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Homogenization with uncertain input parameters

Luděk Nechvátal (2010)

Mathematica Bohemica

We homogenize a class of nonlinear differential equations set in highly heterogeneous media. Contrary to the usual approach, the coefficients in the equation characterizing the material properties are supposed to be uncertain functions from a given set of admissible data. The problem with uncertainties is treated by means of the worst scenario method, when we look for a solution which is critical in some sense.

Hyperbolic Equations in Uniform Spaces

J. W. Cholewa, Tomasz Dlotko (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

The paper is devoted to the Cauchy problem for a semilinear damped wave equation in the whole of ℝ ⁿ. Under suitable assumptions a bounded dissipative semigroup of global solutions is constructed in a locally uniform space H ̇ ¹ l u ( ℝ ⁿ ) × L ̇ ² l u ( ℝ ⁿ ) . Asymptotic compactness of this semigroup and the existence of a global attractor are then shown.

Improved estimates for the Ginzburg-Landau equation : the elliptic case

Fabrice Bethuel, Giandomenico Orlandi, Didier Smets (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the G L -energy E ε and the parameter ε . These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.

Indirect stabilization of locally coupled wave-type systems

Fatiha Alabau-Boussouira, Matthieu Léautaud (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not...

Indirect stabilization of locally coupled wave-type systems

Fatiha Alabau-Boussouira, Matthieu Léautaud (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We study in an abstract setting the indirect stabilization of systems of two wave-like equations coupled by a localized zero order term. Only one of the two equations is directly damped. The main novelty in this paper is that the coupling operator is not assumed to be coercive in the underlying space. We show that the energy of smooth solutions of these systems decays polynomially at infinity, whereas it is known that exponential stability does not...

Inertial manifolds for retarded second order in time evolution equations in admissible spaces

Cung The Anh, Le Van Hieu (2013)

Annales Polonici Mathematici

Using the Lyapunov-Perron method, we prove the existence of an inertial manifold for the process associated to a class of non-autonomous semilinear hyperbolic equations with finite delay, where the linear principal part is positive definite with a discrete spectrum having a sufficiently large distance between some two successive spectral points, and the Lipschitz coefficient of the nonlinear term may depend on time and belongs to some admissible function spaces.

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