| Contents |
| We discuss the asymptotic behavior of globally defined non-negative weak solutions of the two-phase porous medium equation
\begin{equation*}
\left\{
\begin{array}{rcl}
\partial_t f & = & \partial_x\left(f\partial_x\left( (1+R)f+Rg\right)\right),\\
\partial_t g & = & R_\mu\partial_x\left(g\partial_x\left( f+g\right)\right),
\end{array}
\right.
\qquad (t,x)\in (0,\infty)\times \mathbb{R},
\end{equation*}
whereby $R$ and $R_\mu$ are positive parameters.
This strongly coupled degenerate parabolic system was obtained as the thin film approximation of the Muskat problem.
The existence of globally defined non-negative weak solutions is established by interpreting the system of evolution equations as a
gradient flow for the $L_2-$Wasserstein distance.
This system possesses a rich landscape of self-similar symmetric and non-symmetric Barenblatt type profiles.
We show that any non-negative global solution converges in the large towards a self-similar solution of the problem. |
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