We show that the domain of the Ornstein-Uhlenbeck operator on $L^p$$({ℝ}^N,\mu dx...$

We establish existence of mild solutions for the semilinear first order functional abstract Cauchy problem and we prove that the set of mild solutions of this problem is connected in the space of continuous functions.

We consider numerical approximation to the solution of non-autonomous evolution equations. The order of convergence of the simplest possible Magnus method is investigated.

Si considera, in uno spazio di Hilbert $H$ l'operatore lineare ${\mathfrak{M}}_0\phi =1/2\text{Tr}\left[D^2\phi \right]+〈x,AD\phi 〉-〈DU\left(x\right),D\phi 〉$, dove $A$ è un operatore negative autoaggiunto e $U$ è un potenziale che soddisfa a opportune condizioni di integrabilità. Si dimostra con un metodo analitico che ${\mathfrak{M}}_0$ è essenzialmente autoaggiunto in uno spazio $L^2\left(H,\nu \right)$ e si caratterizza il dominio della sua chiusura $\mathfrak{M}$ come sottospazio di $W^{2,2}\left(H,\nu \right)$. Si studia inoltre la «spectral gap property» del semigruppo generato da $\mathfrak{M}$.

We give a new proof, based on analytic semigroup methods, of a maximal regularity result concerning the classical Cauchy-Dirichlet's boundary value problem for second order parabolic equations. More specifically, we find necessary and sufficient conditions on the data in order to have a strict solution $u$ which is bounded with values in $C^{2+\theta }\left(\overline{\mathrm{\Omega }}\right)$(0 < < 1), with${\partial }_{t}u$ bounded with values in $C^{\theta }\left(\overline{\mathrm{\Omega }}\right)$.

We show how results by Diekmann et al. (2007) on the qualitative behaviour of solutions of delay equations apply directly to a resource-consumer model with age-structured consumer population.

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.