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A selfsimilar solution to the focusing problem for the porous medium equation. (English) Zbl 0780.35079
The authors deal with the slow diffusion equation $$\partial u/\partial t=\Delta (u^ m)$$; $$u=u(x,t)$$; $$x\in\mathbb{R}^ d$$, $$m>1$$. They construct a selfsimilar solution of the second kind (it cannot be determined a priori from dimensional considerations) for the radially symmetric focusing problem.
In this case, we have an initial distribution of gas in the exterior of some compact set, and at finite time $$T$$ the gas will reach all points of the initially empty region $$R$$. This solution shows that in more than one space dimension, the velocity of the gas is infinite at the centre of $$R$$ at the focusing time $$T$$.