Some a Priori Estimates for a Singular Evolution Equation Arising in Thin-Film Dynamics

Abstract

This paper considers the singular evolution equation $u_t - \Delta \ln u = 0$, particularly the corresponding Cauchy problem. This equation arises in the study of thin film dynamics and as the formal limit as $m \to 0$ of the porous-medium equation. Through the use of local sup-estimates, similar to those for the porous-medium equation (see E. DiBenedetto and Y. L. Kwong, Intrinsic Harnack estimates and extinction profile for certain parabolic equations, Trans. Amer. Math. Soc., 330 (1992), pp. 783–811), and a global Harnack-type inequality, a critical decay rate for solutions in three or more space dimensions as $|x| \to \infty $ is found. In particular, if the initial data decays faster than this critical rate, there is no solution to the Cauchy problem.