This paper is devoted to the study of the approximation problem for the abstract hyperbolic differential equation u'(t) = A(t)u(t) for t ∈ [0,T], where A(t):t ∈ [0,T] is a family of closed linear operators, without assuming the density of their domains.

We show that the result of Kato on the existence of a semigroup solving the Kolmogorov system of equations in l₁ can be generalized to a larger class of the so-called Kantorovich-Banach spaces. We also present a number of related generation results that can be proved using positivity methods, as well as some examples.

Let ϰ be a positive, continuous, submultiplicative function on ${ℝ}^{+}$ such that $li{m}_{t\to \infty }{e}^{-\omega t}{t}^{-\alpha }\varkappa \left(t\right)=a$ for some ω ∈ ℝ, α ∈ $\overline{{ℝ}^{+}}$ and $a\in {ℝ}^{+}$. For every λ ∈ (ω,∞) let ${\varphi }_{\lambda }\left(t\right)=e^{-\lambda t}$ for $t\in {ℝ}^{+}$. Let $L_{\varkappa }^1\left({ℝ}^{+}\right)$ be the space of functions Lebesgue integrable on ${ℝ}^{+}$ with weight $\varkappa$, and let E be a Banach space. Consider the map ${\varphi }_{•}:(\omega ,\infty )\ni \lambda \to {\varphi }_{\lambda }\in L_{\varkappa }^1\left({ℝ}^{+}\right)$. Theorem 5.1 of the present paper characterizes the range of the linear map $T\to T{\varphi }_{•}$ defined on $L(L_{\varkappa }^1\left({ℝ}^{+}\right);E)$, generalizing a result established by B. Hennig and F. Neubrander for $\varkappa \left(t\right)=e^{\omega t}$. If ϰ ≡ 1 and E =ℝ then Theorem 5.1 reduces to D. V. Widder’s characterization...

This paper deals with the evolution Fokker-Planck-Smoluchowski configurational probability diffusion equation for the FENE dumbbell model in dilute polymer solutions. We prove the exponential convergence in time of the solution of this equation to a corresponding steady-state solution, for arbitrary velocity gradients.

Asymptotic convergence theorems for semigroups of nonnegative operators on a Banach lattice, on C(X) and on $L^p\left(X\right)$ (1 ≤ p ≤ ∞) are proved. The general results are applied to a class of semigroups generated by some differential equations.

We prove uniqueness of the invariant measure and the exponential convergence to equilibrium for a stochastic dissipative system whose drift is perturbed by a bounded function.

We prove uniqueness of the invariant measure and the exponential convergence to equilibrium for a stochastic dissipative system whose drift is perturbed by a bounded function.

The problem to be treated in this note is concerned with the asymptotic behaviour of stochastic semigroups, as the time becomes very large. The subject is largely motived by the Theory of Markov processes. Stochastic semigroups usually arise from pure probabilistic problems such as random walks stochastic differential equations and many others.An outline of the paper is as follows. Section one deals with the basic definitions relative to K-positivity and stochastic semigroups. Asymptotic behaviour...

We prove a priori estimates for a linear system of partial differential equations originating from the equations for the flow of a barotropic compressible viscous fluid under the influence of the gravity it generates. These estimates will be used in a forthcoming paper to prove the nonlinear stability of the motionless, spherically symmetric equilibrium states of barotropic, self-gravitating viscous fluids with respect to perturbations of zero total angular momentum. These equilibrium states as...

A new theorem on asymptotic stability and sweeping of substochastic semigroups is proved, and applied semigroups generated by birth-death processes.

We study the asymptotic behaviour of solutions of a transport equation. We give some sufficient conditions for the complete mixing property of the Markov semigroup generated by this equation.

We present a new necessary and sufficient condition for the asymptotic stability of Markov operators acting on the space of signed measures. The proof is based on some special properties of the total variation norm. Our method allows us to consider the Tjon-Wu equation in a linear form. More precisely a new proof of the asymptotic stability of a stationary solution of the Tjon-Wu equation is given.

We study the asymptotic behaviour of the Markov semigroup generated by an integro-partial differential equation. We give new sufficient conditions for asymptotic stability of this semigroup.