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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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##### References:
 [1] Kawashima, Japan J. Math. (N.S.) 7 pp 1– (1981) [2] Kato, Perturbation Theory for Linear Operators (1976) · Zbl 0342.47009 · doi:10.1007/978-3-642-66282-9 [3] DOI: 10.1002/cpa.3160030302 · Zbl 0039.10403 · doi:10.1002/cpa.3160030302 [4] DOI: 10.1073/pnas.68.8.1686 · Zbl 0229.35061 · doi:10.1073/pnas.68.8.1686 [5] DOI: 10.1007/BF03167068 · Zbl 0634.76120 · doi:10.1007/BF03167068 [6] DOI: 10.3792/pjaa.58.384 · Zbl 0522.76098 · doi:10.3792/pjaa.58.384 [7] Liu, Mem. Math. Soc. 328 pp 56– (1985) [8] Nishida, Patterns and Waves, Qualitative Analysis of Nonlinear Differential Equations 18 (1986) [9] DOI: 10.1002/cpa.3160300605 · Zbl 0358.35014 · doi:10.1002/cpa.3160300605 [10] DOI: 10.1002/cpa.3160100406 · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 [11] Shizuta, Hokkaido Math. J. 14 pp 249– (1985) · Zbl 0587.35046 · doi:10.14492/hokmj/1381757663
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