ISBN 978-80-227-2624-5 ON A NONLINEAR NONLOCAL ODE AND ITS APPLICATIONS∗

Abstract. We consider a limit case of a system of two equations arising in magnetic recording for a one-dimensional domain. The system models the tape deflection when it is driven over a magnetic head profile, which is a known function. The system is reduced to a second order nonlinear equation, where the unknown u appears evaluated in a finite set of distinguished points {xi}ni=1 of the domain. Key words. Nonlocal terms, lubrication theory, Reynolds equation, nonlinear elliptic equations AMS subject classifications. 34B99, 34B60, 74F17 1. Modelling and main results. The problem consists in a magnetic tape which is driven with constant velocity over a magnetic head and its motion entrains air in the space between the head and the tape. The unknowns of the problem are the position of the tape u and the pressure of the air p in the region between the tape and the head. The position of the tape is modelled by the beam equation and the pressure (when the tape is close to the head) satisfies the compressible Reynolds equation. The head presents trenches to control the position of the tape. If the trench is deep enough, Reynolds equation can not be used to model the problem and the pressure in the trench is considered equal to the atmospheric pressure. After nondimensionalization one obtains the following system of equations for the position of the tape “u ” and the pressure of the air “p ” (see [1], [3, Chap. 6] for details) ∂(ph) ∂