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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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##### References:
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