Asymptotic Behavior of Solutions to the Generalized BBM-Burgers Equation

Asymptotic Behavior of Solutions to the Generalized BBM-Burgers Equation We investigate the asymptotic behavior of solutions of the initial-boundary value problem for the generalized BBM-Burgers equation u t + f(u) x = u xx + u xxt on the half line with the conditions u(0, t) = u −, u(∞, t) = u + and u − < u +, where the corresponding Cauchy problem admits the rarefaction wave as an asymptotic states. In the present problem, because of the Dirichlet boundary, the asymptotic states are divided into five cases depending on the signs of the characteristic speeds f′(u ±) of boundary state u − = u(0) and the far fields states u + = u(∞). In all cases both global existence of the solution and asymptotic behavior are shown under the smallness conditions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Asymptotic Behavior of Solutions to the Generalized BBM-Burgers Equation

Asymptotic Behavior of Solutions to the Generalized BBM-Burgers Equation

Acta Mathematicae Applicatae Sinica, English Series Vol. 21, No. 1 (2005) 31–42 Asymptotic Behavior of Solutions to the Generalized BBM-Burgers Equation 1 2 Mi-na Jiang , Yan-ling Xu Laboratory of Nonlinear Analysis, Department of Mathematics, Central China Normal University, Wuhan 430079, China (E-mail: jmn3911@sina.com) Department of Information and Computer Science, College of Science, Huazhong Agriculture University, Wuhan 430079, China Abstract We investigate the asymptotic behavior of solutions of the initial-boundary value problem for the generalized BBM-Burgers equation u + f (u) = u + u on the half line with the conditions u(0,t)= t x xx xxt u ,u(∞,t)= u and u <u , where the corresponding Cauchy problem admits the rarefaction wave as an − + − + asymptotic states. In the present problem, because of the Dirichlet boundary, the asymptotic states are divided into five cases depending on the signs of the characteristic speeds f (u ) of boundary state u = u(0) and ± − the far fields states u = u(∞). In all cases both global existence of the solution and asymptotic behavior are shown under the smallness conditions. Keywords BBM-Burgers equation; stationary solution; rarefaction wave; a priori estimate; L -energy method 2000 MR Subject Classification 35B40; 35B45; 35Q53; 35L65 1 Introduction We consider the initial-boundary value problem on the half line R R R =(0, ∞) for the generalized BBM-Burgers equation: u + f (u) = u + u,x ∈ R R R ,t> 0, t x xx xxt u(x, t)| = u,t> 0, x=0 − (1.1) u ,x =0, ⎪ − u(x, t)| = u (x)= t=0 0 u ,x →∞. where f (u) is a smooth convex function and u aretwo constants. Theequation of type [1] (1.1) is related to the well-known BBM equation advocated by Benjamin-Bona-Mahony as a refinement of the KdV equation. All kinds of generalized BBM-Burgers equations have been [6,7,12] 2 ∞ studied by many authors .The L and L norms decay rates have been...
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Publisher
Springer-Verlag
Copyright
Copyright © 2005 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-005-0212-4
Publisher site
See Article on Publisher Site

Abstract

We investigate the asymptotic behavior of solutions of the initial-boundary value problem for the generalized BBM-Burgers equation u t + f(u) x = u xx + u xxt on the half line with the conditions u(0, t) = u −, u(∞, t) = u + and u − < u +, where the corresponding Cauchy problem admits the rarefaction wave as an asymptotic states. In the present problem, because of the Dirichlet boundary, the asymptotic states are divided into five cases depending on the signs of the characteristic speeds f′(u ±) of boundary state u − = u(0) and the far fields states u + = u(∞). In all cases both global existence of the solution and asymptotic behavior are shown under the smallness conditions.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2005

References

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