Microlocal defect measures for a degenerate thermoelasticity system

2013 ◽  
Vol 13 (1) ◽  
pp. 145-181
Author(s):  
Amel Atallah-Baraket ◽  
Clotilde Fermanian Kammerer
Keyword(s):  
Wave Equation ◽  
Energy Density ◽  
Heat Equation ◽  
Geometric Optic ◽  
Damped Wave Equation ◽  
Order Operator ◽  
Heat Operator ◽  
Open Set ◽  
First Case ◽  
Set Of Points

AbstractIn this paper, we study a system of thermoelasticity with a degenerate second-order operator in the heat equation. We analyze the evolution of the energy density of a family of solutions. We consider two cases: when the set of points where the ellipticity of the heat operator fails is included in a hypersurface and when it is an open set. In the first case, and under special assumptions, we prove that the evolution of the energy density is that of a damped wave equation: propagation along the rays of the geometric optic and damping according to a microlocal process. In the second case, we show that the energy density propagates along rays which are distortions of the rays of the geometric optic.

scholarly journals On the strongly damped wave equation and the heat equation with mixed boundary conditions

2000 ◽  
Vol 5 (3) ◽  
pp. 175-189
Author(s):  
Aloisio F. Neves
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Mixed Boundary ◽  
Strong Solutions ◽  
Mixed Boundary Conditions ◽  
Global Attractors ◽  
Damped Wave Equation ◽  
Strongly Damped Wave Equation ◽  
Strongly Damped

We study two one-dimensional equations: the strongly damped wave equation and the heat equation, both with mixed boundary conditions. We prove the existence of global strong solutions and the existence of compact global attractors for these equations in two different spaces.

scholarly journals Global existence and dynamic structure of solutions for damped wave equation involving the fractional Laplacian

2021 ◽  
Vol 54 (1) ◽  
pp. 245-258
Author(s):  
Younes Bidi ◽  
Abderrahmane Beniani ◽  
Khaled Zennir ◽  
Ahmed Himadan
Keyword(s):  
Wave Equation ◽  
Global Solution ◽  
Fractional Laplacian ◽  
Necessary Conditions ◽  
Dynamic Structure ◽  
Decay Estimates ◽  
Damped Wave Equation ◽  
Structure Of Solutions ◽  
Nonlinear Source ◽  
Galerkin Approximations

Abstract We consider strong damped wave equation involving the fractional Laplacian with nonlinear source. The results of global solution under necessary conditions on the critical exponent are established. The existence is proved by using the Galerkin approximations combined with the potential well theory. Moreover, we showed new decay estimates of global solution.

scholarly journals Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets

2021 ◽  
Author(s):  
Jiahao Qiu ◽  
Jianjie Zhao
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Minimal System ◽  
Combinatorial Properties ◽  
Return Times ◽  
Residual Set ◽  
Closed Sets ◽  
Open Set ◽  
Geometric Progressions ◽  
Set Of Points

AbstractIn this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

SOME COMMENTS ON THE DYNAMICS OF EXTENDED OBJECTS

1998 ◽  
Vol 13 (17) ◽  
pp. 2979-2990 ◽  
Author(s):  
U. KHANAL
Keyword(s):  
Wave Equation ◽  
Energy Density ◽  
Pure Shear ◽  
Equation Of Motion ◽  
Conducting Medium ◽  
Constant Density ◽  
Metric Tensor ◽  
World Volume ◽  
The World ◽  
Hilbert Action

A variational method is used to investigate the dynamics of extended objects. The stationary world volume requires the internal coordinates to propagate as free waves. Stationarity of the action which is the integral of a variable energy density over the world volume leads to the wave equation in a medium, with conductivity given by the gradient of the logarithm of reciprocal energy density, constant density corresponding to free space. The Einstein–Hilbert action for the world curvature gives an equation of motion which, in world space with the Einstein tensor proportional to the metric tensor, reduces to the free wave equation. A similar method applied to the action consisting of the surface area enclosing an incompressible world volume undergoing pure shear again yields the wave equation in a conducting medium. Simultaneous stationarity of the volume can be imposed with a stationary area only in the case of pure shear; stationary Einstein–Hilbert action can also be included and lead to an equation of motion which has a similar interpretation of the wave in the conducting medium. Some Green functions applicable to the medium with constant conductivity are also presented.

Golbal nonexistence in a nonlinearly damped wave equation

2001 ◽  
Vol 80 (3-4) ◽  
pp. 269-277 ◽  
Author(s):  
Salim A. Messaoudi
Keyword(s):  
Wave Equation ◽  
Damped Wave Equation
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