In this paper, we study the time asymptotic behavior of the solution to an abstract Cauchy problem on Banach spaces without restriction on the initial data. The abstract results are then applied to the study of the time asymptotic behavior of solutions of an one-dimensional transport equation with boundary conditions in $L_1$-space arising in growing cell populations and originally introduced by M. Rotenberg, J. Theoret. Biol. 103 (1983), 181–199.

A condition on a family $B\left(t\right):t\in [0,T]$ of linear operators is given under which the inhomogeneous Cauchy problem for u"(t)=(A+ B(t))u(t) + f(t) for t ∈ [0,T] has a unique solution, where A is a linear operator satisfying the conditions characterizing infinitesimal generators of cosine families except the density of their domains. The result obtained is applied to the partial differential equation $$u_{tt}=u_{xx}+b(t,x)u_x(t,x)+c(t,x)u(t,x)+f(t,x)for(t,x)\in [0,T]\times [0,1],u(t,0)=u(t,1)=0fort\in [0,T],u(0,x)=u_0\left(x\right),u_t(0,x)=v_0\left(x\right)forx\in [0,1]$$ in the space of continuous functions on [0,1].

We give here a survey of some recent results on applications of topological quasi *-algebras to the analysis of the time evolution of quantum systems with infinitely many degrees of freedom.

2000 Mathematics Subject Classification: Primary 47A48, Secondary 60G12.In this paper classes of K^r -operators are considered – the classes of bounded and unbounded operators A with equal domains of A and A*, finite dimensional imaginary parts and presented as a coupling of a dissipative operator and an antidissipative one with real absolutely continuous spectra and the class of unbounded dissipative K^r -operators A with different domains of A and A* and with real absolutely continuous spectra....

In the first note we show for a strongly continuous family of operators ${\left(T\left(t\right)\right)}_{t\ge 0}$ that if every orbit $t\mapsto T\left(t\right)x$ is differentiable for $t>t_x$, then all orbits are differentiable for $t>t_0$ with $t_0$ independent of $x$. In the second note we give an example of an eventually differentiable semigroup which is not differentiable on the same interval in the operator norm topology.