Given a compact Hausdorff space and a strongly continuous semigroup of linear isometries of the Banach space of all complex-valued, continuous functions on , the semiflow induced by on is investigated. In the particular case in which is a compact, connected, differentiable manifold, a class of semigroups preserving the differentiable structure of is characterized.
Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral...
This is the second instalment of my previous paper with the same title, [1]. This paper consists of two different parts. The first part is devoted to improvements of the results developed in [1]. These improvements are described in section 0.1 below and developed in sections 1 to 5, and 9 to 10; they are in fact technically distinct from [1] and rely on a systematic use of microlocalisation in the context of Hörmander-Weyl calculus. These paragraphs can therefore be read quite independently from...
We study asymptotic behavior of -semigroups T(t), t ≥ 0, such that ∥T(t)∥ ≤ α(t), where α(t) is a nonquasianalytic weight function. In particular, we show that if σ(A) ∩ iℝ is countable and Pσ(A*) ∩ iℝ is empty, then , ∀x ∈ X. If, moreover, f is a function in which is of spectral synthesis in a corresponding algebra with respect to (iσ(A)) ∩ ℝ, then , where . Analogous results are obtained also for iterates of a single operator. The results are extensions of earlier results of Katznelson-Tzafriri,...
Existence of a mild solution to a semilinear Cauchy problem with an almost sectorial operator is studied. Under additional regularity assumptions on the nonlinearity and initial data we also prove the existence of a classical solution to this problem. An example of a parabolic problem in Hölder spaces illustrates the abstract result.
This paper is devoted to the investigation of the abstract semilinear initial value problem du/dt + A(t)u = f(t,u), u(0) = u₀, in the "parabolic" case.
The paper is devoted to a careful analysis of the shape-preserving properties of the strongly continuous semigroup generated by a particular second-order differential operator, with particular emphasis on the preservation of higher order convexity and Lipschitz classes. In addition, the asymptotic behaviour of the semigroup is investigated as well. The operator considered is of interest, since it is a unidimensional Black-Scholes operator so that our results provide qualitative information on the...
Let be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure on We prove a sharp estimate of the operator norm of the imaginary powers of on
We prove a characterisation of sets with finite perimeter and functions in terms of the short time behaviour of the heat semigroup in . For sets with smooth boundary a more precise result is shown.