  We analyze the contact Hele-Shaw problem with zero surface tension of a free
boundary in a thin domain $\Omega^{\varepsilon}(t).$ Under suitable conditions
on the given data, the one-valued local classical solvability of the problem
for each fixed value of the parameter $\varepsilon$ is proved.
  Using the multiscale analysis, we study the asymptotic behavior of this
problem as $\varepsilon \to 0,$ i.e., when the thin domain
$\Omega^{\varepsilon}(t)$ is shrunk into the interval $(0, l).$ Namely, we find
exact representation of the free boundary for $t\in[0,T],$ derive the
corresponding limit problem $(\varepsilon= 0),$ define other terms of the
asymptotic approximation and prove appropriate asymptotic estimates that
justify this approach.
  We also establish the preserving geometry of the free boundary near corner
points for $t\in[0,T]$ under assumption that free and fixed boundaries form
right angles at the initial time $t=0$.
