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

By Andrea L. Bertozzi, M.C. Pugh. 2000

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Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions (2008, methodology): “As seen above, the balance happens precisely for our chosen exponent m = md .I n the NLS equation this happens for the so-called pseudo-conformal nonlinearity [14, Chapter 6]. In the UTF equation this happens in the so-called marginal case [2, 3 ].”
Hausdorff Dimension of Ruptures for Solutions of a Semilinear Elliptic Equation with Singular Nonlinearity (2006, background): “For backgrounds on (3), we refer to [2, 3, 14–16, 21] and the references therein.”
Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation (2002, methodology): “However, in the present study the nonlinear in f term inH(f ) is due to the fourth-order term rather than to the second-order term as in [11].”
Asymptotic equipartition and long time behavior of solutions of a thin-film equation☆ (2007, background): “The results in [4], where the issue of finite time blow up of solutions for the thin-film type equations has been discussed, show that in general it is possible for classical solutions to break down in finite time.”