A nonlinear integral equation occurring in a singular free boundary problem

Abstract:

We study the Cauchy problem \[ \left \{ \begin {gathered} {u_t} = \phi {({u_x})_x},\qquad (x,t) \in {\mathbf {R}} \times {{\mathbf {R}}_ + }, \hfill \\ u( \cdot ,0) = f \hfill \\ \end {gathered} \right .\] 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 \[ s(t) = - \kappa \sqrt t + o\left ( {\sqrt t } \right ),\qquad t \to {0^ + },\] where the constant $\kappa = 0.9034 \ldots$ is the (numerical) solution of a particular nonlinear equation. Moreover, we show that for any $\alpha \in (0,1/2)$, \[ \left | {\frac {{{d^2}}} {{d{t^2}}}f(s(t))} \right | = O({t^{\alpha - 1}}),\qquad t \to {0^ + }.\] The proof involves the analysis of a nonlinear singular integral equation.