Global existence for a thin film equation with subcritical mass

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 $.