and (1.6) hold. Let

wi = li (u)ux (i = 1, . . . , n),

(2.1)

where li (u) = (li1 (u), . . . , lin (u)) denotes the ith left eigenvector. By (1.5), it follows from (1.12) and (2.1) that u=

n X

v k r k ( u)

(2.2)

k=1

and ux =

n X

wk rk (u).

(2.3)

k=1

Let d ∂ ∂ = + λi (u) di t ∂t ∂x be the directional derivative along the ith characteristic. We have (see [7,13,17]) dv i di t

=

n X

βijk (u)vj wk (i = 1, . . . , n),

(2.4)

(2.5)

j,k=1

where

βijk (u) = (λk (u) − λi (u))li (u)∇ rj (u)rk (u).

(2.6)

Hence, we have

βiji (u) ≡ 0,

∀i, j.

(2.7)

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

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On the other hand, we have (see [7,13,17]) dwi di t

=

n X

γijk (u)wj wk (i = 1, . . . , n),

(2.8)

j,k=1

where

γijk (u) =

1 2

{(λj (u) − λk (u))li (u)∇ rk (u)rj (u) − ∇λi (u)rj (u)δik + (j|k)},

(2.9)

in which (j|k) denotes all the terms obtained by changing j and k in the previous terms. Hence, we have

γijj (u) ≡ 0,

∀i 6= j (i, j = 1, . . . , n)

(2.10)

and

γiii (u) ≡ −∇λi (u)ri (u) (i = 1, . . . , n).

(2.11)

When the ith characteristic λi (u) is linearly degenerate in the sense of Lax, we have

γiii (u) ≡ 0.

(2.12)

Noting (2.3), by (2.8) we have (see [8]) n X ∂wi ∂(λi (u)wi ) def + = Γijk (u)wj wk = Gi (t , x), ∂t ∂x j,k=1

(2.13)

equivalently, d[wi (dx − λi (u)dt )] =

n X

Γijk (u)wj wk dt ∧ dx = Gi (t , x)dt ∧ dx,

(2.14)

j,k=1

where

Γijk (u) =

1 2

(λj (u) − λk (u))li (u)[∇ rk (u)rj (u) − ∇ rj (u)rk (u)].

(2.15)

Hence, we have

Γijj (u) ≡ 0,

∀i, j.

(2.16)

3. Proof of Theorem 1.1 By the existence and uniqueness of local C 1 solution for quasilinear hyperbolic systems (see [24]), there exists T0 > 0 such that the mixed initial–boundary value problem (1.1), (1.10) and (1.11) admits a unique C 1 solution u = u(t , x) on the domain def

D(T0 ) = {(t , x)|0 ≤ t ≤ T0 , x ≥ 0}.

(3.1) 0

Thus, in order to prove Theorem 1.1 it suffices to establish a uniform a priori estimate for the C norm of u and ux on any given domain of existence of the C 1 solution u = u(t , x). Noting (1.2) and (1.7), we have

λ1 (0) < · · · < λm (0) < 0 < λm+1 (0) < · · · < λn (0).

(3.2)

Thus, there exist sufficiently small positive constants δ and δ0 such that

λi+1 (u) − λi (v) ≥ δ0 ,

∀|u|, |v| ≤ δ(i = 1, . . . , n − 1)

(3.3)

and

|λi (0)| ≥ δ0 (i = 1, . . . , n). For the time being it is supposed that on the domain of existence of the C initial–boundary value problem (1.1), (1.10) and (1.11), we have

|u(t , x)| ≤ δ.

(3.4) 1

solution u = u(t , x) to the mixed (3.5)

At the end of the proof of Lemma 3.5, we will explain that this hypothesis is reasonable. Thus, in order to prove Theorem 1.1, we only need to establish a uniform a priori estimate for the C 0 norm of v = (v1 , . . . , vn )T and w = (w1 , . . . , wn )T on the domain of existence of the C 1 solution u = u(t , x).

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Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

For any fixed T > 0, let U∞ (T ) = sup sup |u(t , x)|,

(3.6)

V∞ (T ) = sup sup |v(t , x)|,

(3.7)

W∞ (T ) = sup sup |w(t , x)|,

(3.8)

0≤t ≤T x∈R+

0≤t ≤T x∈R+

0≤t ≤T x∈R+

+∞

Z

W1 (T ) = sup 0≤t ≤T

|w(t , x)|dx,

(3.9)

0

e1 (T ) = maxi=1,...,n maxj6=i sup W Cj

Z

|wi |dt ,

(3.10)

Cj

where | · | stands for the Euclidean norm in Rn , while Cj stands for any given jth characteristic on the domain [0, T ] × R+ . Clearly, V∞ (T ) is equivalent to U∞ (T ). First we recall some basic L1 estimates. They are essentially due to Schartzman [25,26] and Zhou [17]. Lemma 3.1. Let φ = φ(t , x) ∈ C 1 satisfy

φt + (λ(t , x)φ)x = F (t , x),

0 ≤ t ≤ T , x ∈ R,

φ(0, x) = g (x),

where λ ∈ C . Then 1

+∞

Z

|φ(t , x)|dx ≤ −∞

Z

+∞

T

Z

|g (x)|dx +

−∞

+∞

Z

|F (t , x)|dxdt ,

∀t ≤ T ,

(3.11)

−∞

0

provided that the right hand side of the inequality is bounded. Lemma 3.2. Let φ = φ(t , x) and ψ = ψ(t , x) be C 1 functions satisfying

φt + (λ(t , x)φ)x = F1 (t , x),

0 ≤ t ≤ T , x ∈ R,

φ(0, x) = g1 (x),

and

ψt + (µ(t , x)ψ)x = F2 (t , x),

0 ≤ t ≤ T , x ∈ R,

ψ(0, x) = g2 (x),

respectively, where λ, µ ∈ C such that there exists a positive constants δ0 independent of T verifying 1

µ(t , x) − λ(t , x) ≥ δ0 ,

0 ≤ t ≤ T , x ∈ R.

Then T

Z

+∞

Z

|φ(t , x)||ψ(t , x)|dxdt ≤ C −∞

0

+∞

Z

|g1 (x)|dx + −∞

Z

|g2 (x)|dx +

−∞

+∞

Z

|F1 (t , x)|dxdt



−∞

0

+∞

×

T

Z

T

Z

+∞

Z

0



|F2 (t , x)|dxdt ,

(3.12)

−∞

provided that the two factors on the right hand side of the inequality is bounded. In the present situation, similar to the above basic L1 estimates (3.11) and (3.12), we have the following uniform a priori estimates on the L1 norm of the solution. Lemma 3.3. Under the assumptions of Theorem 1.1, on any given domain of existence {(t , x)|0 ≤ t ≤ T , x ≥ 0} of the C 1 solution u = u(t , x) to the mixed initial–boundary value problem (1.1), (1.10) and (1.11), there exists a positive constant K1 independent of ε , T and M such that +∞

Z

+∞

 Z |wi (t , x)|dx ≤ K1 W1 (0) +

0

(|α 0 (t )| + |h0 (t )|)dt + 0

T

Z 0

+∞

Z

 |G(t , x)|dxdt ,

∀t ≤ T ,

(i = 1, . . . , n)

0

(3.13) provided that the right hand side of the inequality is bounded and G = (G1 , G2 , . . . , Gn ). Proof. To estimate

R +∞ 0

|wi (t , x)|dx, we need only to estimate

L

Z

|wi (t , x)|dx 0

for any given L > 0 and then let L → +∞.

