In this article we discuss numerical scheme for the approximation of the Willmore flow of graphs. The scheme is based on the finite difference method. We improve the scheme we presented in Oberhuber [Obe-2005-2,Obe-2005-1] which is based on combination of the forward and the backward finite differences. The new scheme approximates the Willmore flow by the central differences and as a result it better preserves symmetry of the solution. Since it requires higher regularity of the solution, additional...

Our aim in this paper is to study the long time behavior of a class of doubly nonlinear parabolic equations. In particular, we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension.

We consider a fully practical finite element approximation of the following degenerate system$$\frac{\partial }{\partial t}\rho \left(u\right)-\nabla .\left(\alpha \left(u\right)\nabla u\right)\ni \sigma \left(u\right){\left|\nabla \phi \right|}^2,\nabla .\left(\sigma \left(u\right)\nabla \phi \right)=0$$subject to an initial condition on the temperature, $u$, and boundary conditions on both $u$ and the electric potential, $\phi$. In the above $\rho \left(u\right)$ is the enthalpy incorporating the latent heat of melting, $\alpha \left(u\right)\>0$ is the temperature dependent heat conductivity, and $\sigma \left(u\right)\ge 0$ is the electrical conductivity. The latter is zero in the frozen zone, $u\le 0$, which gives rise to the degeneracy in this Stefan system. In addition to showing stability...

We consider a fully practical finite element approximation of the following degenerate system $$\frac{\partial }{\partial t}\rho \left(u\right)-\nabla .\left(\alpha \left(u\right)\nabla u\right)\ni \sigma \left(u\right){\left|\nabla \phi \right|}^2,\nabla .\left(\sigma \left(u\right)\nabla \phi \right)=0$$ subject to an initial condition on the temperature, u, and boundary conditions on both u and the electric potential, ϕ. In the above p(u) is the enthalpy incorporating the latent heat of melting, α(u) > 0 is the temperature dependent heat conductivity, and σ(u) > 0 is the electrical conductivity. The latter is zero in the frozen zone, u ≤ 0, which gives rise to the degeneracy in this Stefan...

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure, and a second energy...

This note is concerned with proving the finite speed of propagation for some non-local porous medium equation by adapting arguments developed by Caffarelli and Vázquez (2010).

In this paper we deal with a problem of segmentation (including missing boundary completion) and subjective contour creation. For the corresponding models we apply the semi-implicit finite volume numerical schemes leading to methods which are robust, efficient and stable without any restriction to a time step. The finite volume discretization enables to use the spatial adaptivity and thus improve significantly the computational time. The computational results related to image segmentation with partly...

We study the Cauchy problem in the hyperbolic space ${ℍ}^n(n\ge 2)$ for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space ${ℝ}^n$ new phenomena arise, which depend on the properties of the diffusion process in ${ℍ}^n$. We also investigate a family of travelling wave solutions, named...

Hölder continuity and, basing on this, full regularity and global existence of weak solutions is studied for a general nondiagonal parabolic system of nonlinear differential equations with the matrix of coefficients satisfying special structure conditions and depending on the unknowns. A technique based on estimating a certain function of unknowns is employed to this end.