Geodesic
Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 8, pp. 1458-1487

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The time asymptotic behavior of a solution to the initial Cauchy problem for a quasilinear parabolic equation is investigated. Such equations arise, for example, in traffic flow modeling. The main result of this paper is the proof of the previously formulated conjecture that, if a monotone initial function has limits at plus and minus infinity, then the solution to the Cauchy problem converges in form to a system of traveling and rarefaction waves; furthermore, the phase shifts of the traveling waves may depend on time. It is pointed out that the monotonicity condition can be replaced with the boundedness condition. 
@article{ZVMMF_2008_48_8_a8,
     author = {A. V. Gasnikov},
     title = {Convergence in the form of a solution to the {Cauchy} problem for a~quasilinear parabolic equation with a~monotone initial condition to a~system of waves},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1458--1487},
     publisher = {mathdoc},
     volume = {48},
     number = {8},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a8/}
}
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A. V. Gasnikov. Convergence in the form of a solution to the Cauchy problem for a quasilinear parabolic equation with a monotone initial condition to a system of waves. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 8, pp. 1458-1487. http://geodesic.mathdoc.fr/item/ZVMMF_2008_48_8_a8/