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arXiv · 1008.0385

Research Announcement: Finite-time Blow Up and Long-wave Unstable Thin Film Equations

Marina Chugunova·M. C. Pugh·Roman M. Taranets
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Abstract

We study short--time existence, long--time existence, finite speed of propagation, and finite--time blow--up of nonnegative solutions for long-wave unstable thin film equations $h_t = -a_0(h^n h_{xxx})_x - a_1(h^m h_x)_x$ with $n>0$, $a_0 > 0$, and $a_1 >0$. The existence and finite speed of propagation results extend those of [Comm Pure Appl Math 51:625--661, 1998]. For $0<n<2$ we prove the existence of a nonnegative, compactly--supported, strong solution on the line that blows up in finite time. The construction requires that the initial data be nonnegative, compactly supported in $\R^1$, be in $H^1(\R^1)$, and have negative energy. The blow-up is proven for a large range of $(n,m)$ exponents and extends the results of [Indiana Univ Math J 49:1323--1366, 2000].

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Marina Chugunova, M. C. Pugh, Roman M. Taranets. 2010-08-02. Research Announcement: Finite-time Blow Up and Long-wave Unstable Thin Film Equations. https://arxiv.org/abs/1008.0385

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