Exponential decay for semilinear distributed systems with damping

The stability of second order abstract distributed systems with damping and nonlinear perturbations is considered. Sufficient conditions, including unique continuation property assumptions, are formulated to obtain (local, non-uniform and uniform) exponential stability. Applications to the wave and Euler-Bernoulli equations are given.     

Stabilization of linear control systems in Hilbert space with nonlinear feedback

This paper considers the problem of stabilization of linear control systems in Hilbert space with nonlinear feedback. Extensions of known results in the linear feedback case to the nonlinear one are given.     

Exponential decay bounds for nonlinear heat problems with Robin boundary conditions

We investigate the behavior of the solution of a nonlinear heat problem, when Robin conditions are prescribed on the boundary ∂Ω × (t > 0), Ω a bounded R ² domain. We determine conditions on the geometry and data sufficient to preclude the blow up of the solution and to obtain an exponential decay bound for the solution and its gradient. Supported by the University of Cagliari.     

Exponential decay for a viscoelastic problem with a singular kernel

In this paper we consider a problem which arises in viscoelasticity. We prove exponential decay of solutions for the problem with a memory term involving a kernel which is singular at zero. This is established by introducing an appropriate Lyapunov type functional and using the energy method. This work extends earlier results.      

On two-point boundary value problems for systems of higher-order ordinary differential equations with singularities

The sufficient conditions of solvability and unique solvability of the two-point boundary value problems of Vallèe-Poussin and Cauchy-Niccoletti have been found for a system of ordinary differential equations of the form

Periodic boundary value problem of functional differential equations with perturbation

By coincidence degree, the existence of solution to the periodic boundary value problem of functional differential equations with perturbation     

Perfectly matched layers in 1-d : energy decay for continuous and semi-discrete waves

In this paper we investigate the efficiency of the method of perfectly matched layers (PML) for the 1-d wave equation. The PML method furnishes a way to compute solutions of the wave equation for exterior problems in a finite computational domain by adding a damping term on the matched layer. In view of the properties of solutions in the whole free space, one expects the energy of solutions obtained by the PML method to tend to zero as t → ∞, and the rate of decay can be understood as a measure of the efficiency of the method. We prove, indeed, that the exponential decay holds and characterize the exponential decay rate in terms of the parameters and damping potentials entering in the implementation of the PML method. We also consider a space semi-discrete numerical approximation scheme and we prove that, due to the high frequency spurious numerical solutions, the decay rate fails to be uniform as the mesh size parameter h tends to zero. We show however that adding a numerical viscosity term allows us to recover the property of exponential decay of the energy uniformly on h. Although our analysis is restricted to finite differences in 1-d, most of the methods and results apply to finite elements on regular meshes and to multi-dimensional problems. This work started while the first author was visiting the Department of Mathematics of the Universidad Autónoma de Madrid, in the frame of the European program “New materials, adaptive systems and their nonlinearities: modeling, control and numerical simulation” HPRN-CT-2002-00284. The work was finished while both authors visited the Isaac Newton Institute of Cambridge within the Program “Highly Oscillatory Problems”.     

Unitary invariants of spectral measures with the CGS-property

A «CGS-property» for the spectral measures is introduced and the classical results of determining complete systems of unitary invariants for self-adjoint and bounded normal operators on separable Hilbert spaces are extended to the class of spectral measures with this property. As a consequence, the above mentioned results are extended to unbounded normal operators on separable Hilbert spaces. Moreover, three different kinds of multiplicity are defined and it is shown that for the measures with the «CGS-property» they all coincide. In the last section some analogues of the multiplicity functions defined by Stone [14] are related to the total multiplicity.     

Asymptotic behavior of solutions to a wave equation with a non linear dissipative term in ℝ<Superscript><Emphasis Type=\"Italic\">n</Emphasis></Superscript>

In this note we investigate the asymptotic behavior of solutions to the wave equation:u\"-Δu+g(u')=0 in ℝⁿxℝ₊.     

Feedback stabilization of some classes of nonlinear transport systems

It is shown that for some cases, the control system of the linearized transport equation with a controllable unstable part is exponentially stabilizable. The same result holds true for nonlinear transport phenomena if it can be found as simulation orbit of perturbed transport equation.     

Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation

This work is concerned with obtention of energy decay estimates for Petrowsky equation with a nonlinear dissipation which is active only in an interior subset of the domain. We prove that the piecewise multiplier method as introduced by [20] and [22] for the wave equation can be extended to the Petrowsky equation. Moreover, we also apply some recent results by the author to obtain precise decay rate estimates for the energy, without specifying the growth of the nonlinear dissipation close to the origin by means of convex properties and nonlinear integral inequalities for the energy of the solutions.     

Localization of blow-up points for a parabolic system with a nonlinear boundary condition

The authors localize the blow-up points of positive solutions of the systemu _t =Δu,v _t =Δv with conditions at the boundary of a bounded smooth domain Θ under some restrictions off andg and the initial data (Δu ₀, Δν₀>c>0). If Θ is a ball, the hypothesis on the initial data can be removed. Supported by Universidad de Buenos Aires under grant EX071 and CONICET.     

Global solution of nonlinear degenerate systems of filtration type in several space variables

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Optimal decay rates of solutions to dissipative nonlinear evolution equations with ellipticity

We derive the optimal decay rates of solution to the Cauchy problem for a set of nonlinear evolution equations with ellipticity and dissipative effects

Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes

In this article, we derive uniform admissibility and observability properties for the finite element space semi-discretizations of , where A ₀ is an unbounded self-adjoint positive definite operator with compact resolvent. To address this problem, we present a new spectral approach based on several spectral criteria for admissibility and observability of such systems. Our approach provides very general admissibility and observability results for finite element approximation schemes of , which stand in any dimension and for any regular mesh (in the sense of finite elements). Our results can be combined with previous works to derive admissibility and observability properties for full discretizations of . We also present applications of our results to controllability and stabilization problems. The author was partially supported by the “Agence Nationale de la Recherche” (ANR), Project C-QUID, number BLAN-3-139579.     

On the long time behaviour of solutions to dissipative wave equations in \mathbb{R}^{2}

In this paper we present a parabolic approach to studying the diffusive long time behaviour of solutions to the Cauchy problem:

A note on the behavior of blow-up solutions for one-phase Stefan problems

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On the classification of Galois objects over the quantum group of a nondegenerate bilinear form

We study Galois and bi-Galois objects over the quantum group of a nondegenerate bilinear form, including the quantum group (SL(2)). We obtain the classification of these objects up to isomorphism and some partial results for their classification up to homotopy.     

Existence theorems for nonlinear functional differential equations of neutral type

Conditions are found upon satisfaction of which the differential equation