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Transactions of the American Mathematical Society

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

 

 

The interfaces of one-dimensional flows in porous media


Author: Juan L. Vázquez
Journal: Trans. Amer. Math. Soc. 285 (1984), 717-737
MSC: Primary 35R35; Secondary 76S05
MathSciNet review: 752500
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Abstract: The solutions of the equation $ {u_t} = {({u^m})_{x\,x}}$ for $ x \in {\mathbf{R}},0 < t < T,m > 1$, where $ u(x,0)$ is a nonnegative Borel measure that vanishes for $ x > 0$ (and satisfies a growth condition at $ - \infty $), exhibit a finite, monotone, continuous interface $ x = \zeta (t)$ that bounds to the right the region where $ u > 0$. We perform a detailed study of $ \zeta$: initial behaviour, waiting time, behaviour as $ t \to \infty $. For certain initial data the solutions blow up in a finite time $ {T^{\ast}}$: we calculate $ {T^{\ast}}$ in terms of $ u(x,0)$ and describe the behaviour of $ \zeta $ as $ t\, \uparrow \,{T^{\ast}}$.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0752500-8
Keywords: Flows in porous media, interfaces, blow-up time, waiting time, asymptotic behaviour
Article copyright: © Copyright 1984 American Mathematical Society