Abstract. 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)where n > 0 is a fixed exponent, with bounded smooth compactly supported initial data. Dealing with the CP (for, at least, n ∈ (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 ε-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 ε-regularization problems of the formwhere φε(u) for ε ∈ (0, 1] is an analytic ε-regularization of the problem (0.1), such that φ0(u) = |u| n and φ1(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 + and n → 0 + , and describe branching of solutions of the TFE-4 from the solutions of the notorious bi-harmonic equationwhich 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. Finally, Riemann-like problems occurring in a boundary layer construction, that occur close to nodal sets of the solutions, as ε → 0 + , are discussed in other to get uniqueness results for the TFE-4 (0.1).

No abstract

Abstract. Countable families of global-in-time and blow-up similarity signchanging patterns of the Cauchy problem for the fourth-order thin film equation (TFE-4)are studied. The similarity solutions are of standard "forward" and "backward" formsand α ∈ R is a parameter (a "nonlinear eigenvalue"). The sign "+", i.e., t > 0, corresponds to global asymptotics as t → +∞, while "−" (t < 0) yields blow-up limits t → 0 − describing possible "micro-scale" (multiple zero) structures of solutions of the PDE. To get a countable set of nonlinear pairs {fγ, αγ}, a bifurcation-branching analysis is performed by using a homotopy path n → 0 + in (0.1), where B ± 0 (α, f ) become associated with a pair {B, B * } of linear non-self-adjoint operatorswhich are known to possess a discrete real spectrum,|γ|≥0 (γ is a multiindex in R N ). These operators occur after corresponding global and blow-up scaling of the classic bi-harmonic equation ut = −Δ 2 u. This allows us to trace out the origin of a countable family of n-branches of nonlinear eigenfunctions by using simple or semisimple eigenvalues of the linear operators {B, B * } leading to important

Abstract. Fundamental global similarity solutions of the tenth-order thin film equationwhere n > 0 are studied. The main approach consists in passing to the limit n → 0 + by using Hermitian non-self-adjoint spectral theory corresponding to the rescaled linear poly-harmonic equation

Abstract. Fourth-order semilinear parabolic equations of the Cahn-Hilliard-type (0.1)are considered in a smooth bounded domain Ω ⊂ R N with Navier-type boundary conditions on ∂Ω, or Ω = R N , where p > 1 and γ are given real parameters. The sign " + " in the "diffusion term" on the right-hand side means the stable case, while " − " reflects the unstable (blow-up) one, with the simplest, so called limit, canonical model for γ = 0,The following three main problems are studied: (i) for the unstable model (0.1), with the −∆(|u| p−1 u), existence and multiplicity of classic steady states in Ω ⊂ R N and their global behaviour for large γ > 0; (ii) for the stable model (0.2), global existence of smooth solutions u(x, t) in R N × R + for bounded initial data u 0 (x) in the subcritical case p ≤ p * = 1 + 4 (N −2)+ ; and (iii) for the unstable model (0.2), a relation between finite time blow-up and structure of regular and singular steady states in the supercritical range. In particular, three distinct families of Type I and II blow-up patterns are introduced in the unstable case.