On the Asymptotic Stability of Wave Equations Coupled by Velocities of Anti-symmetric Type

In this paper, we study the Lp-asymptotic stability of the one dimensional linear damped wave equation with Dirichlet boundary conditions in <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math>. The damping term is assumed to be linear and localized  to an arbitrary open sub-interval of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>. We prove that the  semi-group <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>p</mi></msub><mo>(</mo><mi>t</mi><msub><mo>)</mo><mrow><mi>t</mi><mo>≥</mo><mn>0</mn></mrow></msub></math> associated with the previous equation is well-posed and exponentially stable. The proof relies on the multiplier method and depends on whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>≥</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn></math>.

Download Full-text

Asymptotic stability of wave equations coupled by velocities

Mathematical Control and Related Fields ◽

10.3934/mcrf.2016010 ◽

2016 ◽

Vol 6 (3) ◽

pp. 429-446 ◽

Cited By ~ 4

Author(s):

Yan Cui ◽

Zhiqiang Wang

Keyword(s):

Asymptotic Stability ◽

Wave Equations

Download Full-text

Asymptotic stability of wave equations with memory and frictional boundary dampings

Applicationes Mathematicae ◽

10.4064/am35-3-1 ◽

2008 ◽

Vol 35 (3) ◽

pp. 247-258 ◽

Cited By ~ 1

Author(s):

Fatiha Alabau-Boussouira

Keyword(s):

Asymptotic Stability ◽

Wave Equations ◽

With Memory

Download Full-text

Remarks on nonlinear elastic waves with null conditions

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2020039 ◽

2020 ◽

Vol 26 ◽

pp. 121

Author(s):

Dongbing Zha ◽

Weimin Peng

Keyword(s):

Initial Data ◽

Global Existence ◽

Elastic Waves ◽

Wave Equations ◽

Alternative Proof ◽

Classical Solutions ◽

Small Initial Data ◽

Hyperelastic Materials ◽

Variational Structure ◽

Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

Download Full-text

On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

GUB Journal of Science and Engineering ◽

10.3329/gubjse.v5i1.47898 ◽

2018 ◽

Vol 5 (1) ◽

pp. 31-36

Author(s):

Md Monirul Islam ◽

Muztuba Ahbab ◽

Md Robiul Islam ◽

Md Humayun Kabir

Keyword(s):

Solitary Wave ◽

Acoustic Waves ◽

Water Waves ◽

Wave Equations ◽

Rectangular Channel ◽

Soliton Solutions ◽

Boussinesq Equations ◽

Travelling Wave ◽

Ion Acoustic Waves ◽

Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

Download Full-text