(3.14)

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

1355

(i) For i = 1, . . . , m, for any fixed point (T , L), we draw the ith backward characteristic x = xi (t ) (0 ≤ t ≤ T ) passing through this point:

(

dxi (t )

= λi (u(t , xi (t ))),

t ≤ T,

dt xi (T ) = L.

(3.15)

From (2.13), we have

|wi |t + (λi (u)|wi |)x = sgn(wi )Gi (t , x).

(3.16)

Then, noting (1.7), we get d

xi (t )

Z

dt

xi (t )

∂ |wi (t , x)|dx + x0i (t )|wi (t , xi (t ))| ∂ t 0 Z xi (t ) Z xi (t ) (λi (u)|wi |)x dx + x0i (t )|wi (t , xi (t ))| sgn(wi )Gi (t , x)dx − =

|wi (t , x)|dx =

0

Z

0

0 xi (t )

Z =

sgn(wi )Gi (t , x)dx − (λi (u(t , xi (t ))) − x0i (t ))|wi (t , xi (t ))| + λi (u(t , 0))|wi (t , 0)|

0 xi (t )

Z =

sgn(wi )Gi (t , x)dx + λi (u(t , 0))|wi (t , 0)| ≤

xi (t )

Z

0

|Gi (t , x)|dx.

(3.17)

0

Therefore it follows from (3.17) that xi (t )

Z

|wi (t , x)|dx ≤

xi (0)

Z

0

0

|wi (0, x)|dx +

≤

Z

0

xi (t )

Z

0

+∞

Z

T

Z

|wi (0, x)|dx +

|Gi (t , x)|dxdt

0

T

0

+∞

Z

|G(t , x)|dxdt .

(3.18)

0

Hence, noting (3.15), we have L

Z

|wi (t , x)|dx ≤ W1 (0) +

T

Z

0

0

+∞

Z

|G(t , x)|dxdt .

(3.19)

0

Letting L → +∞, we immediately get the assertion in (3.13). (ii) For i = m + 1, . . . , n, let x = xi (t , 0) (0 ≤ t ≤ T ) be the ith forward characteristic passing through origin O(0, 0). Then, passing through the point (T , L) (L > xi (T , 0)), we draw the ith backward characteristic x = xi (t ) (0 ≤ t ≤ T ) which intersects the x-axis at a point (0, xi0 ). By exploiting the same arguments as in (i), we can deduce that d dt

xi (t )

Z

|wi (t , x)|dx =

xi (t )

Z

0

sgn(wi )Gi (t , x)dx + λi (u(t , 0))|wi (t , 0)|

0 xi (t )

Z ≤

|Gi (t , x)|dx + λi (u(t , 0))|wi (t , 0)|.

(3.20)

0

Thus, noting (3.5), it follows from (3.20) that xi (t )

Z

|wi (t , x)|dx ≤

0

xi0

Z

|wi (0, x)|dx +

0

T

Z

λi (u(t , 0))|wi (t , 0)|dt + 0

+∞

Z

|wi (0, x)|dx +

≤ C1 0

T

Z

Z

0 T

Z

|wi (t , 0)|dt + 0

T

Z 0

+∞

Z

xi (t )

|Gi (t , x)|dxdt

0



|G(t , x)|dxdt ,

(3.21)

0

where here and henceforth, Ci (i = 1, 2, . . .) will denote positive constants independent of ε , T and M. Similar to Lemma 3.2 in [1], by differentiating the nonlinear boundary condition (1.11) with respect to t, we get x=0:

m k X ∂vs ∂ Gs ∂vr X ∂ Gs = (α(t ), v1 , . . . , vm ) + (α(t ), v1 , . . . , vm )αi0 (t ) + h0s (t ) (s = m + 1, . . . , n). (3.22) ∂t ∂v ∂ t ∂αi r r =1 i=1

By (1.1), (1.3) and (2.3), it is easy to see that n X ∂vi ∂ = (li (u)u) = −λi (u)wi + aik (u)wk (i = 1, . . . , n), ∂t ∂t k=1

(3.23)

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Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

where aik (u) = −λk (u)rkT (u)∇ li (u)u.

(3.24)

Thus, noting (1.7) and (3.5), for sufficiently small δ > 0, we easily see from (3.22) and (3.23) that x = 0 : ws =

m X

fsj (t , u)wj +

j=1

k X

f si (t , u)αi0 (t ) +

i=1

n X

e fsl (t , u)h0l (t ) (s = m + 1, . . . , n),

(3.25)

l=m+1

where fsj , f si and e fsl are continuous functions of t and u. Noting (3.5), by (3.25) we have T

Z

|wi (t , 0)|dt = 0

m Z X r =1

+

|fir (t , u(t , 0))wr (t , 0)|dt + 0

j =1

Z n X

|f ij (t , u(t , 0))αj0 (t )|dt 0

T

|e fil (t , u(t , 0))h0l (t )|dt

( m Z X r =1

T

0

l=m+1

≤ C2

k Z X

T

T

Z

|wr (t , 0)|dt +

0

)

+∞

(|α (t )| + |h (t )|)dt . 0

0

(3.26)

0

Then, passing through the point A(T , 0), we draw the rth characteristic cr (r ∈ {1, . . . , m}) which intersects the x-axis at point B(0, xB ). We rewrite (2.14) as d(|wr (t , x)|(dx − λr (u)dt )) = sgn(wr )Gr dxdt .

(3.27)

By (3.27), using Stokes’ formula on the domain AOB, we have

Z

T 0

Z |wr (t , 0)|(−λr (u)dt ) ≤

xB

|wr (0, x)|dx +

Z Z |Gr |dxdt

0

AOB

≤ W1 (0) +

T

Z

+∞

Z

0

|G(t , x)|dxdt .

(3.28)

0

Noting (3.4) and (3.5), for sufficiently small δ > 0, it is easy to see that

|λr (u)| ≥

δ0 2

.

(3.29)

Therefore, it follows from (3.28) that T

Z

T

 Z |wr (t , 0)|dt ≤ C3 W1 (0) +

0

+∞

Z

0

 |G(t , x)|dxdt .

