Geodesic
Time-asymptotic behaviour of a~solution of the Cauchy initial-value problem for a~conservation law with non-linear divergent viscosity
Izvestiya. Mathematics , Tome 73 (2009) no. 6, pp. 1111-1148

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We study the time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity. We shall prove that when a bounded measurable initial function has limits at $\pm\infty$, a solution of the Cauchy initial-value problem converges uniformly to a system of waves consisting of travelling waves and rarefaction waves, where the phase shifts of the travelling waves are allowed to depend on time. The rate of convergence is estimated under additional conditions on the initial function.
Keywords: conservation law with non-linear divergent viscosity, equation of Burgers type, asymptotics of solutions, convergence on the phase plane, travelling wave, rarefaction wave, system of waves, maximum principle, comparison principle (on the phase plane), inequality of Kolmogorov type.
Mots-clés : convergence in form 
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     author = {A. V. Gasnikov},
     title = {Time-asymptotic behaviour of a~solution of the {Cauchy} initial-value problem for a~conservation law with non-linear divergent viscosity},
     journal = {Izvestiya. Mathematics },
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A. V. Gasnikov. Time-asymptotic behaviour of a~solution of the Cauchy initial-value problem for a~conservation law with non-linear divergent viscosity. Izvestiya. Mathematics , Tome 73 (2009) no. 6, pp. 1111-1148. http://geodesic.mathdoc.fr/item/IM2_2009_73_6_a2/