- February 2015
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- This paper is devoted to some aspects of well-posedness of the Cauchy problem (the CP, for short) for a quasilinear degenerate fourth-order parabolic thin film equation (the TFE-4) u(t) = -del center dot (vertical bar u vertical bar(n) del Delta u) in R-N x R+, u(x, 0) = u(0)(x) in R-N, (0.1) where n > 0 is a fixed exponent, with bounded smooth compactly supported initial data. Dealing with the CP (for, at least, n is an element of (0, 3/2)) requires introducing classes of infinitely changing sign solutions that are oscillatory close to finite interfaces. The main goal of the paper is to detect proper solutions of the CP for the degenerate TFE-4 by uniformly parabolic analytic epsilon-regularizations at least for values of the parameter n sufficiently close to 0. Firstly, we study an analytic "homotopy'' approach based on a priori estimates for solutions of uniformly parabolic analytic epsilon-regularization problems of the form u(t) = -del center dot (phi(epsilon)(u)del Delta u) in R-N x R+, where phi(epsilon)(u) for epsilon is an element of (0, 1] is an analytic e-regularization of the problem (0.1), such that phi(0)(u) = vertical bar u vertical bar(n) and phi(1)(u) = 1, using a more standard classic technique of passing to the limit in integral identities for weak solutions. However, this argument has been demonstrated to be non-conclusive, basically due to the lack of a complete optimal estimate-regularity theory for these types of problems. Secondly, to resolve that issue more successfully, we study a more general similar analytic "homotopy transformation'' in both the parameters, as epsilon -> 0(+) and n -> 0(+), and describe branching of solutions of the TFE-4 from the solutions of the notorious bi-harmonic equation u(t) = -Delta(2)u in R-N x R, u(x, 0) = u(0)(x) in R-N, which describes some qualitative oscillatory properties of CP-solutions of (0.1) for small n > 0 providing us with the uniqueness of solutions for the problem (0.1) when n is close to 0...
- thin film equation; the cauchy problem; finite interfaces; oscillatory sign-changing behaviour; analytic epsilon-regularization; uniqueness