IUMJ

Title: Selfsimilar blowup of unstable thin-film equations

Authors: D. Slepcev and M. C. Pugh

Issue: Volume 54 (2005), Issue 6, 1697-1738

Abstract:

We study selfsimilar blowup of long-wave unstable thin-film equations with critical powers of nonlinearities: $u_t = -(u^nu_{xxx} + u^{n+2}u_x)_x$. We show that the equation cannot have selfsimilar solutions (with zero contact angles) that blow up in finite time if $n \geq 3/2$. We show that for $0 < n < 3/2$ there are compactly supported, symmetric, selfsimilar solutions (with zero contact angles) that blow up in finite time. Moreover, there exist families of these solutions with any number of local maxima. We also study the asymptotic behaviour of these selfsimilar solutions as $n$ approaches $3/2$ and obtain a sharp lower bound on the height of solutions one time unit before the blowup. We also prove qualitative properties of solutions; for example, the profile of a selfsimilar solution that blows up in finite time can always be bounded from below by a compactly supported steady state.