We study strictly parabolic stochastic partial differential equations on R d , d ≥ 1, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random , we give sufficient conditions on the correlation of the noise ensuring Hölder continuity for the trajectories of the solution of the equation. For self-adjoint operators with...

The paper is concerned with the optimal harvesting of a marine park, which is described by a parabolic heat equation with Neumann boundary conditions and a nonlinear source term. We consider a cost functional, which is linear with respect to the control; hence the optimal solution can belong to the class of measure-valued control strategies. For each control function, we prove existence, and st...

In this paper we study the numerical solution for an p−Laplacian type of evolution system Ht + ∇ × [|∇ × H|p−2∇ × H] = F(x, t), p > 2 in two space dimensions. For large p this system is an approximation of Bean’s critical-state model for type-II superconductors. By introducing suitable transformation, the system is equivalent to a nonlinear parabolic equation. For the nonlinear parabolic proble...

We introduce a stochastic version of the classical Perron’s method to construct viscosity solutions to linear parabolic equations associated to stochastic differential equations. Using this method, we construct easily two viscosity (sub and super) solutions that squeeze in between the expected payoff. If a comparison result holds true, then there exists a unique viscosity solution which is a ma...

We study the relative value iteration for the ergodic control problem under a nearmonotone running cost structure for a nondegenerate diffusion controlled through its drift. This algorithm takes the form of a quasilinear parabolic Cauchy initial value problem in R. We show that this Cauchy problem stabilizes, or in other words, that the solution of the quasilinear parabolic equation converges f...

We prove L 1 {contraction and thus uniqueness for solutions of the equation modelling un-stationary saturated{unsaturated water ow in porous media. Due to three diierent types of boundary conditions, describing contact with an impervious layer, with water reservoirs of variable level and with the atmosphere, this elliptic{parabolic equation comes as a varia-tional inequality with time{dependent...

We introduce a stochastic version of the classical Perron’s method to construct viscosity solutions to linear parabolic equations associated to stochastic differential equations. Using this method, we construct easily two viscosity (sub and super) solutions that squeeze in between the expected payoff. If a comparison result holds true, then there exists a unique viscosity solution which is a ma...

Abstract. A set of equations is derived from the Boltzmann kinetic equation describing charge transport in semiconductors. The unknowns of these equations depend on the space-time coordinates and the electron energy. The non-parabolic and parabolic band approximation are treated in detail. In these cases, the set of equations is equivalent to those obtained in the spherical-harmonics expansion....

A semilinear second-order parabolic equation is considered in a regular and a singularly perturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the backward Euler, Crank–Nicolson, and discontinuous Galerkin dG(r) methods are addressed. For their full discretizations, we employ elliptic reconstructions...

We study accuracy of alternating direction implicit (ADI) methods for parabolic equations. The original ADI method applied to parabolic equations is a perturbation of the Crank-Nicolson di erence equation and has second-order accuracy both in space and time. The perturbation error is on the same order as the discretization error, in terms of mathematical description. However, we often observe i...