Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system

We show that the weak solutions of the nonlinear hyperbolic system {Mathematical expression} converge, as ε tends to zero, to the solutions of the reduced problem {Mathematical expression}. Then they satisfy the nonlinear parabolic equation {Mathematical expression}. The limiting procedure is carried out using the techniques of "Compensated Compactness". Some connections with the theory of nonlinear heat conduction and the theory of nonlinear diffusion in a porous medium are suggested. The main result is stated in th. (2.9).