In this paper we study a linear population dynamics model. In this model, the birth process is described by a nonlocal term and the initial distribution is unknown. The aim of this paper is to use a controllability result of the adjoint system for the computation of the density of individuals at some time $T$.

The internal and boundary exact null controllability of nonlinear convective heat equations with homogeneous Dirichlet boundary conditions are studied. The methods we use combine Kakutani fixed point theorem, Carleman estimates for the backward adjoint linearized system, interpolation inequalities and some estimates in the theory of parabolic boundary value problems in Lk.

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically ${ℝ}_{+}$ or ${ℝ}^N$. Considering an unbounded and disconnected control region of the form $\omega :={\cup }_n{\omega }_n$, we prove two null controllability results: under some technical assumption on the control parts ${\omega }_n$, we prove that every initial datum in some weighted $L^2$ space can be controlled to zero by usual control functions, and every initial datum in $L^2\left(\Omega \right)$ can...

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically ${ℝ}_{+}$ or ${ℝ}^N$. Considering an unbounded and disconnected control region of the form $\omega :={\cup }_n{\omega }_n$, we prove two null controllability results: under some technical assumption on the control parts ${\omega }_n$, we prove that every initial datum in some weighted L2 space can be controlled to zero by usual control functions, and every initial datum in L2(Ω)...

Nel presente articolo si illustrano alcuni dei principali metodi numerici per l'approssimazione di modelli matematici legati ai fenomeni di transizione di fase. Per semplificare e contenere l'esposizione ci siamo limitati a discutere con un certo dettaglio i metodi più recenti, presentandoli nel caso di problemi modello, quali il classico problema di Stefan e l'evoluzione di superficie per curvatura media, solo accennando alle applicazioni e modelli più generali.

We consider the numerical solution of diffusion problems in $(0,T)\times \Omega$ for $\Omega \subset {ℝ}^d$ and for $T\>0$ in dimension $d\ge 1$. We use a wavelet based sparse grid space discretization with mesh-width $h$ and order $p\ge 1$, and $hp$ discontinuous Galerkin time-discretization of order $r=O\left(\left|logh\right|\right)$ on a geometric sequence of $O\left(\left|logh\right|\right)$ many time steps. The linear systems in each time step are solved iteratively by $O\left(\left|logh\right|\right)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an $L^2\left(\Omega \right)$-error of $O\left(N^{-p}\right)$ for $u(x,T)$ where $N$ is the total number of operations,...

We consider the numerical solution of diffusion problems in (0,T) x Ω for $\Omega \subset {ℝ}^d$ and for T > 0 in dimension dd ≥ 1. We use a wavelet based sparse grid space discretization with mesh-width h and order pd ≥ 1, and hp discontinuous Galerkin time-discretization of order $r=O\left(\left|logh\right|\right)$ on a geometric sequence of $O\left(\left|logh\right|\right)$ many time steps. The linear systems in each time step are solved iteratively by $O\left(\left|logh\right|\right)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L2(Ω)-error of O(N-p) for u(x,T)...

This paper presents two observability inequalities for the heat equation over $\Omega \times (0,T)$. In the first one, the observation is from a subset of positive measure in $\Omega \times (0,T)$, while in the second, the observation is from a subset of positive surface measure on $\partial \Omega \times (0,T)$. It also proves the Lebeau-Robbiano spectral inequality when $\Omega$ is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.

We deal with a Penrose-Fife type model for phase transition. We assume a rather general constitutive low for the heat flux and treat the Dirichlet and Neumann boundary condition for the temperature. Some of our proofs apply to different types of boundary conditions as well and improve some results existing in the literature.