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The initial trace of a solution of the porous medium equation. (English) Zbl 0556.76084
Let u(x,t) be a continuous weak solution of the porous medium equation \(\partial u/\partial t=\Delta (u^ m)\) in \({\mathbb{R}}^ d\times (0,T]\) for some \(T>0\). We show that corresponding to u there is a unique nonnegative Borel measure \(\rho\) on \({\mathbb{R}}^ d\) which is the initial trace of u. Moreover, we show that the initial trace \(\rho\) must belong to a certain growth class. Roughly speaking, this growth restriction shows that there are no solutions of the porous medium equation whose pressure grows, on average, more rapidly then \(| x|^ 2\) as \(| x| \to \infty\).

MSC:
76S05 Flows in porous media; filtration; seepage
35Q99 Partial differential equations of mathematical physics and other areas of application
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