Ján Filo

Diffusivity versus Absorption through the Boundary

for all 0 1 then we do expect to find that any solution becomes unbounded in finite time. See [ 13, 15, 63. It is our goal to prove an analogous result for m f 1. For a = 1 one can find that if v is a positive solution of

Blow-up on the boundary: a survey

where m, p > 0 and Ω is either a smoothly bounded domain in R or Ω = R+ = {(x1, x′) : x′ ∈ RN−1, x1 > 0}, ν is the outward normal. Over the past two decades this problem has received considerable interest. For Ω bounded, m = 1 and p > 1 it was shown by Levine and Payne ([LP1]) in 1974 and by Walter ([Wa]) in 1975 that there are solutions which blow up in finite time. This means that lim sup t→T max Ω u(x, t) =∞ for some T <∞.

A Free Boundary Problem in Dermal Drug Delivery

In this paper we study a free boundary problem in a multicomponent domain. Our study was motivated by the mathematical modeling of dermal and transdermal drug delivery, where the multilayered skin model was considered. At the interface connecting two components the conservation of the flux and Nernsts distribution law hold and it is supposed that in any component there is a positive minimum concentration at which the diffusion front can proceed. The existence of a solution and uniqueness in special cases are shown.

L.∞-estimate for nonlinear diffusion equation

In this paper, the nonlinear parabolic equation of the form in the cylinder bounded being nonlinear r-Laplacian, with Dirichlet boundary conditions is considered. It is proved that the value of ess plays a key role for obtaining the L∞ estimate of solutions.

An approximation of the axially symmetric flow through a pipe-like domain with a moving part of a boundary

The purpose of this work is to study the existence of solutions for approximation of an unsteady fluid-structure interaction problem. We consider a perturbed Navier-Stokes equation in the cylindrical coordinate system assuming axially symmetric flow. A priori unknown part of the boundary (that interacts with the fluid) is governed with a linear equation of fifth order. We prove the existence of at least one weak solution as long as the boundary does not touch the axis of symmetry. An explicit expression for a class of divergence-free functions is given.MSC:35Q30, 35Q35, 35D05, 74F10, 76D03.