Energy decay for a damped nonlinear hyperbolic equation

Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

(x 1 , . . . , x n ) ∈ Rn and t > 0, k 1 > 0 and k 2 > 0 are constants. The nonlinear termf(u) = O(u1+θ) and θ is a positive integer. Equation (1) is a model in variational form for the neoHookean elastomer rod and describes the motion of a neoHookean elastomer rod with internal damping; for more detailed physical background, we refer to [1]. In [1], the authors have studied a general class of ...

Energy Decay of Damped Systems

We present a new and simple bound for the exponential decay of second order systems using the spectral shift. This result is applied to finite matrices as well as to partial differential equations of Mathematical Physics. The type of the generated semigroup is shown to be bounded by the upper real part of the numerical range of the underlying quadratic operator pencil.

Stability 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, ...

Energy decay for the damped wave equation under a pressure condition

We establish the presence of a spectral gap near the real axis for the damped wave equation on a manifold with negative curvature. This results holds under a dynamical condition expressed by the negativity of a topological pressure with respect to the geodesic flow. As an application, we show an exponential decay of the energy for all initial data sufficiently regular. This decay is governed by...

Local decay for the damped wave equation in the energy space