Ukrainian Mathematical Bulletin

Volume 1, Issue 3, 2004 pp. 407-450.

On the Thin-Film Equation with Nonlinear Convection in Multidimensional Domains

Authors: Andrey E. Shishkov, Roman M. Taranets
Author institution: Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, R. Luxembourg Str., 74, Donetsk 83114, Ukraine

Summary: We investigate the global solvability of the Cauchy problem for a multidimensional thin-film equation with nonlinear convection. Preliminarily, we construct a nonnegative local generalized ``strong'' solution of the Neumann problem in a bounded domain as the limit of the sequence of solutions of the corresponding regularized boundary-value problems. We establish the finiteness of the speed of support propagation for arbitrary ``strong'' solutions of the Neumann problem. Using this property, we construct a nonnegative global ``strong'' solution of the Cauchy problem with an arbitrary finite initial function under optimal conditions on parameters of nonlinearity of the equation.

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