###
Finite-time Blow-up of Solutions of Some Long-wave Unstable Thin Film Equations

* with A. L. Bertozzi,
Indiana Univ. Math. J. 49(2000)4:1323-1366.
*

to appear in Indiana University Mathematics Journal
### Abstract

We consider the family of long-wave unstable lubrication equations

*h*_{t} = -(*h h*_{xxx})_{x} - (*h*^{m} *h*_{x})_{x}

with .
Given a fixed ,
we prove the existence of a
weak solution that becomes singular in finite time. Specifically,
given compactly supported nonnegative initial data with negative
energy, there is a time
,
determined by *m* and the
*H*^{1} norm of the initial data, and a compactly supported nonnegative
weak solution such that
.
We discuss the relevance of these singular solutions to an earlier
conjecture [Comm Pure Appl Math 51:625-661, 1998] on when
finite-time singularities are possible for long-wave unstable
lubrication equations.

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We thank Andrew J. Bernoff for pointing out the formal second moment
argument upon which the finite-time blow-up proof hinges. M. P. thanks
Richard S. Laugesen for useful mathematical conversations.

A. B. was supported by an ONR Young Investigator/PECASE award and an
Alfred P. Sloan Research Fellowship. M. P. was supported by NSF grant
number DMS-9971392 and an Alfred P. Sloan Research Fellowship.