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

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 ${\mathbb{R}}^1$, be in $H^1({\mathbb{R}}^1)$, 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].