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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- Gasnikov, A.V. Comput. Math. and Math. Phys. (2008) 48: 1376. doi:10.1134/S0965542508080095
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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.