(3.30)

0

Combining (3.21) with (3.26) and (3.30), we obtain xi ( t )

Z



|wi (t , x)|dx ≤ C4 W1 (0) +

0

+∞

Z

(|α (t )| + |h (t )|)dt + 0

0

T

Z

0

+∞

Z



|G(t , x)|dxdt .

0

0

Z

+∞

(3.31)

Thus, noting (3.15), we have L

Z

+∞

 Z |wi (t , x)|dx ≤ C4 W1 (0) +

0

(|α 0 (t )| + |h0 (t )|)dt + 0

T

Z 0

 |G(t , x)|dxdt .

(3.32)

0

Letting L → +∞, we immediately get the assertion in (3.13). The proof of Lemma 3.3 is finished.



Lemma 3.4. Under the assumptions of Theorem 1.1, on any given domain of existence {(t , x)|0 ≤ t ≤ T , x ≥ 0} of the C 1 solution u = u(t , x) to the mixed initial–boundary value problem (1.1), (1.10) and (1.11), there exists a positive constant K2 independent of ε , T and M such that T

Z 0

+∞

Z

+∞

 Z |wi (t , x)||wj (t , x)|dxdt ≤ K2 W1 (0) +

0

(|α 0 (t )| + |h0 (t )|)dt 0

T

Z

+∞

Z

|G(t , x)|dxdt

+ 0

2

,

∀i 6= j (i, j = 1, . . . , n),

0

provided that the right hand side of the inequality is bounded and G = (G1 , G2 , . . . , Gn ).

(3.33)

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

1357

Proof. To estimate T

Z

+∞

Z

0

|wi (t , x)||wj (t , x)|dxdt ,

(3.34)

0

it is enough to estimate T

Z 0

L

Z

|wi (t , x)||wj (t , x)|dxdt ,

(3.35)

0

for any given L > 0 and then let L → +∞. (i) For i, j ∈ {1, . . . , m} and i 6= j, without loss of generality, we suppose that i < j, passing through the point (T , L), we draw the ith backward characteristic x = xi (t ) (0 ≤ t ≤ T ) which intersects the x-axis at a point (0, xi0 ). In the meantime, passing through the point (T , 0), we draw the jth characteristic x = xj (t ) (0 ≤ t ≤ T ) which intersects the x-axis at a point (0, xj0 ) with xj0 ≤ xi0 . We introduce the ‘‘continuous Glimm’s functional’’ (cf. [5,25,17]) Q (t ) =

Z Z xj (t )
|wj (t , x)||wi (t , y)|dxdy.

(3.36)

Then, it is easy to see that dQ (t ) dt

Z xi (t ) |wj (t , x)|dx − x0j (t )|wj (t , xj (t ))| |wi (t , x)|dx xj (t ) x (t ) Z Z j ∂ ∂ (|wj (t , x)|)|wi (t , y)|dxdy + |wj (t , x)| (|wi (t , y)|)dxdy ∂ t ∂ t xj (t )
Z

xi (t )

xj (t )
xj (t )
= (x0i (t ) − λi (u(t , xi (t ))))|wi (t , xi (t ))|

xi ( t )

Z

xj (t )

+ (λj (u(t , xj (t ))) − x0j (t ))|wj (t , xj (t ))|

|wj (t , x)|dx

xi (t )

Z

xj ( t )

|wi (t , x)|dx

xi (t )

Z

(λj (u(t , x)) − λi (u(t , x)))|wi (t , x)||wj (t , x)|dx Z Z Z Z + sgn(wj )Gj (t , x)|wi (t , y)|dxdy + −

xj ( t )

xj (t )
xi (t )

Z ≤ −δ0

xj ( t ) xi (t )

Z

xj ( t )

xi (t )

Z ≤ −δ0

|wi (t , x)||wj (t , x)|dx +

|Gi (t , x)|dx

+

xj ( t )

|Gi (t , x)|dx

+

Z

xi (t )

xj (t )

0

xi (t )

Z

xj ( t )

|Gj (t , x)|dx

Z

|Gj (t , x)|dx

Z

xi (t ) xj (t )

|wj (t , x)|sgn(wi )Gi (t , y)dxdy

|wi (t , x)|dx

|wj (t , x)|dx

|wi (t , x)||wj (t , x)|dx +

+∞

Z

xj (t )
+∞

Z 0

+∞

|wi (t , x)|dx 0

+∞

Z

|wj (t , x)|dx.

(3.37)

0

It then follows from Lemma 3.3 that dQ (t ) dt

+ δ0

Z

+∞

Z ≤ C5 0

xi (t ) xj ( t )

|wi (t , x)||wj (t , x)|dx +∞

 Z |G(t , x)|dx W1 (0) +

(|α 0 (t )| + |h0 (t )|)dt + 0

Z 0

T

Z 0

+∞

 |G(t , x)|dxdt .

(3.38)

1358

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

Therefore

δ0

T

Z

xi (t )

Z

|wi (t , x)||wj (t , x)|dxdt ≤ Q (0) + C5

xj (t )

0

T

Z 0

 ×

W1 (0) +

+∞

Z

|G(t , x)|dxdt 0

+∞

Z

T

Z

(|α (t )| + |h (t )|)dt + 0

0



|G(t , x)|dxdt .

(3.39)

0

0

0

+∞

Z

Noting Q (0) ≤

+∞

Z

|wi (0, x)|dx

+∞

Z

|wj (0, x)|dx,

(3.40)

0

0

we have

δ0

T

Z

xi (t )

Z

xj (t )

0

+∞

 Z |wi (t , x)||wj (t , x)|dxdt ≤ C6 W1 (0) +

(|α 0 (t )| + |h0 (t )|)dt +

T

Z

|G(t , x)|dxdt

2

.

(3.41)

0

0

0

+∞

Z

It then follows that T

Z

xi ( t )

Z

xj (t )

0

+∞

 Z |wi (t , x)||wj (t , x)|dxdt ≤ C7 W1 (0) +

(|α 0 (t )| + |h0 (t )|)dt +

T

Z

0

+∞

Z

|G(t , x)|dxdt

0

0

Z

+∞

2

.

(3.42)

Hence T

Z 0

L

Z



|wi (t , x)||wj (t , x)|dxdt ≤ C7 W1 (0) +

+∞

Z

0

(|α (t )| + |h (t )|)dt + 0

0

0

T

Z 0

|G(t , x)|dxdt

2 (3.43)

0

and the desired conclusion follows by taking L → +∞. (ii) For i ∈ {1, . . . , m} and j ∈ {m + 1, . . . , n}, passing through the point (T , 0), we draw the ith backward characteristic x = xi (t ) (0 ≤ t ≤ T ) which intersects the x-axis at a point (0, xi0 ). Let x = xj (t , xi0 ) (0 ≤ t ≤ T ) be the jth forward characteristic passing through point (0, xi0 ). Then, we draw the jth backward characteristic x = xj (t ) (0 ≤ t ≤ T ) passing through the point (T , L) (L > xj (T , xi0 )). We introduce the ‘‘continuous Glimm’s functional’’ Q (t ) =

Z Z xi (t )
|wi (t , x)||wj (t , y)|dxdy.

