منابع مشابه
Waves, damped wave and observation
This talk describes some applications of two kinds of observation estimate for the wave equation and for the damped wave equation in a bounded domain where the geometric control condition of C. Bardos, G. Lebeau and J. Rauch may failed. 1 The wave equation and observation We consider the wave equation in the solution u = u(x, t) ∂ t u−∆u = 0 in Ω× R , u = 0 on ∂Ω× R , (u, ∂tu) (·, 0) = (u...
متن کاملA unifying fractional wave equation for compressional and shear waves.
This study has been motivated by the observed difference in the range of the power-law attenuation exponent for compressional and shear waves. Usually compressional attenuation increases with frequency to a power between 1 and 2, while shear wave attenuation often is described with powers less than 1. Another motivation is the apparent lack of partial differential equations with desirable prope...
متن کاملDamped Wave Equation with a Critical Nonlinearity
We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity { ∂2 t u+ ∂tu−∆u+ λu 2 n = 0, x ∈ Rn, t > 0, u(0, x) = εu0 (x) , ∂tu(0, x) = εu1 (x) , x ∈ Rn, where ε > 0, and space dimensions n = 1, 2, 3. Assume that the initial data u0 ∈ H ∩H, u1 ∈ Hδ−1,0 ∩H−1,δ, where δ > n 2 , weighted Sobolev spaces are H = { φ ∈ L; ...
متن کاملStability of Travelling Waves for a Damped Hyperbolic Equation
We consider a nonlinear damped hyperbolic equation in R, 1 ≤ n ≤ 4, depending on a positive parameter ǫ. If we set ǫ = 0, this equation reduces to the well-known Kolmogorov-Petrovski-Piskunov equation. We remark that, after a change of variables, this hyperbolic equation has the same family of one-dimensional travelling waves as the KPP equation. Using various energy functionals, we show that, ...
متن کاملFinite Energy Travelling Waves for Nonlinear Damped Wave Equations
E(u,ut) = f |Vu|2 + m\u\2 + |ut|2 dx — [ |u|"+1 dx, (1.3) 2 JRn a+ 1 JR„ which represents a Lyapunov function of the problem, i.e., it is decreasing along any nonstationary trajectory of (1.1). Employing the potential-well arguments of PAYNE-SATTINGER [16] one observes that any solution of (1.1) emanating from sufficiently small initial data exists globally for all t e R+ and tends to zero with...
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ژورنال
عنوان ژورنال: Journal of Mathematical Physics
سال: 2013
ISSN: 0022-2488,1089-7658
DOI: 10.1063/1.4794076