In this paper we model and solve the problem of shaping and placing in an optimal way sensors for a wave equation with constant damping in a bounded open connected subset Ω of IRn. Sensors are modeled by subdomains of Ω of a given measure L|Ω|, with 0 < L < 1. We prove that, if L is close enough to 1, then the optimal design problem has a unique solution, which is characterized by a finite numb...

is referred to as the decay rate associated with a. If a is to be introduced in order to absorb an initial disturbance then one naturally wishes to strike upon that a with the least possible (most negative) decay rate. The mathematical attraction here lies in the oftnoted fact that, with respect to damping, ‘more is not better.’ More precisely, for constant a, the decay rate is not a decreasing...

The issue of stablity of solutions to nonlinear wave equations has been addressed by many authors. So many results concerning energy decay have been established. Here in this paper we consider the following nonlinearly damped wave equation utt −∆u+ a(1 + |ut|)ut = bu|u|p−2, a, b > 0, in a bounded domain and show that, for suitably chosen initial data, the energy of the solution decays exponenti...

Helicon waves in a plasma confined by a cylinder are treated. The undamped normal modes of the helicon (H) and Trivelpiece–Gould ~TG! waves have distinctly different wave patterns at high magnetic fields but at low fields have similar patterns and therefore interact strongly. Damping of these modes, their excitation by antennas, and the rf plasma absorption efficiency are considered. Nonuniform...

The weak well-posedness, with the mixed boundary conditions, of strongly damped linear wave equation and non Westervelt is proved in a large natural class Sobolev admissible non-smooth domains. In framework uniform domains $$\mathbb {R}^2$$ or {R}^3$$ we also validate approximation solution on fractal domain by solutions prefractals using Mosco convergence corresponding variational forms.

Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constr...

According to the Smolukowski-Kramers approximation, we show that the solution of the semi-linear stochastic damped wave equations μutt (t, x) = u(t, x)−ut (t, x)+ b(x, u(t, x)) + QẆ(t), u(0) = u0, ut (0) = v0, endowed with Dirichlet boundary conditions, converges as μ goes to zero to the solution of the semi-linear stochastic heat equation ut (t, x) = u(t, x)+b(x, u(t, x))+QẆ(t), u(0) = u0, end...

We extend some previous results for the damped wave equation in bounded domains in R to the unbounded case. In particular, we show that if the damping term is of the form αa with bounded a taking on negative values on a set of positive measure, then there will always exist unbounded solutions for sufficiently large positive α. In order to prove these results, we generalize some existing results...

The perturbation theory based on the Riemann-Hilbert problem is constructed for the nonlinear Schrödinger equation. The second-order equations for the spectral data describing the soliton-wave interaction are obtained. The theory is applied to the parametrically driven, damped nonlinear Schrödinger equation. The parametrically driven NLS soliton is shown to become unstable, when the driving str...

In this paper optimal control problems governed by the wave equation with control constraints are analyzed. Three types of control action are considered: distributed control, Neumann boundary control and Dirichlet control, and proper functional analytic settings for them are discussed. For treating inequality constraints semismooth Newton methods are discussed and their convergence properties a...