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S. Antontsev\\
Universidad da Beira Interior, \\
Covilh$\widetilde{a}$, Portugal\\

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and\\

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S. Shmarev\\
Universidad de Oviedo, Spain\\


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{\bf Parabolic equations with anisotropic non-uniform degeneracy}
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Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with
Lipschitz-continuous boundary $\Gamma$ and $Q=\Omega\times (0,T]$.
We study the parabolic equations with anisotropic non-uniform
degeneracy
\begin{equation}\label{eq1}
%\begin{split}
%& u_t
u_t-\sum\limits_{i=1}^nD_{i}\left(a_i(x,t,u)|D_{i}u|^{p_i(x,t)-2}D_iu\right)
 +c(x,t,u)|u|^{\sigma(x,t)-2}u=f(x,t)\quad\ {\rm in}\ Q,
\end{equation}
\begin{equation}\label{eq2}
u_{t}-\sum\limits_{i=1}^{n}D_{i}\left(a_{i}(x,t,u)
|u|^{\gamma_i(x,t)}D_i\,u +b_i(x,t,u)|u|^{\delta_i(x,t)}u\right)
+c(x,t,u)|u|^{\sigma(x,t)}u=f(x,t)
\end{equation}
under the boundary and initial conditions
\begin{equation}\label{eq3}
{u=0\ {\rm on}\ \Gamma}, \qquad {u(x,0)=u_{0}(x)\ {\rm in}\
\Omega}.
\end{equation}
The coefficients $\gamma_i$, $\delta_i$, $a_i$, $b_i$, $c$, $p_i$
and $\sigma$ are given functions of their arguments. Such
equations emerge from the mathematical modelling of
electro-rheological fluids, fluids with temperature-dependent
viscosity, the processes of filtration in inhomogenenous
anisotropic media. It is assumed that the coefficients $a_i$, $c$
are bounded, $a_i>0$, $c\geq 0$, and the exponents $p_i(x,t)>1$,
$\sigma(x,t)>1$ are  continuous in $Q$ with a logarithmic module
of continuity.

We prove the existence and uniqueness of weak solutions of
problems (\ref{eq1}), (\ref{eq3}) and (\ref{eq2}), (\ref{eq3}) and
study the localization (vanishing) properties of weak solutions
and the effect of finite time stabilization of solutions to a
stationary profile. The study of the localization properties is
performed with the method of local energy estimates
\cite{ADS-2002}. The detailed proofs can be found in
\cite{anzh-2004,an-ro-2005,ansh-2004-p4,AS-2005}
\begin{thebibliography}{1}

\bibitem{anzh-2004}
{\sc S.~Antontsev and V.~Zhikov}, {\em Higher integrability for
parabolic
  equations of $p(x,t)-$ Laplacian type}, Preprint 5, Universidade da Beira
  Interior, Portugal, www.ubi.pt/externos/noe.html,  (2004), pp.~0--30.

\bibitem{ADS-2002}
{\sc S.~N. Antontsev, J.~I. D{\'\i}az, and S.~Shmarev}, {\em
Energy Methods for
  Free Boundary Problems:Applications to Non-linear PDEs and Fluid Mechanics},
  Bikh\"auser, Boston, 2002.
\newblock Progress in Nonlinear Differential Equations and Their Applications,
  Vol. 48.

\bibitem{an-ro-2005}
{\sc S.~N. Antontsev and F.~Rodrigues}, {\em On stationary
thermo-rheological
  viscous flows}, Ann. Univ. Ferrara, Sez. , Sc. Mat.
\newblock To appear.

\bibitem{ansh-2004-p4}
{\sc S.~N. Antontsev and S.~I. Shmarev}, {\em Existence and
  uniqueness of solutions of degenerate parabolic equations with variable
  exponents of nonlinearity}, Preprint 3, Universidade da Beira Interior,
  Portugal, www.ubi.pt/externos/noe.html,  (2004), pp.~1--16.

\bibitem{AS-2005}
\leavevmode\vrule height 2pt depth -1.6pt width 23pt, {\em A model
porous
  medium equation with variable exponent of nonlinearity: existence, uniqueness
  and localization properties of solutions}, Journal of Nonlinear Analysis, 60
  (2005), pp.~515--545.

\end{thebibliography}

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