Droplet Spreading Under Weak Slippage-Existence for the Cauchy Problem

In this paper, we consider the thin film equation u t + div(|u| n ∇Δu) = 0 in the multi-dimensional setting and solve the Cauchy problem in the parameter regime n ∈ [2, 3). New interpolation inequalities applied to the energy estimate enable us to control third order derivatives of appropriate powers of solutions. In such a way, a natural solution concept - reminiscent of that one used by Bernis and Friedman [Bernis, F., Friedman, A., ( 1990 ). Higher order nonlinear degenerate parabolic equations. J. Differential Equations 83:179-206] in space dimension N = 1 − becomes available for the first time in the multi-dimensional setting. In addition, we provide the key integral estimate to establish results on the qualitative behavior of solutions like finite speed of propagation or occurrence of a waiting time phenomenon.