scholarly journals Initial Boundary Value Problem and Asymptotic Stabilization of the Two-Component Camassa-Holm Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Xiju Zong ◽  
Xingong Cheng ◽  
Zhonghua Wang ◽  
Zhenlai Han
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Result ◽  
Asymptotic Stabilization ◽  
Global Asymptotic Stabilization ◽  
Holm Equation ◽  
Two Component ◽  
Initial Boundary Value

The nonhomogeneous initial boundary value problem for the two-component Camassa-Holm equation, which describes a generalized formulation for the shallow water wave equation, on an interval is investigated. A local in time existence theorem and a uniqueness result are achieved. Next by using the fixed-point technique, a result on the global asymptotic stabilization problem by means of a boundary feedback law is considered.

scholarly journals An extension of Gronwall inequality in the theory of bodies with voids

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1161-1167
Author(s):  
Marin Marin ◽  
Praveen Ailawalia ◽  
Ioan Tuns
Keyword(s):  
Boundary Value Problem ◽  
Internal State ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Result ◽  
Internal Variables ◽  
State Variables ◽  
Gronwall’S Inequality ◽  
Initial Boundary Value

Abstract In this paper, we obtain a generalization of the Gronwall’s inequality to cover the study of porous elastic media considering their internal state variables. Based on some estimations obtained in three auxiliary results, we use this form of the Gronwall’s inequality to prove the uniqueness of solution for the mixed initial-boundary value problem considered in this context. Thus, we can conclude that even if we take into account the internal variables, this fact does not affect the uniqueness result regarding the solution of the mixed initial-boundary value problem in this context.

An initial boundary value problem of Camassa–Holm equation

2000 ◽  
Vol 41 (12) ◽  
pp. 8279-8285 ◽  
Author(s):  
Keng-Huat Kwek ◽  
Hongjun Gao ◽  
Weinian Zhang ◽  
Chaochun Qu
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Holm Equation ◽  
Initial Boundary Value

scholarly journals Initial Boundary Value Problem of the General Three-Component Camassa-Holm Shallow Water System on an Interval

2013 ◽  
Vol 2013 ◽  
pp. 1-15
Author(s):  
Lixin Tian ◽  
Qingwen Yuan ◽  
Lizhen Wang
Keyword(s):  
Boundary Value Problem ◽  
Shallow Water ◽  
Blow Up ◽  
Boundary Value ◽  
Water System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Result ◽  
Shallow Water System ◽  
Initial Boundary Value

We study the initial boundary value problem of the general three-component Camassa-Holm shallow water system on an interval subject to inhomogeneous boundary conditions. First we prove a local in time existence theorem and present a weak-strong uniqueness result. Then, we establish a asymptotic stabilization of this system by a boundary feedback. Finally, we obtain a result of blow-up solution with certain initial data and boundary profiles.

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