Selfsimilar Blowup of Unstable Thin-film Equations

with D. Slepcev Indiana Univ. Math. J. 54(2005)6:1697-1738

Abstract


We study selfsimilar blowup of long-wave unstable thin-film equations with critical powers of nonlinearities:
ut = -(u^n uxxx + u^(n+2) ux)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. Asymptotic behaviour of blowup selfsimilar solutions as $n$ approaches $3/2$ is also investigated, and a sharp lower bound on the height of solutions one time unit before the blowup is obtained. We also prove qualitative properties of solutions; for example, a compactly supported selfsimilar solution that blows up in finite time always has larger mass than the compactly supported steady states.


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