Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches

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) equation(0.1)ut=−∇⋅(|u|n∇Δu)in  RN×R+,u(x,0)=u0(x)in  RN, where n>0n>0 is a fixed exponent, with bounded smooth compactly supported initial data. Dealing with the CP (for, at least, n∈(0,32)) 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 εε-regularizations at least for values of the parameter nn sufficiently close to 0.Firstly, we study an analytic “homotopy” approach based on a priori   estimates for solutions of uniformly parabolic analytic εε-regularization problems of the form ut=−∇⋅(ϕε(u)∇Δu)in  RN×R+, where ϕε(u)ϕε(u) for ε∈(0,1]ε∈(0,1] is an analytic εε-regularization of the problem (0.1), such that ϕ0(u)=|u|nϕ0(u)=|u|n and ϕ1(u)=1ϕ1(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 ε→0+ε→0+ and n→0+n→0+, and describe branching of solutions of the TFE-4 from the solutions of the notorious bi-harmonic equationut=−Δ2uin  RN×R,u(x,0)=u0(x)in  RN, which describes some qualitative oscillatory properties of CP-solutions of (0.1) for small n>0n>0 providing us with the uniqueness of solutions for the problem (0.1) when nn is close to 0.Finally, Riemann-like problems   occurring in a boundary layer construction, that occur close to nodal sets of the solutions, as ε→0+ε→0+, are discussed in other to get uniqueness results for the TFE-4 (0.1).