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Transactions of the American Mathematical Society

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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
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(\vert x\vert,\,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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