A free-boundary problem is considered for the thin film equation \(u_t =\) \(-(u^n u_{xxx})_x\) with the interface conditions \(u= \) \(u^n u_x =0\) at \(x=s(t)\) and \(x=-S(t)\), where \(u\) denotes the thickness of a film. The condition \(u^n u_x =0\) represents conservation of mass. The initial condition is \(u_0 (x)=\) \(I(x)+\varepsilon\), where \(I(x)=0 \) for \(|x|\geq 0\) and \(I(x)>0\) for \(|x|<0\); \(s(0)=\) \(S(0)=a\). An asymptotic analysis is performed in the limit \(\varepsilon\to 0\) with \(t=O(1)\) for different values of \(n\). Additional free boundary conditions are discussed in relation to the non-uniqueness problem.