𝔖 Bobbio Scriptorium
✦   LIBER   ✦

On the Linear Damped Wave Equation

✍ Scribed by Julián López-Gómez

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
864 KB
Volume
134
Category
Article
ISSN
0022-0396

No coin nor oath required. For personal study only.

✦ Synopsis

In this work we estimate the spectrum of the linear damped wave semigroup under homogeneous Dirichlet boundary conditions by using the principal eigenvalue of an elliptic operator related to the equation. Our estimate is optimal for real eigenvalues. Then, we analyze the behavior of the estimate as the damping amplitude grows to infinity. When the damping changes of sign we extend a result of Freitas [5] to show that the semigroup possesses at least two real eigenvalues greater than one if the amplitude is sufficiently large. In particular, the trivial state is unstable, in strong contrast with the sign definited case. Finally, we characterize the limiting behavior of the real eigenpairs which originate the inestability of the trivial state. This analysis is based upon the behavior of the principal eigenpair of a singular perturbation problem at the singular limit. Our theory is of interest by itself and it has many applications to reaction diffusion systems (c.f. for instance [6]).

📜 SIMILAR VOLUMES

Attractors for a damped wave equation on
Attractors for a damped wave equation on ℝ3 with linear memory
✍ Vittorino Pata 📂 Article 📅 2000 🏛 John Wiley and Sons 🌐 English ⚖ 164 KB 👁 1 views

A damped semilinear hyperbolic equation on 1 with linear memory is considered in a history space setting. Viewing the past history of the displacement as a variable of the system, it is possible to express the solution in terms of a strongly continuous process of continuous operators on a suitable H

Periodic problem for a nonlinear-damped
Periodic problem for a nonlinear-damped wave equation on the boundary
✍ G. O. Antunes; H. R. Crippa; M. D. G. da Silva 📂 Article 📅 2009 🏛 John Wiley and Sons 🌐 English ⚖ 177 KB

We employ elliptic regularization and monotone method. We consider X⊂R n (n 1) an open bounded set that has regular boundary C and Q = X×(0,T), T>0, a cylinder of R n+1 with lateral boundary R = C×(0,T).

Non-homogeneous non-linear damped wave e
Non-homogeneous non-linear damped wave equations in unbounded domains
✍ Reinhard Racke 📂 Article 📅 1990 🏛 John Wiley and Sons 🌐 English ⚖ 431 KB

## Abstract We present a global existence theorem for solutions of __u__^__tt__^ − ∂~__i__~__a__~__ik__~ (__x__)∂~__k__~__u__ + u~t~ = ƒ(__t__, __x__, __u__, __u__~__t__~, ∇__u__, ∇__u__~__t__~, ∇^2^__u__), __u__(__t__ = 0) = __u__^0^, __u__(=0)=__u__^1^, __u__(__t, x__), __t__ ⪖ 0, __x__ϵΩ.Ω equal

Dimension of the Global Attractor for St
Dimension of the Global Attractor for Strongly Damped Nonlinear Wave Equation
✍ Shengfan Zhou 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 109 KB

The existence and estimate of the upper bound of the Hausdorff dimension of the global attractor for the strongly damped nonlinear wave equation with the Dirichlet boundary condition are considered by introducing a new norm in the phase space. The gained Hausdorff dimension decreases as the damping