Global Classical Solutions to the Mixed Initial-boundary Value Problem for a Class of Quasilinear Hyperbolic Systems of Balance Laws

T i i r (u) r (u) 1 (i 1,...., n) ≡ = (1.6) Where ij δ stands for the Kronecker's symbol. Clearly, all i ij ij (u), l (u) and r (u)(i, j 1,..., n) λ = (1.7) have the same regularity as A(u), i.e., C2 regularity. We assume that on the domain under consideration, each characteristic with positive velocity is weakly linearly degenerate and the eigenvalues of A(u) f(u) = ∇ (1.8) satisfy the non-characteristic condition. r(u) 0 s(u) (r 1,....,m;s m 1,..., n) λ < < λ = = +  (1.9) r(u) 0 s(u) (r 1,....,m;s m 1,..., n) λ < < λ = = + (1.10) We are concerned with the existence and uniqueness of global C1 solutions to the mixed initial-boundary value problem for system (1.1) in the half space D {(t, x) | t 0, x 0} = ≥ ≥ (1.11) with the initial condition: t 0 : u (x)(x 0) = = φ ≥ (1.12) and the nonlinear boundary condition:


Introduction and Main Result
Consider the following quasilinear hyperbolic system of balance laws in one space dimension: where L>0 is a constant; u=(u 1 ,…., u n ) T is the unknown vector function of (t, x), f(u) is a given C 3 vector function of u.
We assume that on the domain under consideration, each characteristic with positive velocity is weakly linearly degenerate and the eigenvalues of A(u) f(u) = ∇ (1.8) satisfy the non-characteristic condition.
r(u) 0 s(u) (r 1,...., m;s m 1,..., n) λ < < λ = = + (1.9) r(u) 0 s(u) (r 1,...., m;s m 1,..., n) λ < < λ = = + (1.10) We are concerned with the existence and uniqueness of global C 1 solutions to the mixed initial-boundary value problem for system (1.1) in the half space with the initial condition: t 0 : u (x)(x 0) == ϕ ≥ (1.12) and the nonlinear boundary condition: are all C 1 functions with respect to their arguments, which satisfy the conditions of C 1 compatibility at the point (0; 0). Also, we assume that there exists a constant µ >0 such that For the special case where (1.1) is a quasilinear hyperbolic system of conservation laws, i.e., L=0, such kinds of problems have been extensively studied (for instance, [1][2][3][4][5][6][7][8] and the references therein). In particular, Li and Wang proved the existence and uniqueness of global C 1 solutions to the mixed initial boundary value problem for first order quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space {(t, x | t 0, x 0 |)} ≥ ≥ . On the other hand, for quasilinear hyperbolic systems of balance laws, many results on the existence of global solutions have also been obtained by Liu, et al., (for instance, see [8][9][10][11][12][13][14] and the references therein), and some methods have been established. So the following question arises naturally: when can we obtain the existence and uniqueness of semiglobal C1 solutions for quasilinear hyperbolic systems of balance laws? It is well known that for first-order quasilinear hyperbolic systems of balance laws, generically speaking, the classical solution exists only locally in time and the singularity will appear in a finite time even if the data are sufficiently smooth and small [15][16][17][18][19][20]. However, in some cases global existence in time of classical solutions can be obtained. In this paper, we will generalize the results in [21] to a nonhomogeneous quasilinear hyperbolic system, the analysis relies on a careful study of the interaction of the nonhomogeneous term. Our main results can be stated as follows: Theorem 1.1. Suppose that the non-characteristic condition (1.10) holds and system (1.1) is strictly hyperbolic. Suppose furthermore that for j = m + 1,…., n; each j-characteristic field with positive velocity is weakly linearly degenerate. Suppose finally that The rest of this paper is organized as follows. In Section 2, we give the main tools of the proof that is several formulas on the decomposition of waves for system (1.1). Then, the main result will be proved in Section 3. Finally, an application is given in Section 4 [22].

