Finite Speed of Propagation for a Class of Stochastic Thin-Film Equations

We present an energy method to prove a qualitative result on finite speed of propagation for energy-dissipating solutions to stochastic thin-film equations with conservative nonlinear multiplicative noise under periodic boundary conditions. Physically, the equations under consideration correspond to weak slippage, i.e. the case of mobility exponents $n\in (2,3)$. Analytically, our approach is based on two main ingredients. First, a new weighted decay estimate for appropriate powers of the solution which is set up in combination with Bernis inequalities and energy estimates. Secondly, tailored filtering techniques inspired by the methods which were presented for stochastic porous-medium equations with source-term noise in (J. Fischer and G. Grün, SIAM J. Math. Anal. 47:825-854, 2015).

This is joint work with L. Klein (Erlangen).