Abstract:
In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case ht + x(hn xxxh) + x(hn+2xh) = 0; where n 1. There exists a critical mass Mc = 2 p 6 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 Mc. For n 4, entropy weak solutions are positive and unique. For n = 1, a finite 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 infinity as the time goes to infinity or h(tk) 0 in L1(R) for some subsequence tk 1.
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