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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Concavity of solutions of the porous medium equation
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by Philippe Bénilan and Juan Luis Vázquez PDF
Trans. Amer. Math. Soc. 299 (1987), 81-93 Request permission

Abstract:

We consider the problem \[ \left ( {\text {P}} \right )\quad \quad \left \{ {\begin {array}{*{20}{c}} {{u_t} = {{({u^m})}_{xx}},} \\ {u(x,0) = {u_0}(x)} \\ \end {array} } \right .\quad \begin {array}{*{20}{c}} {{\text {with}} x \in {\mathbf {R}}, t > 0} \\ {{\text {for}} x \in {\mathbf {R}}} \\ \end {array} \] 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 \[ {({u^{m - 1}})_{xx}} \leq - \frac {C} {{1 + C(m(m + 1)/(m - 1))t}}\] 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
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 299 (1987), 81-93
  • MSC: Primary 35K60; Secondary 76S05
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0869400-8
  • MathSciNet review: 869400