We establish comparison and existence theorems of viscosity solutions of the initial-boundary value problem for some singular degenerate parabolic partial di/erential equations with nonlinear oblique derivative boundary conditions. The theorems cover the capillary problem for the mean curvature 1ow equation and apply to more general Neumann-type boundary problems for parabolic equations in the ...

In this paper, we study time discretizations of fully nonlinear parabolic differential equations. Our analysis uses the fact that the linearization along the exact solution is a uniformly sectorial operator. We derive smooth and nonsmooth-data error estimates for the backward Euler method, and we prove convergence for strongly A(θ)stable Runge–Kutta methods. For the latter, the order of converg...

A convergence analysis is presented for the implicit Euler and Lie splitting schemes when applied to nonlinear parabolic equations with delay. More precisely, we consider a vector field which is the sum of an unbounded dissipative operator and a delay term, where both point delays and distributed delays fit into the framework. Such equations are frequently encountered, e.g. in population dynami...

We consider the exponential reaction–diffusion equation in space-dimension n ∈ (2, 10). We show that for any integer k ≥ 2 there is a backward selfsimilar solution which crosses the singular steady state k-times. The sameholds for the power nonlinearity if the exponent is supercritical in the Sobolev sense and subcritical in the Joseph–Lundgren sense. © 2008 Elsevier Ltd. All rights reserved.

The Perona–Malik nonlinear parabolic problem, which is widely used in image processing, is investigated in this paper from the numerical point of view. An explicit finite volume numerical scheme for this problem is presented and consistency property is proved.

We consider a family of multi-phase Stefan problems for a certain 1-d model of cell-to-cell adhesion and diffusion, which takes the form of a nonlinear forward-backward parabolic equation. In each material phase the cell density stays either high or low, and phases are connected by jumps across an 'unstable' interval. We develop an existence theory for such problems which allows for the annihil...

This paper concerns the weak solutions to a Cauchy problem in RN for a degenerate nonlinear parabolic equation. We obtain the Hölder regularity of the weak solutions to this problem.