Eva Fašangová

Asymptotic behaviour of a semilinear viscoelastic beam model

Abstract. The asymptotic behaviour of a model for a viscoelastic beam with nonlinear load is studied. Under the assumptions that the energy is coercive and the solution set of the stationary problem is discrete, the convergence of the solutions of the dynamical problem to a steady state are shown. To prove this statement, a method known for problems in finite dimensions is combined with results about the corresponding linear problem, energy estimates for the nonlinear problem, and harmonic analysis of vector-valued functions.

The Sector of Analyticity of the Ornstein–Uhlenbeck Semigroup on Lp Spaces with Respect to Invariant Measure

The sector of analyticity of the Ornstein-Uhlenbeck semigroup is computed on the space with respect to its invariant measure . If denotes the generator of the Ornstein-Uhlenbeck semigroup, then the angle of the sector of analyticity in is minus the spectral angle of being the matrix determining the Gaussian measure . The angle of analyticity in is then given by the formula.

Evolution Equations with Dissipation of Memory Type

Well-posedness of semilinear evolution equations with linear dissipation of fractional order or memory type is studied in a Hilbert space framework. Under the main assumptions that the energy is coercive, the set of stationary solutions is discrete, the underlying linear problem is stable, and if the nonlinearity is subject to a compactness condition, convergence of the mild solutions to a steady state is shown. The methods employed include stability of the linear problem, energy estimates, and harmonic analysis of vector-valued functions. The results apply to a variety of problems in mathematical physics, like semilinear beam and plate models, and semilinear wave equations.

Equality of two spectra arising in harmonic analysis and semigroup theory

We show that a new notion of a spectrum of a function u E L∞ (R + , X) (X is a Banach space), defined by B. Basit and the first author, coincides with the Arveson spectrum of some shift group, provided u is uniformly continuous. We apply this result to prove a new version of a tauberian theorem.

Attractor for a beam equation with weak damping

We prove that there is a global attractor for in where the damping function P is monotone and allowed to be bounded.

A Banach Algebra Approach to the Weak Spectral Mapping Theorem for Locally Compact Abelian Groups

We give a general version of the weak spectral mapping theorem for non-quasianalytic representations of locally compact abelian groups which are weakly continuous in the sense of Arveson, based on a Banach algebra approach.