A numerical support tracking method for a porous media equation with strong absorption

Kenji Tomoeda

Faculty of Engineering, Osaka Institute of Technology, 5-16-1, Omiya, Ahahi-ku, Osaka, 535-8585, Japan

tomoeda@ge.oit.ac.jp

Contributed talk

A porous media equation is well-known as the representative model which describes several phenomena caused by the effect of nonlinear diffusion. Among such phenomena, `` total extinction in a finite time" is an interesting one caused by the interaction between diffusion and absorption in the fields of fluid dynamics, plasma physics and population dynamics. Here the volumetric absorption is given by evaporation, radiation, death and so on.

To investigate such a phenomenon, we use the following one-dimensional initial value problem which is written in the form of the porous media equation with absorption:

$$v_t=(v^m)_{xx}-cv^p,\quad t>0, x\in\mbox{\bf R}^{\rm 1},$$ (1)

$$v(0,x)=v^0(x),\quad x\in\mbox{\bf R}^{\rm 1},$$ (2)

where $m>1$, $c>0$, $p>0$ and $m+p\ge2$, and $v^0(x)\in C(\mbox{\bf R}^{\rm 1})$ is bounded and nonnegative and has compact support. Kalashinikov [1] proved that `` total extinction in a finite time" occurs only in the case where $0<p<1$. Taking this property into consideration, we may expect that the support splitting phenomena appear in the case where $v^0(x)$ has two local maxima. This motivates us to investigate such phenomena in both numerical and analytical points of view.

In this talk we propose a numerical support tracking method in the specific case where $m+p=2\ (0<p<1)$, and state some sufficient conditions imposed on $v^0(x)$ with two local maxima under which the support behaves as follows:

(i)

The support splitting phenomena appear for the initially connected support([3],[4]);

(ii)

The support becomes connected for the initially disconnected support, and thereafter the support splitting phenomena appear;

(iii)

The support splitting phenomena never appear for $t>0$.

Our mathematical methods are based on finite difference methods, the comparison theorem and Kersner's exact solution([2]).