(3.44)

Then, it is easy to see that dQ (t ) dt

Z xj (t ) |wi (t , x)|dx − x0i (t )|wi (t , xi (t ))| |wj (t , x)|dx xi (t ) x (t ) Z Z i ∂ ∂ (|wi (t , x)|)|wj (t , y)|dxdy + |wi (t , x)| (|wj (t , y)|)dxdy ∂ t ∂ t xi (t )
Z

xj (t )

xi (t )
xi (t )
= (x0j (t ) − λj (u(t , xj (t ))))|wj (t , xj (t ))|

xj (t )

Z

xi ( t )

+ (λi (u(t , xi (t ))) − xi (t ))|wi (t , xi (t ))| 0

Z

xj ( t )

− x (t )

Z iZ +

Z

|wi (t , x)|dx

xj (t ) xi (t )

|wj (t , x)|dx

(λi (u(t , x)) − λj (u(t , x)))|wi (t , x)||wj (t , x)|dx Z Z sgn(wi )Gi (t , x)|wj (t , y)|dxdy +

xi (t )
xi (t )
|wi (t , x)|sgn(wj )Gj (t , y)dxdy

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368 xj (t )

Z ≤ −δ0

|wi (t , x)kwj (t , x)|dx +

xi (t ) xj (t )

Z

|Gj (t , x)|dx

+ xi ( t )

xj (t )

Z ≤ −δ0

xj (t )

xi (t )

|Gi (t , x)|dx

Z

|Gi (t , x)|dx

Z

|Gj (t , x)|dx

Z

xj (t ) xi (t )

1359

|wj (t , x)|dx

|wi (t , x)|dx Z

+∞

+∞

|wj (t , x)|dx

0

0

+∞

Z

xi (t )

|wi (t , x)kwj (t , x)|dx +

xi (t )

+

xj ( t )

Z

Z

+∞

|wi (t , x)|dx.

(3.45)

0

0

It then follows from Lemma 3.3 that dQ (t )

+ δ0

dt

Z

xi ( t )

+∞

Z

xj (t )

≤ C8

|wi (t , x)||wj (t , x)|dx +∞

 Z |G(t , x)|dx W1 (0) +

(|α 0 (t )| + |h0 (t )|)dt +

T

0

0

0

Z

+∞

Z

 |G(t , x)|dxdt .

(3.46)

0

Hence

δ0

T

Z

xj (t )

Z

|wi (t , x)||wj (t , x)|dxdt

xi ( t )

0

≤ Q (0) + C8

Z

T

+∞

Z

0

0

≤ (W1 (0))2 + C8 ≤ C9 W1 (0) +

+∞

Z

0



T

Z

|G(t , x)|dxdt W1 (0) + 0

(|α 0 (t )| + |h0 (t )|)dt +

(|α 0 (t )| + |h0 (t )|)dt + 0

0

|G(t , x)|dxdt

2



+∞

Z

0

+∞

Z

T

Z

0 T

Z

|G(t , x)|dxdt

0

+∞

Z

+∞

Z

0



+∞

Z

(|α 0 (t )| + |h0 (t )|)dt + 0

T

Z

+∞

 Z |G(t , x)|dxdt W1 (0) +

|G(t , x)|dxdt



0

.

(3.47)

0

It then follows that T

Z

xj (t )

Z

xi ( t )

0

+∞

 Z |wi (t , x)||wj (t , x)|dxdt ≤ C10 W1 (0) +

(|α 0 (t )| + |h0 (t )|)dt +

T

Z

0

+∞

Z

|G(t , x)|dxdt

0

0

Z

+∞

2

.

(3.48)

Therefore T

Z

L

Z

0



|wi (t , x)||wj (t , x)|dxdt ≤ C10 W1 (0) +

+∞

Z

0

(|α (t )| + |h (t )|)dt + 0

0

T

Z

0

0

|G(t , x)|dxdt

2 (3.49)

0

and the desired conclusion follows by taking L → +∞. (iii) For j ∈ {1, . . . , m} and i ∈ {m + 1, . . . , n}, passing through the point (T , 0), we draw the jth backward characteristic x = xj (t ) (0 ≤ t ≤ T ) which intersects the x-axis at a point (0, xj0 ). Let x = xi (t , xj0 ) (0 ≤ t ≤ T ) be the ith forward characteristic passing through the point (0, xj0 ). Then, we draw the ith backward characteristic x = xi (t )(0 ≤ t ≤ T ) passing through the point (T , L)(L > xi (T , xj0 )). We introduce the ‘‘continuous Glimm’s functional’’ Q (t ) =

Z Z xj (t )
|wj (t , x)||wi (t , y)|dxdy.

(3.50)

By exploiting the same arguments as in (i), we can deduce that T

Z

xi (t )

Z

xj (t )

0



|wi (t , x)||wj (t , x)|dxdt ≤ C11 W1 (0) +

+∞

Z

(|α (t )| + |h (t )|)dt + 0

0

T

Z

0

+∞

Z

|G(t , x)|dxdt

0

0

Z

+∞

2

.

(3.51)

Hence T

Z 0

L

Z

+∞

 Z |wi (t , x)||wj (t , x)|dxdt ≤ C11 W1 (0) +

0

and the desired conclusion follows by taking L → +∞.

(|α 0 (t )| + |h0 (t )|)dt + 0

T

Z 0

|G(t , x)|dxdt 0

2 (3.52)

1360

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

(iv) For i, j ∈ {m + 1, . . . , n} and i 6= j, without loss of generality, we suppose that i < j. Let x = xi (t , 0) (0 ≤ t ≤ T ) be the ith forward characteristic passing through origin O(0, 0). Then, we draw the ith backward characteristic x = xi (t ) (0 ≤ t ≤ T ) passing through the point (T , L) (L > xi (T , 0)). We introduce the ‘‘continuous Glimm’s functional’’

Z Z

Q (t ) =

0
|wj (t , x)||wi (t , y)|dxdy.

