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.

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

The self-consistent chemotaxis-fluid system $$\left\{\begin{array}{cc}{n}_t+u·\nabla n=\Delta n-\nabla ·\left(n\nabla c\right)+\nabla ·\left(n\nabla \phi \right),\hfill & x\in \Omega ,t>0,\hfill \\ c_t+u·\nabla c=\Delta c-nc,\hfill & x\in \Omega ,t>0,\hfill \\ u_t+\kappa (u·\nabla )u+\nabla P=\Delta u-n\nabla \phi +n\nabla c,\hfill & x\in \Omega ,t>0,\hfill \\ \nabla ·u=0,\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$$ is considered under no-flux boundary conditions for $n,c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $\Omega \subset {ℝ}^N$$(N=2,3...$