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Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications. (English) Zbl 0653.35066
The large-time behaviour of solutions to the initial value problem \[ f\quad 0(u)_ t+f(u)_ x=(G(u)u_ x)_ x;\quad u(0,x)=u_ 0(x), \] is studied. Here u(t,x) is a vector valued function ranging in \({\mathbb{R}}^ m,\) f 0 and f are smooth mappings \({\mathbb{R}}\) \(m\to {\mathbb{R}}^ m,\) while G is a smooth matrix valued function on \({\mathbb{R}}^ m.\) In particular, it is shown that the solution approaches a superposition of the nonlinear and the linear diffusion waves representing self-similar solutions of the Burgers equations and the linear heat equation.
Reviewer: S.V.Duzhin

MSC:
35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
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