Towards optimal regularity for the fourth-order thin film equation in RN: Graveleau-type focusing self-similarity

Abstract

An approach to some "optimal" (more precisely, non-improvable) regularity of solutions of the thin film equation ut = -del.(vertical bar u vertical bar(n)del Delta u) in R-N x R+, u(x,0) = u(0)(x) in R-N, where n is an element of (0,2) is a fixed exponent, with smooth compactly supported initial data u(0)(x), in dimensions N >= 2 is discussed. Namely, a precise exponent for the Hoder continuity with respect to the spatial radial variable vertical bar x vertical bar is obtained by construction of a Graveleau-type focusing self-similar solution. As a consequence, optimal regularity of the gradient del u in certain L-P spaces, as well as a Holder continuity property of solutions with respect to x and t, are derived, which cannot be obtained by classic standard methods of integral identities inequalities. Several profiles for the solutions in the cases n = 0 and n > 0 are also plotted. In general, we claim that, even for arbitrarily small n > 0 and positive analytic initial data u(0)(x), the solutions u(x,t) cannot be better than C-x(2-epsilon)-smooth, where epsilon(n) = O(n) as n -> 0.