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Long-wave instabilities and saturation in thin film equations

* with A. Bertozzi, CPAM 51(1998)625-661.
*

### Abstract

Hocherman and Rosenau conjectured that long-wave unstable Cahn-Hilliard
type interface models develop finite-time singularities when the
nonlinearity in the destabilizing term grows faster at large
amplitudes than the nonlinearity in the stabilizing term
(Phys. D 67:113-125, 1993). We disprove this conjecture for a class of equations,
often used to model thin films in a lubrication context, by showing
that in fact the destabilizing term can be stronger than previously
conjectured yet
the solution still remains globally bounded.
For example, they conjecture that for the long-wave unstable equation
*h*_{t} = - (*h*^{n} *h*_{xxx})_{x} - (*h*^{m} *h*_{x})_{x},

*m*>*n* leads to
blow-up. Using a conservation of volume constraint, we conjecture a
different critical exponent for possible singularities
of non-negative solutions.
We prove that nonlinearities with exponents below the conjectured
critical exponent yield globally bounded solutions.
Specifically, for the above
equation, solutions are bounded if *m*<*n*+2. The bound is proved using
energy methods and is then used to prove the existence of
non-negative weak solutions in the sense of distributions.
We present preliminary numerical evidence suggesting
that *m*>*n*+2
can allow blow-up.

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A. B. was supported by an ONR Young Investigator/PECASE award and an
Alfred P. Sloan Research Fellowship. M. P. was supported by an NSF
postdoctoral fellowship while at the Courant Institute and the Ambrose
Monell Foundation while at the Institute for Advanced Study.