, Volume 48, Issue 8, pp 1376-1405
Date: 20 Aug 2008

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

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Abstract

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.

Original Russian Text © A.V. Gasnikov, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 8, pp. 1458–1487.