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

摘要

The time asymptotic behavior of a solution to the initial Cauchy problem for a quasilinearparabolic equation is investigated. Such equations arise, for example, in traffic flow modeling. The mainresult of this paper is the proof of the previously formulated conjecture that, if a monotone initial func-tion has limits at plus and minus infinity, then the solution to the Cauchy problem converges in form toa system of traveling and rarefaction waves; furthermore, the phase shifts of the traveling waves maydepend on time. It is pointed out that the monotonicity condition can be replaced with the boundednesscondition.