Classical solutions of hyperbolic initial boundary-value problems with state-dependent delays

We consider the initial boundary-value problem for a system of quasilinear partial functional differential equations of the first order

$ {*{20}{c}} {{partial_t}{z_i}left( {t,x} right) + sumlimits_{j = 1}^n {{rho_{ij}}left( {t,x,Vleft( {z;t,x} right)} right){partial_{{x_j}}}{z_i}left( {t,x} right) = {G_i}left( {t,x,Vleft( {z;t,x} right)} right),} } hfill & {1 leq i leq m,} hfill $ begin{array}{*{20}{c}} {{partial_t}{z_i}left( {t,x} right) + sumlimits_{j = 1}^n {{rho_{ij}}left( {t,x,Vleft( {z;t,x} right)} right){partial_{{x_j}}}{z_i}left( {t,x} right) = {G_i}left( {t,x,Vleft( {z;t,x} right)} right),} } hfill & {1 leq i leq m,} hfill end{array}
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