We consider the Burgers equation and prove a property which seems to have been unobserved until now: there is no limitation on the growth of the nonnegative initial datum u0(x) at infinity when the problem is formulated on unbounded intervals, as, e.g. (0 +∞), and the solution is unique without prescribing its behaviour at infinity. We also consider the associate stationary problem. Finally, some applications to the linear heat equation with boundary conditions of Robin type are also given.

Inspired by the growing use of non linear discretization techniques for the linear diffusion equation in industrial codes, we construct and analyze various explicit non linear finite volume schemes for the heat equation in dimension one. These schemes are inspired by the Le Potier’s trick [C. R. Acad. Sci. Paris, Ser. I 348 (2010) 691–695]. They preserve the maximum principle and admit a finite volume formulation. We provide a original functional setting for the analysis of convergence of such methods....

The existence of a solution of the two - dimensional heat conduction equation in a semi-infinite strip, under mixed boundary condition, is discussed.

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)...