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Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system. (English) Zbl 0617.35078

We show that the weak solutions of the nonlinear hyperbolic system \[ \epsilon u_ t^{\epsilon}+p(v^{\epsilon})_ x=-u^{\epsilon},\quad v_ t^{\epsilon}-u_ x^{\epsilon}=0, \] converge, as \(\epsilon\) tends to zero, to the solutions of the reduced problem \(u+p(v)_ x=0\), \(v_ t-u_ x=0\). Then they satisfy the nonlinear parabolic equation \(v_ t+p(v)_{xx}=0.\)
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.

MSC:

35L60 First-order nonlinear hyperbolic equations
35B25 Singular perturbations in context of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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References:

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