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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:

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