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

 

A nonlinear integral equation occurring in a singular free boundary problem


Authors: Klaus Höllig and John A. Nohel
Journal: Trans. Amer. Math. Soc. 283 (1984), 145-155
MSC: Primary 35R35; Secondary 35K55, 45G10
MathSciNet review: 735412
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Abstract: We study the Cauchy problem

$\displaystyle \left\{ \begin{gathered}{u_t} = \phi {({u_x})_x},\qquad (x,t) \in... ...{{\mathbf{R}}_ + }, \hfill \\ u( \cdot ,0) = f \hfill \\ \end{gathered} \right.$

with the piecewise linear constitutive function $ \phi (\xi ) = {\xi _ + } = \max (0,\xi )$ and with smooth initial data $ f$ which satisfy $ xf'(x) \geqslant 0$, $ x \in {\mathbf{R}}$, and $ f''(0) > 0$. We prove that free boundary $ s$, given by $ {u_x}(s{(t)^ + },t) = 0$, is of the form

$\displaystyle s(t) = - \kappa \sqrt t + o\left( {\sqrt t } \right),\qquad t \to {0^ + },$

where the constant $ \kappa = 0.9034 \ldots $ is the (numerical) solution of a particular nonlinear equation. Moreover, we show that for any $ \alpha \in (0,1/2)$,

$\displaystyle \left\vert {\frac{{{d^2}}} {{d{t^2}}}f(s(t))} \right\vert = O({t^{\alpha - 1}}),\qquad t \to {0^ + }.$

The proof involves the analysis of a nonlinear singular integral equation.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0735412-5
PII: S 0002-9947(1984)0735412-5
Keywords: Cauchy problem, parabolic, nonlinear, free boundary regularity, nonlinear singular integral equation
Article copyright: © Copyright 1984 American Mathematical Society