In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin lm equation in one-dimensional case
ht + @x(h n @xxxh) + @x(h n+2@xh) = 0; where n 1. There exists a critical mass Mc = 2 p 6 3 found by Witelski etal. (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
Mc. For n 4, entropy weak solutions are positive and unique. For n = 1,a nite time blow-up occurs for solutions with initial mass larger than Mc.For the Cauchy problem with n = 1 and initial mass less than Mc, we show that at least one of the following long-time behavior holds: the second moment goes to in nity as the time goes to in nity or h(; tk) * 0 in L1 (R) for some subsequence tk ! 1.