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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The initial trace of a solution of the porous medium equation


Authors: D. G. Aronson and L. A. Caffarelli
Journal: Trans. Amer. Math. Soc. 280 (1983), 351-366
MSC: Primary 35K55; Secondary 76S05
MathSciNet review: 712265
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Abstract: Let $ u = u(x,t)$ be a continuous weak solution of the porous medium equation in $ {{\mathbf{R}}^d} \times (0,T)$ for some $ T > 0$. We show that corresponding to $ u$ there is a unique nonnegative Borel measure $ \rho $ on $ {{\mathbf{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 $ \vert x{\vert^2}$ as $ \vert x\vert \to \infty $.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0712265-1
Article copyright: © Copyright 1983 American Mathematical Society