
Previous Article
The Riemann problem for scalar CJcombustion model without convexity
 DCDS Home
 This Issue
 Next Article
More results on the decay of solutions to nonlinear, dispersive wave equations
1.  Department of Mathematics and Applied Research Laboratory, The Pennsylvania State University, University Park, PA 16802, United States 
2.  Mathematical Sciences, Loughborough University, Loughborough, Leics. LE11 3TU, United Kingdom 
$u_{t}+u_x+u^pu_{x}\nu u_{x x}u_{x xt}=0$ ($*$)
is considered for $\nu > 0$ and $p \geq 1.$ Complementing recent studies which determined sharp decay rates for these kind of nonlinear, dispersive, dissipative wave equations, the present study concentrates on the more detailed aspects of the longterm structure of solutions. Scattering results are obtained which show enhanced decay of the difference between a solution of ($*$) and an associated linear problem. This in turn leads to explicit expressions for the largetime asymptotics of various norms of solutions of these equations for general initial data for $p > 1$ as well as for suitably restricted data for $p \geq 1.$ Higherorder temporal asymptotics of solutions are also obtained. Our techniques may also be applied to the generalized Kortewegde VriesBurgers equation
$u_{t}+u_x+u^pu_{x}\nu u_{x x}+u_{x x x}=0,$ ($**$)
and in this case our results overlap with those of Dix. The decay of solutions in the spatial variable $x$ for both ($*$) and ($**$) is also considered.
[1] 
Jerry L. Bona, Laihan Luo. Largetime asymptotics of the generalized BenjaminOnoBurgers equation. Discrete and Continuous Dynamical Systems  S, 2011, 4 (1) : 1550. doi: 10.3934/dcdss.2011.4.15 
[2] 
Zhaosheng Feng, Yu Huang. Approximate solution of the BurgersKortewegde Vries equation. Communications on Pure and Applied Analysis, 2007, 6 (2) : 429440. doi: 10.3934/cpaa.2007.6.429 
[3] 
JeanClaude Saut, Yuexun Wang. Long time behavior of the fractional Kortewegde Vries equation with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 11331155. doi: 10.3934/dcds.2020312 
[4] 
Eduardo Cerpa. Control of a Kortewegde Vries equation: A tutorial. Mathematical Control and Related Fields, 2014, 4 (1) : 4599. doi: 10.3934/mcrf.2014.4.45 
[5] 
M. Agrotis, S. Lafortune, P.G. Kevrekidis. On a discrete version of the KortewegDe Vries equation. Conference Publications, 2005, 2005 (Special) : 2229. doi: 10.3934/proc.2005.2005.22 
[6] 
Guolian Wang, Boling Guo. Stochastic Kortewegde Vries equation driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 52555272. doi: 10.3934/dcds.2015.35.5255 
[7] 
Muhammad Usman, BingYu Zhang. Forced oscillations of the Kortewegde Vries equation on a bounded domain and their stability. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 15091523. doi: 10.3934/dcds.2010.26.1509 
[8] 
Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Kortewegde Vries equation. Discrete and Continuous Dynamical Systems  B, 2009, 11 (3) : 655668. doi: 10.3934/dcdsb.2009.11.655 
[9] 
Pierre Garnier. Damping to prevent the blowup of the kortewegde vries equation. Communications on Pure and Applied Analysis, 2017, 16 (4) : 14551470. doi: 10.3934/cpaa.2017069 
[10] 
Ahmat Mahamat Taboye, Mohamed Laabissi. Exponential stabilization of a linear Kortewegde Vries equation with input saturation. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021052 
[11] 
Julie Valein. On the asymptotic stability of the Kortewegde Vries equation with timedelayed internal feedback. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021039 
[12] 
Belkacem SaidHouari. Longtime behavior of solutions of the generalized Kortewegde Vries equation. Discrete and Continuous Dynamical Systems  B, 2016, 21 (1) : 245252. doi: 10.3934/dcdsb.2016.21.245 
[13] 
Mudassar Imran, Youssef Raffoul, Muhammad Usman, Chi Zhang. A study of bifurcation parameters in travelling wave solutions of a damped forced Korteweg de VriesKuramoto Sivashinsky type equation. Discrete and Continuous Dynamical Systems  S, 2018, 11 (4) : 691705. doi: 10.3934/dcdss.2018043 
[14] 
Weiguo Zhang, Yujiao Sun, Zhengming Li, Shengbing Pei, Xiang Li. Bounded traveling wave solutions for MKdVBurgers equation with the negative dispersive coefficient. Discrete and Continuous Dynamical Systems  B, 2016, 21 (8) : 28832903. doi: 10.3934/dcdsb.2016078 
[15] 
Brian Pigott. Polynomialintime upper bounds for the orbital instability of subcritical generalized Kortewegde Vries equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 389418. doi: 10.3934/cpaa.2014.13.389 
[16] 
Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Kortewegde Vries equation on a treeshaped network. Evolution Equations and Control Theory, 2020, 9 (3) : 673692. doi: 10.3934/eect.2020028 
[17] 
Ludovick Gagnon. Qualitative description of the particle trajectories for the Nsolitons solution of the Kortewegde Vries equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 14891507. doi: 10.3934/dcds.2017061 
[18] 
Arnaud Debussche, Jacques Printems. Convergence of a semidiscrete scheme for the stochastic Kortewegde Vries equation. Discrete and Continuous Dynamical Systems  B, 2006, 6 (4) : 761781. doi: 10.3934/dcdsb.2006.6.761 
[19] 
Roberto A. CapistranoFilho, Shuming Sun, BingYu Zhang. General boundary value problems of the Kortewegde Vries equation on a bounded domain. Mathematical Control and Related Fields, 2018, 8 (3&4) : 583605. doi: 10.3934/mcrf.2018024 
[20] 
Qifan Li. Local wellposedness for the periodic Kortewegde Vries equation in analytic Gevrey classes. Communications on Pure and Applied Analysis, 2012, 11 (3) : 10971109. doi: 10.3934/cpaa.2012.11.1097 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]