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Finite-time blow-up of solutions of some long-wave unstable thin film equations

We consider the family of long–wave unstable lubrication equations ht = −(hhxxx)x − (hhx)x with m ≥ 3. Given a fixed m ≥ 3, we prove the existence of a weak solution that becomes singular in finite time. Specifically, given compactly supported nonnegative initial data with negative energy, there is a time T ∗ < ∞, determined by m and the H norm of the initial data, and a compactly supported nonnegative weak solution such that lim supt→T∗ ||h(·, t)||L∞ = lim supt→T∗ ||h(·, t)||H1 = ∞. We discuss the relevance of these singular solutions to an earlier conjecture [Comm Pure Appl Math 51:625–661, 1998] on when finite–time singularities are possible for long–wave unstable lubrication equations.