(3.53)

Then, it is easy to see that dQ (t ) dt

xi (t )

∂ (|wj (t , x)|)|wi (t , y)|dxdy ∂ 0
= xi (t )|wi (t , xi (t ))| 0

Z

Z Z

|wj (t , x)|dx +

0
0
xi (t )

Z

= (xi (t ) − λi (u(t , xi (t ))))|wi (t , xi (t ))| 0

|wj (t , x)|dx + λj (u(t , 0))|wj (t , 0)|

0 xi (t )

Z −

Z0 Z +

0
0
xi ( t )

Z

xi (t )

Z

|Gj (t , x)|dx

xi (t )

|wi (t , x)|dx +

xi (t )

Z

0

≤ λj (u(t , 0))|wj (t , 0)| +∞

Z

|Gj (t , x)|dx 0

xi (t )

Z

|wi (t , x)|dx − δ0

xi (t )

Z

|wj (t , x)|dx

|wi (t , x)||wj (t , x)|dx

0

+∞

Z

|Gi (t , x)|dx

0

0

+

|wi (t , x)||wj (t , x)|dx

0

+∞

Z

|wj (t , x)|sgn(wi )Gi (t , y)dxdy

0

Z

0

xi (t )

Z

|wi (t , x)|dx − δ0

0

+

|wi (t , x)|dx

0

(λj (u(t , x)) − λi (u(t , x)))|wi (t , x)||wj (t , x)|dx Z Z sgn(wj )Gj (t , x)|wi (t , y)|dxdy +

≤ λj (u(t , 0))|wj (t , 0)|

xi (t )

Z

|wi (t , x)|dx +

+∞

Z

0

|Gi (t , x)|dx 0

+∞

Z

|wj (t , x)|dx.

(3.54)

0

It then follows from Lemma 3.3 that dQ (t ) dt

+ δ0

xi (t )

Z 0

 ×

+∞

 Z |wi (t , x)||wj (t , x)|dx ≤ C12 λj (u(t , 0))|wj (t , 0)| +



|G(t , x)|dx 0

W1 (0) +

+∞

Z

T

Z

(|α 0 (t )| + |h0 (t )|)dt + 0

0

+∞

Z

 |G(t , x)|dxdt .

(3.55)

0

Therefore

δ0

T

Z 0



xi (t )

Z

|wi (t , x)||wj (t , x)|dxdt ≤ Q (0) + C12

Z

0

× W1 (0) +

T

λj (u(t , 0))|wj (t , 0)|dt +

0

+∞

Z

(|α (t )| + |h (t )|)dt + 0

0

0

T

Z 0

+∞

Z

0

+∞

Z

T

Z

|G(t , x)|dxdt



0



|G(t , x)|dxdt .

(3.56)

0

By exploiting the same arguments as in Lemma 3.3, we can deduce that T

Z

λj (u(t , 0))|wj (t , 0)|dt ≤ C13 0

T

Z

|wj (t , 0)|dt ≤ C14 0

( m Z X r =1

T

|wr (t , 0)|dt + 0

+∞

 Z ≤ C15 W1 (0) +

(|α 0 (t )| + |h0 (t )|)dt + 0

)

+∞

Z

(|α (t )| + |h (t )|)dt 0

0

0 T

Z 0

+∞

Z 0

 |G(t , x)|dxdt .

(3.57)

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

1361

Then, noting +∞

Z

Q (0) ≤

|wi (0, x)|dx

+∞

Z

|wj (0, x)|dx,

(3.58)

0

0

it follows from (3.56) and (3.57) that

δ0

T

Z

xi (t )

Z



|wi (t , x)||wj (t , x)|dxdt ≤ C16 W1 (0) +

(|α (t )| + |h (t )|)dt + 0

0

T

Z

+∞

Z

|G(t , x)|dxdt

2

. (3.59)

0

0

0

0

0

+∞

Z

It thus follows T

Z

xi (t )

Z

(|α 0 (t )| + |h0 (t )|)dt +

T

Z

0

0

0

+∞

 Z |wi (t , x)||wj (t , x)|dxdt ≤ C17 W1 (0) +

Z

+∞

|G(t , x)|dxdt

0

0

Z

+∞

2

.

(3.60)

Hence T

Z

L

Z

0

+∞

 Z |wi (t , x)||wj (t , x)|dxdt ≤ C17 W1 (0) +

0

(|α 0 (t )| + |h0 (t )|)dt +

T

Z 0

0

|G(t , x)|dxdt

2 (3.61)

0

and the desired conclusion follows by taking L → +∞. The proof of Lemma 3.4 is finished.



Lemma 3.5. Under the assumptions of Theorem 1.1, on any given domain of existence {(t , x)|0 ≤ t ≤ T , x ≥ 0} of the C 1 solution u = u(t , x) to the mixed initial–boundary value problem (1.1), (1.10) and (1.11), there exists a positive constant K3 independent of ε , T and M, such that the following uniform a priori estimates hold:

e1 (T ) ≤ K3 ε, W1 (T ), W

(3.62)

U∞ (T ), V∞ (T ) ≤ K3 ε

(3.63)

W ∞ ( T ) ≤ K3 M .

(3.64)

and

Proof. We introduce QW (T ) =

n XZ X j=1 i6=j

T

+∞

Z

0

|wi (t , x)||wj (t , x)|dxdt .

(3.65)

0

By (2.13), it follows from Lemma 3.4 that QW (T ) ≤ C18



W1 (0) +

+∞

Z

(|α 0 (t )| + |h0 (t )|)dt +

Z

0

0

T

Z

+∞

|G(t , x)|dxdt

2

.

(3.66)

0

Noting (2.16), we have T

Z 0

+∞

Z

|G(t , x)|dxdt ≤ C19 QW (T ).

(3.67)

0

It follows from (3.66) and (3.67) that QW (T ) ≤ C20



W1 (0) +

+∞

Z

(|α 0 (t )| + |h0 (t )|)dt + QW (T )

2

,

(3.68)

0

Thus, noting (1.18), we have QW (T ) ≤ C21 ε 2 .

(3.69)

We now estimate W1 (T ). By Lemma 3.3, we have +∞

Z



|wi (t , x)|dx ≤ K1 W1 (0) + 0

+∞

Z

(|α (t )| + |h (t )|)dt + 0

0

0

T

Z 0

+∞

 Z ≤ C22 W1 (0) +

0

+∞

Z

|G(t , x)|dxdt



0

 (|α 0 (t )| + |h0 (t )|)dt + QW (T ) .

(3.70)

1362

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

Therefore, noting (1.18), (3.69) and (3.70) we obtain W1 (T ) ≤ C23 ε.

(3.71)

e1 (T ). We next estimate W e1 (T ), we need to estimate To estimate W Z

|wi (t , x)|dt , Cj

where Cj stands for any given jth characteristic on the domain [0, T ] × R+ . Without loss of generality, we assume that Cj intersects the x-axis with point A(0, α), and intersects the line t = T with point B. (i) For i = 1, . . . , m, passing through point B, we draw the ith backward characteristic Ci which intersects the x-axis at a point C (0, β). For fixing the idea, suppose that α < β . We rewrite (2.14) as d(|wi (t , x)|(dx − λi (u)dt )) = sgn(wi )Gi dxdt .

(3.72)

By (3.72), using Stokes’ formula on the domain ABC, we have

Z Z Z Z β |wi (0, x)|dx + |Gi |dxdt |wi (t , x)|(λj (u) − λi (u))dt ≤ Cj α ABC ≤ W1 (0) + C24 QW (T ).

