scholarly journals Asymptotic behavior of solutions of 1D-Burgers equation with quasi-periodic forcing

1998 ◽  
Vol 11 (2) ◽  
pp. 219 ◽  
Author(s):  
Ya. G. Sinai
Keyword(s):  
Asymptotic Behavior ◽  
Burgers Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Periodic Forcing

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO CONSERVATION LAWS WITH PERIODIC FORCING

2006 ◽  
Vol 03 (02) ◽  
pp. 387-401 ◽  
Author(s):  
DEBORA AMADORI ◽  
DENIS SERRE
Keyword(s):  
Asymptotic Behavior ◽  
Steady State ◽  
Initial Data ◽  
Conservation Law ◽  
Inflection Point ◽  
Global Minimum ◽  
Asymptotic Behavior Of Solutions ◽  
Periodic Forcing ◽  
Scalar Conservation Law ◽  
Scalar Conservation

We consider the asymptotic behavior of the solution of a forced scalar conservation law, where both the forcing and the initial data are periodic. We prove that there exists a steady state toward which the solution converges in the L1 norm, as t → ∞. We do not assume any smallness or smoothness of the initial data. The limit steady state can be discontinuous, as effect of resonance, and it can be identified when the potential of the forcing term has a unique global minimum, thanks to the conservation of mass. The flux is assumed to be strictly convex; the relevance of this assumption is justified by the construction of solutions with a lattice of periods for a flux with an inflection point.

scholarly journals Asymptotic behavior of solutions toward the rarefaction waves to the Cauchy problem for the generalized Benjamin–Bona–Mahony–Burgers equation with dissipative term

2021 ◽  
pp. 1-19
Author(s):  
Natsumi Yoshida
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Rarefaction Wave ◽  
Fourth Order ◽  
Burgers Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Far Field ◽  
Rarefaction Waves ◽  
Dissipative Term ◽  
The Cauchy Problem

In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem with the far field condititon for the generalized Benjamin–Bona–Mahony–Burgers equation with a fourth-order dissipative term. When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, it is proved that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity. We can further obtain the same global asymptotic stability of the rarefaction wave to the generalized Korteweg–de Vries–Benjamin–Bona–Mahony–Burgers equation with a fourth-order dissipative term as the former one.

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