On $\alpha$-entropy solutions of a nonlocal thin film equation: existence and finite speed of propagation

Abstract

We consider an initial-boundary value problem for a class of nonlocal thin film equations governed by the spectral fractional Laplacian with homogeneous Neumann boundary conditions. We were the first to establish an $\alpha$-entropy estimate for nonlocal thin film equations, which yields essential a priori bounds for the regularity and long-time behavior of weak solutions. By developing a localized version of this estimate, we prove finite speed of propagation, showing that the support of nonnegative solutions remains compact for positive times. Furthermore, we find a sufficient condition for a waiting time phenomenon, whereby the solution remains identically zero in a region for a nontrivial time interval. These results highlight new features in the interaction between nonlocal effects and classical thin film dynamics.