(3.73)

e1 (T ), j 6= i, thus we have from (3.3) that In the definition of W |λj (u) − λi (u)| ≥ δ0 .

(3.74)

Therefore, it follows that

Z

|wi (t , x)|dt ≤ C25 {W1 (0) + QW (T )}.

(3.75)

Cj

(ii) For i = m + 1, . . . , n, we draw the ith backward characteristic Ci passing through point B. Here, there are only two possibilities: (a) The ith backward characteristic Ci intersects the t-axis at a point C (β, 0). By (3.72), using Stokes’ formula on the domain OABC, we have

Z Z Z β Z Z α ≤ |w ( 0 , x )| dx + |λ ( u ( t , 0 ))||w ( t , 0 )| dt + |Gi |dxdt |w ( t , x )|(λ ( u ) − λ ( u )) dt i i i j i Cj i 0 0 OABC Z β  ≤ W1 (0) + C26 |wi (t , 0)|dt + QW (T ) .

(3.76)

0

Therefore, it follows from (3.74) and (3.76) that

 Z |wi (t , x)|dt ≤ C27 W1 (0) +

Z Cj

β

 |wi (t , 0)|dt + QW (T ) .

(3.77)

0

Noting (3.5), by (3.25), we have β

Z

|wi (t , 0)|dt =

0

β

m Z X r =1

≤ C28

|fir (t , u)wr (t , 0)|dt +

0

β

r =1

|wr (t , 0)|dt +

0

|f ij (t , u)αj (t )|dt + 0

0

j=1

( m Z X

β

k Z X

l=m+1

|e fil (t , u)h0l (t )|dt

0

)

+∞

Z

β

Z n X

(|α 0 (t )| + |h0 (t )|)dt .

(3.78)

0

Then, passing through C (β, 0), we draw the rth characteristic Cr (r ∈ {1, . . . , m}) which intersects the x-axis at point D(0, xD ). By (3.27), using Stokes’ formula on the domain COD, we have

Z

β 0

Z |wr (t , 0)|(−λr (u)dt ) ≤

xD

|wr (0, x)|dx +

0

≤ W1 (0) + C29 QW (T ).

Z Z |Gr |dxdt COD

(3.79)

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

1363

Noting (3.4) and (3.5), for sufficiently small δ > 0, it is easy to see that

|λr (u)| ≥

δ0 2

.

(3.80)

Therefore, it follows that β

Z

|wr (t , 0)|dt ≤ C30 {W1 (0) + QW (T )}.

(3.81)

0

Thus



Z

|wi (t , x)|dt ≤ C31 W1 (0) +

+∞

Z



(|α (t )| + |h (t )|)dt + QW (T ) . 0

0

(3.82)

0

Cj

(b) The ith backward characteristic ci intersects the x-axis at a point (0, β). By exploiting the same arguments as in (i), we can deduce that

Z

|wi (t , x)|dt ≤ C32 {W1 (0) + QW (T )}.

(3.83)

Cj

Combining (3.75), (3.82) and (3.83), we have



e1 (T ) ≤ C33 W1 (0) + W

+∞

Z

 (|α 0 (t )| + |h0 (t )|)dt + CQW (T ) ≤ C ε.

(3.84)

0

We next estimate W∞ (T ). (i) For j = 1, . . . , m, passing through any fixed point (t , x) in the domain {(t , x)|0 ≤ t ≤ T , x ≥ 0}, we draw the jth backward characteristic Cj which intersects the x-axis at a point (0, y). By integrating (2.8) along this characteristic Cj , and recalling (2.10), we have

Z n X γjil (u)wi wl dt |wj (t , x)| ≤ |wj (0, y)| + C j i ,l = 1 Z e1 (T ). ≤ W∞ (0) + |γjjj (u)wj2 |dt + C34 W∞ (T )W

(3.85)

Cj

Then, noting (1.8), (2.12) and (3.84), we get

|wj (t , x)| ≤ C35 {M + ε W∞ (T )}.

(3.86)

(ii) For j = m + 1, . . . , n, for any fixed point (t , x) in the domain {(t , x)|0 ≤ t ≤ T , x ≥ 0}, we draw the jth backward characteristic Cj passing through this point. Here, there are only two possibilities: (a) The jth backward characteristic Cj intersects the x-axis at a point (0, α). By exploiting the same arguments as in (i), we can deduce that

|wj (t , x)| ≤ C36 {M + ε W∞ (T )}.

(3.87)

(b) The jth backward characteristic Cj intersects the t-axis at a point (t0 , 0). By integrating (2.8) along Cj from t0 to t, we obtain

wj (t , x) = wj (t0 , 0) +

Z X n

γjkl (u)wk wl dt .

(3.88)

Cj k,l=1

By (3.25), we have

wj (t0 , 0) =

m X

fjr (t0 , u(t0 , 0))wr (t0 , 0) +

r =1

k X

f ji (t0 , u(t0 , 0))αi0 (t0 ) +

i=1

n X

e fjl (t0 , u(t0 , 0))h0l (t0 ).

(3.89)

l=m+1

By employing the same arguments as in (i), we can obtain

|wr (t0 , 0)| ≤ C37 {M + ε W∞ (T )}.

(3.90)

Thus, noting (1.13), it follows from (3.89) and (3.90) that

|wj (t0 , 0)| ≤ C38 {M + ε W∞ (T )} .

(3.91)

Hence, it follows from (3.88) that

|wj (t , x)| ≤ C39 {M + ε W∞ (T )} .

(3.92)

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Combining (3.86) with (3.87) and (3.92), we obtain W∞ (T ) ≤ C40 M .

(3.93)

We finally estimate U∞ (T ) and V∞ (T ). For the time being we assume that V∞ (T ) ≤ Aε , where A is a fixed positive constant which is determined below. (i) For r = 1, . . . , m, passing through any fixed point (t , x) in the domain {(t , x)|0 ≤ t ≤ T , x ≥ 0}, we draw the rth backward characteristic Cr which intersects the x-axis at a point (0, α). By integrating (2.5) along Cr , and recalling (2.7), we have

Z n X e1 (T ). |vr (t , x)| ≤ |vr (0, α)| + βril (u)vi wl dt ≤ |vr (0, α)| + C41 V∞ (T )W Cr i,l=1

(3.94)

Noting (1.15) and using (1.18), we have

|ϕ(x)| ≤ ε,

∀x ∈ R+ .

(3.95)

Thus, it follows from (3.84), (3.94) and (3.95) that

|vr (t , x)| ≤ C42 ε + C43 Aε 2 .

(3.96)

(ii) For j = m + 1, . . . , n, for any fixed point (t , x) in the domain {(t , x)|0 ≤ t ≤ T , x ≥ 0}, we draw the jth backward characteristic Cj : x = xj (s; t , x) passing through this point. Here, there are only two possibilities: (a) The jth backward characteristic Cj intersects the x-axis at a point (0, α). By exploiting the same arguments as in (i), we can deduce that

|vj (t , x)| ≤ C44 ε + C45 Aε 2 .

