This article is cited in 4 scientific papers (total in 4 papers)
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
A. V. Gasnikov
Moscow Institute of Physics and Technology (MFTI, State University), per. Institutskii 9, Dolgoprudnyi, Moscow oblast, 141700, Russia
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.
conservation law with nonlinear divergent viscosity, convergence in form, traveling wave, rarefaction wave, system of waves, Cauchy problem for a quasilinear parabolic equation.
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Computational Mathematics and Mathematical Physics, 2008, 48:8, 1376–1405
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”, Zh. Vychisl. Mat. Mat. Fiz., 48:8 (2008), 1458–1487; Comput. Math. Math. Phys., 48:8 (2008), 1376–1405
Citation in format AMSBIB
\paper 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
\jour Zh. Vychisl. Mat. Mat. Fiz.
\jour Comput. Math. Math. Phys.
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This publication is cited in the following articles:
Gasnikov A.V., “On the intermediate asymptotic of the solution to the Cauchy problem for a quasilinear equation of parabolic type with a monotone initial condition”, J. Comput. Systems Sci. Internat., 47:3 (2008), 475–484
A. V. Gasnikov, “Time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity”, Izv. Math., 73:6 (2009), 1111–1148
A. V. Gasnikov, “On the velocity of separation between two successive traveling waves in the asymptotics of the solution to the Cauchy problem for a Burgers-type equation”, Comput. Math. Math. Phys., 52:6 (2012), 937–939
Henkin G.M., “Burgers Type Equations, Gelfand's Problem and Schumpeterian Dynamics”, J. Fixed Point Theory Appl., 11:2 (2012), 199–223
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