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Selfsimilar Blowup of Unstable Thin-film Equations

* with D. Slepcev *
Indiana Univ. Math. J. 54(2005)6:1697-1738

### Abstract

We study selfsimilar blowup of long-wave
unstable thin-film equations with critical powers of
nonlinearities:
*u*_{t} = -(*u^n*
*u*_{xxx} + *u^(n+2)*
*u*_{x})_{x}.
We show that the equation cannot have selfsimilar
solutions (with zero contact angles) that blow up in finite time if
$n \geq 3/2$. We show that for $0 < n < 3/2$ there are compactly
supported, symmetric, selfsimilar solutions (with zero contact
angles) that blow up in finite time. Moreover there exist families
of these solutions with any number of local maxima.
Asymptotic
behaviour of blowup selfsimilar solutions as $n$ approaches $3/2$
is also investigated,
and a sharp lower bound on the height of solutions one time
unit before the blowup is obtained. We also prove qualitative properties of
solutions; for example, a compactly supported selfsimilar
solution that blows up in finite time always has larger mass than
the compactly supported steady states.

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