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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Concavity of solutions of the porous medium equation


Authors: Philippe Bénilan and Juan Luis Vázquez
Journal: Trans. Amer. Math. Soc. 299 (1987), 81-93
MSC: Primary 35K60; Secondary 76S05
MathSciNet review: 869400
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Abstract: We consider the problem

$\displaystyle \left( {\text{P}} \right)\quad \quad \left\{ {\begin{array}{*{20}... ...\in {\mathbf{R}},\,t > 0} \\ {{\text{for}}\,x \in {\mathbf{R}}} \\ \end{array} $

where $ m > 1$ and $ {u_0}$ is a continuous, nonnegative function that vanishes outside an interval $ (a,\,b)$ and such that $ (u_0^{m - 1})'' \leq - C \leq 0$ in $ (a,\,b)$. Using a Trotter-Kato formula we show that the solution conserves the concavity in time: for every $ t > 0,\,u(x,t)$ vanishes outside an interval $ \Omega (t) = ({}_{\zeta 1}(t),\,{}_{\zeta 2}(t))$ and

$\displaystyle {({u^{m - 1}})_{xx}} \leq - \frac{C} {{1 + C(m(m + 1)/(m - 1))t}}$

in $ \Omega (t)$. Consequently the interfaces $ x{ = _{\zeta i}}(t)$, $ i = 1,\,2$, are concave curves. These results also give precise information about the large time behavior of solutions and interfaces.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0869400-8
PII: S 0002-9947(1987)0869400-8
Keywords: Concavity, flow in porous media, Trotter-Kato formula, interfaces, asymptotic behavior
Article copyright: © Copyright 1987 American Mathematical Society