Two Results Concerning Asymptotic Behavior of Solutions of the Burgers Equation with Force

2010 ◽  
pp. 501-514
Author(s):  
Ya. G. Sinai
Keyword(s):  
Asymptotic Behavior ◽  
Burgers Equation ◽  
Asymptotic Behavior Of Solutions

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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