We consider a non degenerate quasilinear parabolic stochastic partial differential equation with a uniformly elliptic diffusion matrix. It is driven by a nonlinear noise. We study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine conditions on coefficients and initial data under which the weak solution is Hölder continuous in time an...

The global-in-time existence of nonnegative solutions to a parabolic strongly coupled system with mixed Dirichlet-Neumann boundary conditions is shown. The system describes the time evolution of the electron and hole densities in a semiconductor when electron-hole scattering is taken into account. The parabolic equations are coupled to the Poisson equation for the electrostatic potential. Writt...

This lecture will outline the developments over the past two years of Todd Dupont and the author on problems of flow of fluids in porous media arising from petroleum reservoir engineering. Two specific problems will be discussed and a third mentioned. The first is the single-phase flow of a non-ideal gas. This flow can be described by a nonlinear parabolic equation. The second is the two-phase ...

A free boundary problem for the dynamics of a glasslike binary fluid naturally leads to singular perturbation strongly degenerate parabolic partial differential equation in 1D. We present conjecture an asymptotic formula velocity and prove weak version conjecture. The results are based on analysis family local travelling wave solutions.

A finite volume method for solving the degenerate chemotaxis model is presented, along with numerical examples. This model consists of a degenerate parabolic convection– diffusion PDE for the density of the cell-population coupled to a parabolic PDE for the chemoattractant concentration. It is shown that discrete solutions exist, and the scheme converges.

In this paper, we consider the controllability of a strongly degenerate parabolic equation with a degenerate one-order transport term. Despite the strong degeneracy, we prove a result of well-posedness and null controllability with a Dirichlet boundary control that acts on the degenerate part of the boundary. Then, we study the uniform controllability in the vanishing viscosity limit and prove ...

Consider the Cauchy problem for the nonlinear hyperbolic-parabolic equation: (*) ut + 1 2 a · ∇xu = ∆u+ for t > 0, where a is a constant vector and u+ = max{u, 0}. The equation is hyperbolic in the region [u < 0] and parabolic in the region [u > 0]. It is shown that entropy solutions to (*), that grow at most linearly as |x| → ∞, are stable in a weighted L(IR ) space, which implies that the sol...

Mohammed Kbiri Alaoui Department of Mathematics, King Khalid University, P.O. Box 9004, Abha, Saudi Arabia Correspondence should be addressed to Mohammed Kbiri Alaoui, mka [email protected] Received 31 March 2011; Accepted 28 July 2011 Academic Editor: Mihai Putinar Copyright q 2011 Mohammed Kbiri Alaoui. This is an open access article distributed under the Creative Commons Attribution License, which...