(3.97)

(b) The jth backward characteristic Cj intersects the t-axis at a point (t0 , 0). By integrating (2.5) along Cj , we have

vj (t , x) = vj (t0 , 0) +

Z tX n

βjil (u)vi wl (s, xj (s; t , x))ds.

(3.98)

t0 i ,l = 1

Noting (1.14), by (1.11), it is easy to see that

vj (t0 , 0) =

m X

gjr (t0 )vr (t0 , 0) + hj (t0 ),

(3.99)

r =1

where gjr (t0 ) =

1

Z 0

∂ Gj (α(t0 ), τ v1 (t0 , 0), . . . , τ vm (t0 , 0))dτ . ∂vr

(3.100)

Noting (1.15) and using the assumption that the conditions of C 1 compatibility are satisfied at the point (0, 0), we have hs (0) = 0 (s = m + 1, . . . , n).

(3.101)

Thus, it follows from (1.18) that hs ( t ) ≤ ε

(s = m + 1, . . . , n).

(3.102)

Then, noting (2.7), (3.5), (3.84) and (3.96), we obtain from (3.98) and (3.99) that

( |vj (t , x)| ≤ C46 ε +

m X

) e1 (T ) |vr (t0 , 0)| + V∞ (T )W

r =1

≤ C47 ε + C48 Aε 2 .

(3.103)

Combining (3.96), (3.97) and (3.103), we have V∞ (T ) ≤ C49 ε + C50 Aε 2 ≤ 2C49 ε,

(3.104)

provided that C50 Aε ≤ C49 . Taking A ≥ 4C49 , we have V∞ (T ) ≤ 2C49 ε ≤

A 2

ε.

(3.105)

Hence, noting (3.95), by continuous induction, we get V∞ ( T ) ≤

A 2

ε.

(3.106)

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

1365

Equivalently, U∞ (T ) ≤ C51 Aε.

(3.107)

Taking K3 suitably large and noting (3.71), (3.84), (3.93), (3.106) and (3.107), we obtain (3.62)–(3.64) immediately. Finally, we observe that when ε > 0 is suitably small, from (3.63), we have U∞ (T ) ≤ K3 ε ≤

1 2

δ.

This implies the validity of the hypothesis (3.5). The proof of Lemma 3.5 is finished.



Proof of Theorem 1.1. Under the assumptions of Theorem 1.1, by (3.63) and (3.64), we know that there exists ε > 0 suitably small such that on any given domain of existence D(T ) = {(t , x)|0 ≤ t ≤ T , x ≥ 0} of the C 1 solution u = u(t , x) to the mixed initial–boundary value problem (1.1), (1.10) and (1.11), we have the following uniform a priori estimate for the C 1 norm of the solution: 4

ku(t , .)kC 1 = ku(t , .)kC 0 + kux (t , .)kC 0 ≤ K4 θ . This leads to the conclusion of Theorem 1.1 immediately. The proof of Theorem 1.1 is finished.

(3.108) 

4. An application of Theorem 1.1 In this section, we use the conclusion of Theorem 1.1 to consider the mixed initial–boundary value problem for the system of the motion of the relativistic string in the Minkowski space–time R1+n . Recall Kong et al.’s work [27] at first. We denote by X = (t , x1 , . . . , xn ) points in the (1 + n)-dimensional Minkowski space R1+n . Then the scalar product of two vectors X and Y = (e t , y1 , . . . , yn ) in R1+n is defined by X ·Y =

n X

xi yi − te t,

(4.1)

i=1

in particular, n X

X2 =

x2i − t 2 .

(4.2)

i=1

The Lorentzian metric of R1+n can be written as ds2 =

n X

dx2i − dt 2 .

(4.3)

i=1

To describe the motion of the relativistic string in the (1 + n)-dimensional Minkowski space R1+n , we consider the local equation of an extremal timelike surface S in R1+n taking the following parameter form in a suitable coordinate system (cf. [27]): xi = xi (t , θ ) (i = 1, . . . , n).

(4.4)

Then, in the surface coordinates t and θ , the Lorentzian metric (4.3) is expressed as ds2 = (dt , dθ )M (dt , dθ )T ,

(4.5)

where,

 M =

|xt |2 − 1 hxt , xθ i

 hxt , xθ i , |xθ |2

(4.6)

in which x = (x1 , . . . , xn )T and

hxt , xθ i =

n X

xi,t xi,θ ,

|xt |2 = hxt , xt i and |xθ |2 = hxθ , xθ i.

(4.7)

i=1

Since the surface S is C 2 and timelike, i. e., det M < 0,

(4.8)

equivalently,

hxt , xθ i2 − (|xt |2 − 1)|xθ |2 > 0,

(4.9)

it follows that the area element of the surface S is dA =

p hxt , xθ i2 − (|xt |2 − 1)|xθ |2 dtdθ .

(4.10)

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The surface S is called an extremal surface, if x = x(t , θ ) is the critical point of the area functional I =

Z Z p

hxt , xθ i2 − (|xt |2 − 1)|xθ |2 dtdθ .

(4.11)

The corresponding Euler–Lagrange equation is (cf. [27])

!

|xθ |2 xt − hxt , xθ ixθ p

−

hxt , xθ i2 − (|xt |2 − 1)|xθ |2

t

!

hxt , xθ ixt − (|xt |2 − 1)xθ p hxt , xθ i2 − (|xt |2 − 1)|xθ |2

= 0.

(4.12)

θ

Let u = xt ,

v = xθ ,

(4.13)

where u = (u1 , . . . , un ) and v = (v1 , . . . , vn ) , Then (4.12) can be equivalently rewritten as T

    

T

!

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

−

p

t

!

hu, viu − (|u|2 − 1)v p

hu, vi2 − (|u|2 − 1)|v|2

= 0,

(4.14)

θ

Now let us consider the mixed initial–boundary value problem for system (4.14) with the initial condition t = 0 : u = u0 (θ),

v =e v0 + v0 (θ ) (θ ≥ 0)

(4.15)

and the boundary condition

θ = 0 : u = 0 (t ≥ 0).

(4.16)

q Here, e v0 = (e v10 , . . . ,e vn0 )T is a constant vector with |e v10 )2 + · · · + (e vn0 )2 > 0, (u0 (θ )T , v0 (θ )T ) ∈ C 1 with bounded v0 | = (e C 1 norm, such that

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

(4.17) 1

for some positive constant M (bounded but possibly large). Also, we assume that the conditions of C compatibility are satisfied at the point (0, 0). Let

  U =

u

v

.

