We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in ${ℝ}^n$. The main results are global Schauder estimates, which hold in spite of the unboundedness of the coefficients.

We prove the unique solvability of parabolic equations with discontinuous leading coefficients in $W_p^{1,2}((0,T)\times {ℝ}^d)$. Using this result, we establish the uniqueness of diffusion processes with time-dependent discontinuous coefficients.

Si dà una condizione sufficiente per la esistenza di una soluzione in uno spazio di Gevrey ${\mathrm{\Gamma }}^d({𝐑}^{𝐧})$, $d$ razionale $\ge 1$, $n\ge 2$, di una equazione lineare a derivate parziali a coefficienti costanti $P(D)u=f$, quando $f\in {\mathrm{\Gamma }}^d({𝐑}^{𝐧})$. La dimostrazione completa dei risultati ottenuti è contenuta in una nota dell’autore in corso di pubblicazione su "Astérisque".

A general class of nonconforming meshes has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMA J. Numer. Anal. 30 (2010), 1009–1043). Thanks to the basic ideas developed in the stated reference for stationary problems, we derive a new discretization scheme in order to approximate the nonstationary heat problem. The unknowns of this scheme are the values at the centre of the control volumes, at some internal interfaces, and at the mesh points...

The paper presents the theory of the discontinuous Galerkin finite element method for the space-time discretization of a linear nonstationary convection-diffusion-reaction initial-boundary value problem. The discontinuous Galerkin method is applied separately in space and time using, in general, different nonconforming space grids on different time levels and different polynomial degrees $p$ and $q$ in space and time discretization, respectively. In the space discretization the nonsymmetric interior...

We introduce a solver method for spatially dependent and nonlinear fluid transport. The motivation is from transport processes in porous media (e.g., waste disposal and chemical deposition processes). We analyze the coupled transport-reaction equation with mobile and immobile areas. The main idea is to apply transformation methods to spatial and nonlinear terms to obtain linear or nonlinear ordinary differential equations. Such differential equations can be simply solved with Laplace transformation...

This article investigates the long-time behaviour of parabolic scalar conservation laws of the type ${\partial }_{t}u+{div}_{y}A(y,u)-{\Delta }_{y}u=0$, where $y\in {ℝ}^N$ and the flux $A$ is periodic in $y$. More specifically, we consider the case when the initial data is an $L^1$ disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in $L^1$ norm like a self-similar profile for large times. The proof uses a time and space change of variables which is...

Let X(t) be a diffusion process satisfying the stochastic differential equation dX(t) = a(X(t))dW(t) + b(X(t))dt. We analyse the asymptotic behaviour of p(t) = ProbX(t) ≥ 0 as t → ∞ and construct an equation such that $limsu{p}_{t\to \infty }{t}^{-1}{\int }_0^{t}p\left(s\right)ds=1$ and $limin{f}_{t\to \infty }{t}^{-1}{\int }_0^{t}p\left(s\right)ds=0$.