Finite-time Blow-up of Solutions of Some Long-wave Unstable Thin Film Equations
نویسندگان
چکیده
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 H1 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 (1998), 625-661] on when finite-time singularities are possible for long-wave unstable lubrication equations.
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