(4.18)

We can rewrite system (4.14) as Ut + A(U )Uθ = 0,

(4.19)

where



2hu, vi

In × n − A(U ) =  |v|2 −In×n

 |u|2 − 1 In × n  . |v|2

(4.20)

0

  0

It is easy to see that in a neighborhood of U0 = e v0 , (4.14) is a hyperbolic system with the following real eigenvalues:

λ1 (U ) ≡ · · · ≡ λn (U ) = λ− < 0 < λn+1 (U ) ≡ · · · ≡ λ2n (U ) = λ+ ,

(4.21)

where

λ± =

−hu, vi ±

p

hu, vi2 − (|u|2 − 1)|v|2 . |v|2

(4.22)

The corresponding left and right eigenvectors are li (U ) = (ei , λ+ ei )

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

li (U ) = (ei−n , λ− ei−n )

(i = n + 1, . . . , 2n)

(4.23)

and ri (U ) = (−λ− ei , ei )T

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

ri (U ) = (−λ+ ei−n , ei−n )T

(i = n + 1, . . . , 2n)

(4.24)

respectively, where (i)

ei = (0, . . . , 0, 1 , 0, . . . , 0)

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

(4.25)

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

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When n = 1, (4.14) is a strictly hyperbolic system; while, when n ≥ 2, (4.14) is a non-strictly hyperbolic system with characteristics with constant multiplicity. It is easy to see that all characteristic fields are linearly degenerate in the sense of Lax, i.e.,

∇λi (U )ri (U ) ≡ 0 (i = 1, . . . , 2n),

(4.26)

see [27]. Let Vi = li (U )(U − U0 )

(i = 1, . . . , 2n).

(4.27)

Then, the boundary condition (4.16) can be rewritten as

θ = 0 : Vn+i = −Vi ,

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

(4.28)

By Theorem 1.1 we get Theorem 4.1. Suppose that u0 , v0 are all C 1 functions with respect to their arguments, for which there is a constant M > 0 such that

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

(4.29)

Suppose furthermore that the conditions of C 1 compatibility are satisfied at the point (0, 0). Then there exists a small positive constant ε independent of M such that, if (4.29) holds together with +∞

Z

|u0 (θ )|dθ , 0

0

+∞

Z

|v00 (θ )|dθ ≤ ε,

(4.30)

0

then the mixed initial–boundary value problem (4.14)–(4.16) admits a unique global C 1 solution U = U (t , θ ) in the half space {(t , θ)|t ≥ 0, θ ≥ 0}. Acknowledgment The author thanks Prof. Ta-tsien Li for his valuable lectures and guidance in Fuzhou in December, 2003. The author also wishes to thank Prof. Jiaxing Hong for his guidance and constant encouragement. The author was supported by the National Natural Science Foundation of China (Grant 70371025), the Scientific Research Foundation of the Ministry of Education of China (Grant 02JA790014), the National Natural Science Foundation of Fujian Province (Grant JB07021), and the Science and Technology Developmental Foundation of Fuzhou University (Grant 2004-XQ-16). References [1] T.-T. Li, L.-B. Wang, Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems, Discrete Contin. Dyn. Syst. 12 (2005) 59–78. [2] Z.-Q. Shao, Global weakly discontinuous solutions for hyperbolic conservation laws in the presence of a boundary, J. Math. Anal. Appl. 345 (2008) 223–242. [3] Z.-Q. Shao, Global solution to the generalized Riemann problem in the presence of a boundary and contact discontinuities, J. Elasticity 87 (2007) 277–310. [4] Z.-Q. Shao, Blow-up of solutions to the initial-boundary value problem for quasilinear hyperbolic systems of conservation laws, Nonlinear Anal. 68 (2008) 716–740. [5] A. Bressan, Contractive metrics for nonlinear hyperbolic systems, Indiana Univ. Math. J. 37 (1988) 409–421. [6] A. Bressan, Hyperbolic Systems of Conservation Laws: The One-dimensional Cauchy Problem, Oxford University Press, 2000. [7] 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. [8] L. Hörmander, The Lifespan of Classical Solutions of Nonlinear Hyperbolic Equations, in: Lecture Notes in Mathematics, vol. 1256, Springer, Berlin, 1987. [9] D.-X. Kong, Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data, Chinese Ann. Math. Ser. B 21 (2000) 413–440. [10] D.-X. Kong, Cauchy Problem for Quasilinear Hyperbolic Systems, in: MSJ Memoirs, vol. 6, Mathematical Society of Japan, Tokyo, 2000. [11] T.-T. Li, D.-X. Kong, Y. Zhou, Global classical solutions for general quasilinear non-strictly hyperbolic systems with decay initial data, Nonlinear Stud. 3 (1996) 203–229. [12] T.-T. Li, Y. Zhou, D.-X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Differential 19 (1994) 1263–1317. [13] 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. [14] T.-P. Liu, Development of singularities in the nonlinear waves for quasilinear hyperbolic partial differential equations, J. Differential Equations 33 (1979) 92–111. [15] T.-P. Liu, Quasilinear hyperbolic systems, Comm. Math. Phys. 68 (1979) 141–172. [16] T. Yang, C.-J. Zhu, H.-J. Zhao, Global smooth solutions for a class of quasilinear hyperbolic systems with dissipative terms, Proc. Roy. Soc. Edinburgh 127A (1997) 1311–1324. [17] Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chinese Ann. Math. Ser. B 25 (2004) 37–56. [18] L. Hsiao, P. Marcati, Nonhomogeneous quasilinear hyperbolic system arising in chemical engineering, Ann. Sc. Norm. Super. Pisa Cl. Sci. 15 (1988) 65–97.

1368

Z.-Q. Shao / Nonlinear Analysis 71 (2009) 1350–1368

[19] L. Hsiao, R.-H. Pan, Initial boundary value problem for the system of compressible adiabatic flow through porous media, J. Differential Equations 159 (1999) 280–305. [20] L. Hsiao, R.-H. Pan, The damped p-system with boundary effects, Contemp. Math. 255 (2000) 109–123. [21] T.-T. Li, Y. Jin, Semi-global C 1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems, Chinese Ann. Math. Ser. B 22 (2001) 325–336. [22] T.-T. Li, Y.-J. Peng, The mixed initial-boundary value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics, Nonlinear Anal. 52 (2003) 573–583. [23] F. Liu, Mixed initial boundary value problem for hyperbolic geometric flow on Riemann surfaces, J. Math. Anal. Appl. (2008) doi:10.1016/j.jmaa.2008. 10.043. [24] T.-T. Li, W.-C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Press, 1985. [25] M. Schatzman, Continuous Glimm functional and uniqueness of the solution of Riemann problem, Indiana Univ. Math. J. 34 (1985) 533–589. [26] M. Schatzman, The Geometry of Continuous Glimm Functionals, in: Lectures in Applied Mathematics, vol. 23, American Mathematical Society, Providence, RI, 1986, pp. 417–439. [27] 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.