Keywords: nonlinear diffusion, free boundary, interface, support splitting, support merging, difference scheme.

We are concerned with the dynamical behaviour of non-stationary seepage in a non-linear filtration. The representative filtration is well known as the flow through porous media where the water evaporates. In particular, it is expected that such a seepage exhibits support splitting and merging phenomena, which are caused by the interaction between the nonlinear diffusion and the penetration of the fluid from the boundary on which the flowing tide and the ebbing tide occur. Here the support means the region where the fluid exists.

To model such phenomena in one dimensional space we introduce a model based on the equation described as the nonlinear initial boundary value problem, which is used to describe the flow through porous media with absorption [1,2]. This equation is also used to describe the propagation of thermal waves in plasma physics [3].

From analytical points of view, the existence and uniqueness of a weak solution and the comparison theorem are proved by Bertsch [4]. For the initial-boundary value problem Kersner [5] proved the appearance of support splitting phenomena, but he did not show that support merging phenomena appear after the support splits.

To investigate such phenomena to this initial boundary value problem it is important to construct a numerical method to it and to analyze the profile of the support of the stationary solution of it. We obtain the following results:

1. Numerical solutions given by our difference scheme converge to the exact one as the space mesh width tends to zero;
2. A stationary solution of the initial boundary value problem exists and is unique.
3. Stabilization Theorem; that is, the solution of the initial boundary value problem converges to a stationary solution as time tends to the infinity.

The proof of 1) can be obtained by the similar argument as stated in [6]. We also prove 2) by use of the standard theory of ordinary differential equations. The stabilization theorem is proved by use of the omega-limit theory and some inequality given by Bertsch [4].

We demonstrate some interesting numerical solutions, which show support splitting and merging phenomena.

1

P.Y. Polubarinova-Kochina, "Theory of Ground Water Movement", Princeton Univ. Press, 1962.

2

A.E. Scheidegger, "The Physics of Flow through Porous Media", Third edition, University of Toronto Press, 1974.

3

P. Rosenau, S. Kamin, "Thermal waves in an absorbing and convecting medium", Physica, 8D, 273-283, 1983. doi:10.1016/0167-2789(83)90325-1

4

M. Bertsch, "A class of degenerate diffusion equations with a singular nonlinear term", Nonlinear Anal., 7, 117-127, 1983. doi:10.1016/0362-546X(83)90110-4

5

R. Kersner, "Degenerate parabolic equations with general nonlinearities", Nonlinear Anal., 4, 1043-1062, 1980. doi:10.1016/0362-546X(80)90015-2

6

T. Nakaki, K. Tomoeda, "A finite difference scheme for some nonlinear diffusion equations in an absorbing medium: support splitting phenomena", SIAM J. Numer. Anal., 40, 945-964, 2002. doi:10.1137/S0036142900380303

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