Steady States, Global Existence and Blow-up for Fourth-Order Semilinear Parabolic Equations of Cahn–Hilliard Type

Abstract

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.

Countable families of solutions of a limit stationary semilinear fourth-order Cahn-Hilliard-type equation I. Mountain pass and Lusternik-Schnirel’man patterns in R N $\mathbb{R}^{N}$

“…Assuming that data u 0 (x) are sufficiently fast exponentially decaying at infinity, the same behaviour holds for the unique classic solution of (1.1), at least locally in time, since, for p > 1, u(x, t) may blow-up in finite time; see key references and results in [6] and [14]. The classic Cahn-Hilliard equation describes the dynamics of pattern formation in phase transition in alloys, glasses, and polymer solutions.…”

“…A similar fourth-order problem was studied in [6], where further references can be found. However, in [6] the stationary equation (1.4) was considered in a bounded smooth domain Ω ∈ R N , with homogeneous Navier-type boundary conditions (1.7) u = ∆u = 0 on ∂Ω.…”

“…Previous related results. Recall again that, in [6], existence and multiplicity results were obtained for the steady-states of the unstable C-H equation of the form…”

“…Exponentially decaying patterns in R N . The preliminary conclusions presented here formally allow us to consider our equations (1.4) in the whole R N , unlike as in [6] where the problem was assumed to be in a bounded domain Ω ⊂ R N .…”

Finite-time thin film rupture driven by modified evaporative loss

“…While the derivation of the evaporative flux (3b) given by Ajaev [2] suggests that the physically achievable range of parameters for typical fluids has < 0, in the spirit of other applied studies of singularity formation in physical systems, we will consider the rich and novel dynamics that occur for > 0 to better understand the influence of non-conservative e↵ects and singular potentials on the development of finitetime rupture. This is also relevant to nonlinear interfacial instabilities in other fourth-order nonlinear PDE's with non-conservative contributions [3,37,35].…”