scholarly journals On the Symmetries and Conservation Laws of the Multidimensional Nonlinear Damped Wave Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-11
Author(s):  
Usamah S. Al-Ali ◽  
Ashfaque H. Bokhari ◽  
A. H. Kara ◽  
F. D. Zaman
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Wave Equations ◽  
Lie Symmetries ◽  
Damped Wave Equation ◽  
Variable Damping ◽  
Nonlinear Damped Wave Equations ◽  
Nonlinear Damped Wave Equation ◽  
Similarity Reductions

We carry out a classification of Lie symmetries for the (2+1)-dimensional nonlinear damped wave equationutt+fuut=div(gugrad u)with variable damping. Similarity reductions of the equation are performed using the admitted Lie symmetries of the equation and some interesting solutions are presented. Employing the multiplier approach, admitted conservation laws of the equation are constructed for some new, interesting cases.

Global solutions and finite time blow-up for fourth order nonlinear damped wave equation

2018 ◽  
Vol 59 (6) ◽  
pp. 061503 ◽  
Author(s):  
Runzhang Xu ◽  
Xingchang Wang ◽  
Yanbing Yang ◽  
Shaohua Chen
Keyword(s):  
Wave Equation ◽  
Fourth Order ◽  
Finite Time ◽  
Blow Up ◽  
Global Solutions ◽  
Damped Wave Equation ◽  
Nonlinear Damped Wave Equation

Noether symmetries of vacuum classes of pp-waves and the wave equation

2016 ◽  
Vol 13 (09) ◽  
pp. 1650109 ◽  
Author(s):  
Sameerah Jamal ◽  
Ghulam Shabbir
Keyword(s):  
Wave Equation ◽  
Gravitational Waves ◽  
Wave Equations ◽  
Noether Symmetry ◽  
Symmetry Groups ◽  
Noether Symmetries ◽  
Noether’S Theorem ◽  
Metric Functions ◽  
Symmetry Algebras

The Noether symmetry algebras admitted by wave equations on plane-fronted gravitational waves with parallel rays are determined. We apply the classification of different metric functions to determine generators for the wave equation, and also adopt Noether's theorem to derive conserved forms. For the possible cases considered, there exist symmetry groups with dimensions two, three, five, six and eight. These symmetry groups contain the homothetic symmetries of the spacetime.

scholarly journals Symmetry analysis of the Klein–Gordon equation in Bianchi I spacetimes

2015 ◽  
Vol 12 (03) ◽  
pp. 1550033 ◽  
Author(s):  
A. Paliathanasis ◽  
M. Tsamparlis ◽  
M. T. Mustafa
Keyword(s):  
Wave Equation ◽  
Gordon Equation ◽  
Invariant Solution ◽  
Lie Symmetries ◽  
Conformal Algebra ◽  
Noether Symmetries ◽  
Bianchi I ◽  
Klein Gordon ◽  
Klein Gordon Equation

In this work we perform the symmetry classification of the Klein–Gordon equation in Bianchi I spacetime. We apply a geometric method which relates the Lie symmetries of the Klein–Gordon equation with the conformal algebra of the underlying geometry. Furthermore, we prove that the Lie symmetries which follow from the conformal algebra are also Noether symmetries for the Klein–Gordon equation. We use these results in order to determine all the potentials in which the Klein–Gordon admits Lie and Noether symmetries. Due to the large number of cases and for easy reference the results are presented in the form of tables. For some of the potentials we use the Lie admitted symmetries to determine the corresponding invariant solution of the Klein–Gordon equation. Finally, we show that the results also solve the problem of classification of Lie/Noether point symmetries of the wave equation in Bianchi I spacetime and can be used for the determination of invariant solutions of the wave equation.

scholarly journals Global existence for equivalent nonlinear special scale invariant damped wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sandra Lucente
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Vector Fields ◽  
Wave Equations ◽  
Small Data ◽  
Damped Wave Equation ◽  
Scale Invariant ◽  
Forcing Term ◽  
Time Existence

<p style='text-indent:20px;'>In this paper we give the notion of equivalent damped wave equations. As an application we study global in time existence for the solution of special scale invariant damped wave equation with small data. To gain such results, without radial assumption, we deal with Klainerman vector fields. In particular we can treat some potential behind the forcing term.</p>

scholarly journals Stabilization of damped waves on spheres and Zoll surfaces of revolution

2018 ◽  
Vol 24 (4) ◽  
pp. 1759-1788
Author(s):  
Hui Zhu
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Indicator Function ◽  
Geometric Control ◽  
Damped Wave Equation ◽  
Surfaces Of Revolution ◽  
Strong Stabilization ◽  
Geometric Objects ◽  
Dimension 2 ◽  
Appl Math

We study the strong stabilization of wave equations on some sphere-like manifolds, with rough damping terms which do not satisfy the geometric control condition posed by Rauch−Taylor [J. Rauch and M. Taylor, Commun. Pure Appl. Math. 28 (1975) 501–523] and Bardos−Lebeau−Rauch [C. Bardos, G. Lebeau and J. Rauch, SIAM J. Control Optimiz. 30 (1992) 1024–1065]. We begin with an unpublished result of G. Lebeau, which states that on 𝕊d, the indicator function of the upper hemisphere strongly stabilizes the damped wave equation, even though the equators, which are geodesics contained in the boundary of the upper hemisphere, do not enter the damping region. Then we extend this result on dimension 2, to Zoll surfaces of revolution, whose geometry is similar to that of 𝕊2. In particular, geometric objects such as the equator, and the hemi-surfaces are well defined. Our result states that the indicator function of the upper hemi-surface strongly stabilizes the damped wave equation, even though the equator, as a geodesic, does not enter the upper hemi-surface either.

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