Nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type are considered. A general class of difference methods for the problem is constructed. Theorems on the convergence of difference schemes and error estimates of approximate solutions are presented. The proof of the stability of the difference functional problem is based on a comparison techn...

We consider the Cauchy problem for a doubly nonlinear parabolic equation, obtaining optimal estimates both for the sup norm of nonnegative solutions, and for their support for large times.

In this paper we explain the notion of stochastic backward differential equations and its relationship with classical (backward) parabolic differential equations of second order. The paper contains a mixture of stochastic processes like Markov processes and martingale theory and semi-linear partial differential equations of parabolic type. Some emphasis is put on the fact that the whole theory ...

Given two probability measures μ and ν we consider a mass transportation mapping T satisfying 1) T sends μ to ν, 2) T has the form T = φ ∇φ |∇φ| , where φ is a function with convex sublevel sets. We prove a change of variables formula for T . We also establish some a priori estimates for T , and a new form of the parabolic maximum principle. In addition, we discuss relations to the Monge–Kantor...

We prove the doubling property of L-caloric measure corresponding to the second order parabolic equation in the whole space and in Lipschitz domains. For parabolic equations in the divergence form, a weaker form of the doubling property follows easily from a recent result, the backward Harnack inequality, and known estimates of the Green's function. Our method works for both the divergence and ...

In this note we present a new Rosenbrock solver which is third{order accurate for nonlin-ear parabolic problems. Since Rosenbrock methods suuer from order reductions when they are applied to partial diierential equations, additional order conditions have to be satissed. Although these conditions have been known for a longer time, from the practical point of view only little has been done to con...

The Perona-Malik equation is an ill-posed forward-backward parabolic equation with some application in image processing. In this paper, we study the Perona-Malik type equation on a ball in an arbitrary dimension n and show that there exist infinitely many radial weak solutions to the homogeneous Neumann boundary problem for smooth nonconstant radially symmetric initial data. Our approach is to ...