DCDS-B
Global existence for a thin film equation with subcritical mass
Jian-Guo Liu Jinhuan Wang
Discrete & Continuous Dynamical Systems - B 2017, 22(4): 1461-1492 doi: 10.3934/dcdsb.2017070
In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case
$h_t+\partial_x (h^n\,\partial_{xxx} h)+\partial_x (h^{n+2}\partial_{x} h)=0,$
where
$n≥q 1$
. There exists a critical mass
$M_c=\frac{2\sqrt{6}π}{3}$
found by Witelski et al.(2004 Euro. J. of Appl. Math. 15,223-256) for
$n=1$
. We obtain global existence of a non-negative entropy weak solution if initial mass is less than
$M_c$
. For
$n≥q 4$
, entropy weak solutions are positive and unique. For
$n=1$
, a finite time blow-up occurs for solutions with initial mass larger than
$M_c$
. For the Cauchy problem with
$n=1$
and initial mass less than
$M_c$
, we show that at least one of the following long-time behavior holds:the second moment goes to infinity as the time goes to infinity or
$ h(·, t_k)\rightharpoonup 0$
in
$L^1(\mathbb{R})$
for some subsequence
${t_k} \to \infty $
.
keywords: Long-wave instability free-surface evolution equilibrium the Sz. Nagy inequality long-time behavior

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