Decomposition of Waves
Suppose that on the domain under consideration, system (1.1) is strictly hyperbolic and (1.2)-(1.6) hold.
Suppose that k A(u) C ∈ where k is an integer ¸ 1. By Lemma 2.5 in [23], there exists an invertible k 1 C + transformation u u(u)(u(0) 0) = =  such that in u  -space for each i = 1,…, n, the ith characteristic trajectory passing through u 0 =  coincides with the i u  -axis at least for i | u |  small, namely, be the directional derivative along the ith characteristic. Our aim in this section is to prove several formulas on the decomposition of waves for system (1.1), which will play an important role in our discussion.
On the other hand, we have  Where ijk(u) γ  is given by (2.8) and Proof. Differentiating the first equation of (2.27) with respect to y gives Then, noting (2.6), it follows from (2.31) that Then along the ith and for i = m + 1; : : : ; n; let where n>0 is suitably small (Figure 1).
Noting that n > 0 is small, by (3.3), it is easy to see that where c and C are positive constants independent of T. Proof. Differentiating the first equation of (2.27) with respect to y gives Then, noting (2.19), it follows from (2.44) that

Proof of Theorem 1.1
By the existence and uniqueness of a local C1 solution for quasilinear hyperbolic systems [22], there exists T0>0 such that the mixed initial-boundary value problem (1.1) and (1.12)-(1.13) admits a unique C 1 solution u=u(t, x) on the domain Thus, in order to prove Theorem 1.1 it suffices to establish a uniform a priori estimate for the C 0 norm of u and u x on any given domain of existence of the C 1 solution u = u(t; x).
Thus, there exist sufficiently small positive constants δ and 0 δ such that For the time being it is supposed that on the domain of existence of the C 1 solution u = u(t; x) to the mixed initial-boundary value problem (1.1) and (1.12)-(1.13), we have At the end of the proof of Lemma 3.3, 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 piecewise C 0 norm of v and w defined by (1.14) and (2.1) on the domain of existence of the ) t x ( (0) ) t} Where j C  denotes any given jth characteristic in In the present situation, similar to the corresponding result in [24,[30][31][32][33], we have to the mixed initialboundary value problem (1.1) and (1.12)-(1.13), we have the following uniform a priori estimates: where here and henceforth, ki(i=1; 2,….) are positive constants independent of θ and T.
By integrating (2.6) along j (s; t, x) ξ = ξ and noting (2.9) and (2.11), we have By using Lemma 3.2 and noting (3.54) and (3.57), it is easy to see that By Hadamard's formula, we have Thus, noting the fact that L>0, and using (3.13) and (3.54), we obtain from (3.59) that Similar to Lemma 3.2 in [21], differentiating the nonlinear boundary condition (1.13) with respect to t, we get By (1.1), (1.3) and (2.4), it is easy to see that Therefore it follows from (3.63)-(3.65) that where B1(u) is a matrix whose elements are all C1 functions of u, which satisfy in which Fs(s = m + 1,…,n) are continuous functions of t and u. x 0 : ws Proof. Noting (3.4), it is easy to see that On the other hand, by (3.8), we have Noting the fact that By employing the same arguments as in (i), we can obtain  0) ) y) if t t t (resp.t t t ) which gives a one-to-one correspondence t = t(y) between the segment . Thus, the integral on j C  with respect to t can be reduced to the integral with respect to y. Differentiating (3.80) with respect to t gives i i i (t, y) 1 (u(t, x (t, y))) (u(t, x (t, y)))  it suffices to estimate | q (t, x (t, y)) | | q (t, x (t, y)) | Similarly, we have

T W T W T V T V T U T V T U T V T T V T W T W T V T W T V T U T V T
Hence, noting the fact that L > 0, we obtain from (3.123) that Combining (3.122) and (3.128), we get 2 37 1 1 We next estimate 1 (T) V 

and V1(T).
For i=m + 1,…, n, for any given jth characteristic as in the proof of (3.90), in order to estimate 1 (T) V  it suffices to estimate        Thus, by continuity there exist positive constants k2; k3; k4; k5; k6; k7 and k8 independent of µ, such that (3.47)-(3.53) hold at least for
Finally, we observe that when µ0 > 0 is suitably small, by (3.52) we have  In addition, we assume that By (4.4), it is easy to see that in a neighborhood of 0 0 0 u v ae ö ÷ ç ÷ = ç ÷ ç ÷ ç è ø  system (4.1) is strictly hyperbolic and has the following two distinct real eigenvalues: The corresponding right eigenvectors are T T 1 2 (u) / /( (w)) , (u) / /(1 (w))) r r s s ¢ ¢ - (4.9) It is easy to see that in a neighborhood of 0 0 0 u v ae ö ÷ ç ÷ = ç ÷ ç ÷ ç è ø  all characteristics are linearly degenerate, then weakly linearly degenerate, provided that (w) 0, | w | s¢¢ º " small (4.10) The corresponding left eigenvectors can be taken as Suppose finally that 1 (t) C h Î satisfies (4.14) and that the conditions of C1 compatibility are satisfied at the point (0; 0). Then there is a sufficiently small 0 0 q > such that for any given