Boundary stabilization of elastodynamic systems. Part II: The case of a linear feedback.

We here consider a elastodynamic system damped by a linear feedback of Neumann-type. We prove stabilization results by using the multipliers method and Rellich-type relation given in Part I. Especially, we take in account singularities wich appear when changing boundary conditions

Boundary stabilization of elastodynamic systems. Part I: Rellich-type relations for a problem in elasticity involving singularities.

For the Lamé's system, mixed boundary conditions generate singularities in the solution, mainly when the boundary of the domain is connected. We here prove Rellich relations involving these singularities. These relations are useful in the problem of boundary stabilization of the elastodynamic system when using the multipliers method. This problem is studied in Part II

Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers.

17 pages, 9 figuresWe study the boundary stabilization of the wave equation by means of a linear or non-linear Neumann feedback. The rotated multiplier method leads to new geometrical cases concerning the active part of the boundary where the feedback is applied. Due to mixed boundary conditions, these cases generate singularities. Under a simple geometrical condition concerning the orientation of the boundary, we obtain stabilization results in both cases

Boundary stabilization of the linear elastodynamic system by a Lyapunov-type method

We propose a direct approach to obtain the boundary stabilization of the isotropic linear elastodynamic system by a “natural" feedback; this method uses local coordinates in the expression of boundary integrals as a main tool. It leads to an explicit decay rate of the energy function and requires weak geometrical conditions: for example, the spacial domain can be the difference of two star-shaped sets

Boundary stabilization of the linear elastodynamic system by a Lyapunov-type method

We propose a direct approach to obtain the boundary stabilization of the isotropic linear elastodynamic system by a "natural" feedback; this method uses local coordinates in the expression of boundary integrals as a main tool. It leads to an explicit decay rate of the energy function and requires weak geometrical conditions: for example, the spacial domain can be the difference of two star-shaped sets

Existence and Non-existence of Global Solutions to Initial Boundary Value Problems for Nonlinear Evolution Equations with the Strong Dissipation

The main purpose in this paper is to investigate existence and non-existence of global solutions of the initial Dirichlet-boundary value problem for evolution equations with the strong dissipation. Many authors studied classes consisting of such type of equations for which initial boundary value problems possess global solutions. For this purpose we consider a related problem and seek global solutions and blow-up solutions of it depending on whether it belongs to such classes or not. Key words: evolution equation, strong dissipation, blow-up, global existence 2000 MSC: 35L70, 35L15, 35L20, 58K55 1