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Transactions of the American Mathematical Society

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

 
 

 

The asymptotic behavior of gas in an $n$-dimensional porous medium


Authors: Avner Friedman and Shoshana Kamin
Journal: Trans. Amer. Math. Soc. 262 (1980), 551-563
MSC: Primary 35K05; Secondary 76S05
DOI: https://doi.org/10.1090/S0002-9947-1980-0586735-0
MathSciNet review: 586735
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Abstract: Consider the flow of gas in an n-dimensional porous medium with initial density ${u_0}(x) \geqslant 0$. The density $u(x, t)$ then satisfies the nonlinear degenerate parabolic equation ${u_t} = \Delta {u^m}$ where $m > 1$ is a physical constant. Assuming that $I \equiv \int { {u_0}(x)} dx < \infty$ it is proved that $u(x, t)$ behaves asymptotically, as $t \to \infty$, like the special (explicitly given) solution $V(|x|, t)$ which is invariant by similarity transformations and which takes the initial values $\delta (x)I (\delta (x) =$ the Dirac measure) in the distribution sense.


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Article copyright: © Copyright 1980 American Mathematical Society