NDLI: Boundary-value problem for the Korteweg-de Vries-Burgers type equation

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Homogenization for strongly anisotropic nonlinear elliptic equations

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The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results

Some remarks about stability of nonlinear discrete-time control systems

Boundary-value problem for the Korteweg-de Vries-Burgers type equation

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The blow-up rate for semilinear parabolic problems on general domains

Nonlinear eigenvalue problems for quasilinear operators on unbounded domains

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Boundary-value problem for the Korteweg-de Vries-Burgers type equation

Content Provider	Springer Nature Link
Author	Hayashi, Nakao Kaikina, Elena I. Ruiz Paredes, H. Francisco
Copyright Year	2001
Abstract	We study the following initial-boundary value problem for the Korteweg-de Vries-Burgers equation on half-line¶¶ \cases{ u_{t}+u_{x}^{2}-u_{xx}+u_{xxx}=0, $(x,t)\in {{\bf R}^{+}}\times {{\bf R} ^{+}}$ ,\cr u(x,0)=u_{0}(x), $ x\in {{\bf R}^{+}}$ ,\hspace*{8pc}(1)\cr u(0,t)=0, $ t\in {{\bf R}^{+}}$ .} ¶¶We prove that if the initial data $ u_{0}\in {\bf X} $ , and the norm $ \Vert u_{0}\Vert _{{\bf X}} $ is sufficiently small, where $ {\bf X}=\{\varphi \in {\bf L}^{1}\cap {\bf H}^{1};\Vert \varphi \Vert _{{\bf X}}=\Vert \varphi \Vert _{{\bf L}^{1}}+\Vert \varphi \Vert _{{\bf H}^{1}}$ < $ \infty \} $ , then there exists a unique solution $ u\in {\bf C}([0,\infty );{\bf H}^{1}) $ of the initial-boundary value problem (1), where H k is the Sobolev space with norm $ \Vert \phi \Vert _{{\bf H}^{k}}=\Vert (1-\partial _{x}^{2})^{\frac{k}{2}}\phi \Vert _{{\bf L}^{2}}. $ We also find the large time asymptotics of the solutions under the condition $ x^{1+\delta }u\in {\bf L}^{1}\cap {\bf L}^{2} $ with $ \delta \in (0,1). $ More pricesely, we prove¶¶ $ u(x,t)=\frac{A}{t}e^{-\frac{x^{2}}{4t}}\frac{x}{2\sqrt{t}}+O\Bigg(\min \left( 1,\frac{x}{2\sqrt{t}}\right) t^{-1-\frac{\delta}{2}}\Bigg), $ ¶¶where A will be defined below in Theorem 2.
Starting Page	439
Ending Page	463
Page Count	25
File Format	PDF
ISSN	10219722
Journal	Nonlinear Differential Equations and Applications NoDEA
Volume Number	8
Issue Number	4
e-ISSN	14209004
Language	English
Publisher	Birkhäuser Verlag
Publisher Date	2001-11-01
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation

The initial-boundary value problem for the Korteweg-de Vries equation

Well-posedness of Korteweg-de Vries-Burgers equation on a finite domain

Asymptotics for the Korteweg–de Vries–Burgers Equation

Stability of a traveling-wave solution to the Cauchy problem for the Korteweg-de Vries-Burgers equation

Modelling and nonlinear boundary stabilization of the modified generalized Korteweg–de Vries–Burgers equation

Boundary linear stabilization of the modified generalized Korteweg–de Vries–Burgers equation

Stabilization of the Korteweg-de Vries-Burgers equation with non-periodic boundary feedbacks

Numerical Analysis of the Boundary-Value Problem for a Nonlinear Korteweg–De Vries–Burgers Equation

Boundary-value problem for the Korteweg-de Vries-Burgers type equation

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Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation

The initial-boundary value problem for the Korteweg-de Vries equation

Well-posedness of Korteweg-de Vries-Burgers equation on a finite domain

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Boundary-value problem for the Korteweg-de Vries-Burgers type equation