scholarly journals Local strong solutions to a quasilinear degenerate fourth-order thin-film equation

2020 ◽  
Vol 27 (2) ◽  
Author(s):  
Christina Lienstromberg ◽  
Stefan Müller
Keyword(s):  
Thin Film ◽  
Fourth Order ◽  
Strong Solutions ◽  
Thin Film Equation ◽  
Local Strong Solutions

scholarly journals Blowup and dissipation in a critical-case unstable thin film equation

2004 ◽  
Vol 15 (2) ◽  
pp. 223-256 ◽  
Author(s):  
T. P. WITELSKI ◽  
A. J. BERNOFF ◽  
A. L. BERTOZZI
Keyword(s):  
Thin Film ◽  
Fourth Order ◽  
Critical Case ◽  
Blow Up ◽  
Critical Mass ◽  
Similarity Solutions ◽  
Fluid Layers ◽  
Parabolic Pde ◽  
Thin Film Equation ◽  
Order Degenerate

We study the dynamics of dissipation and blow-up in a critical-case unstable thin film equation. The governing equation is a nonlinear fourth-order degenerate parabolic PDE derived from a generalized model for lubrication flows of thin viscous fluid layers on solid surfaces. %For a special balance between %destabilizing second-order terms and regularizing fourth-order terms, There is a critical mass for blow-up and a rich set of dynamics including families of similarity solutions for finite-time blow-up and infinite-time spreading. The structure and stability of the steady-states and the compactly-supported similarity solutions is studied.

Strong solutions and trajectory attractors to the thin-film equation with absorption

2021 ◽  
Vol 493 (2) ◽  
pp. 124562
Author(s):  
Oleksiy V. Kapustyan ◽  
Pavlo O. Kasyanov ◽  
Roman M. Taranets
Keyword(s):  
Thin Film ◽  
Strong Solutions ◽  
Thin Film Equation ◽  
Trajectory Attractors

Blow-up similarity solutions of the fourth-order unstable thin film equation

2007 ◽  
Vol 18 (2) ◽  
pp. 195-231 ◽  
Author(s):  
J. D. EVANS ◽  
V. A. GALAKTIONOV ◽  
J. R. KING
Keyword(s):  
Thin Film ◽  
Fourth Order ◽  
Blow Up ◽  
Similarity Solutions ◽  
Diffusion Term ◽  
Thin Film Equation ◽  
Self Similar ◽  
Image Position ◽  
General Aspects ◽  
Similar Solutions

We study blow-up behaviour of solutions of the fourth-order thin film equationwhich contains a backward (unstable) diffusion term. Our main goal is a detailed study of the case of the first critical exponentwhereN≥ 1 is the space dimension. We show that the free-boundary problem with zero contact angle and zero-flux conditions admits continuous sets (branches) of blow-up self-similar solutions. For the Cauchy problem inRN×R+, we detect compactly supported blow-up patterns, which have infinitely many oscillations near interfaces and exhibit a “maximal” regularity there. As a key principle, we use the fact that, for small positiven, such solutions are close to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equationwhich are better understood and have been studied earlier [19]. We also discuss some general aspects of formation of self-similar blow-up singularities for other values ofp.

Source-type solutions of the fourth-order unstable thin film equation

2007 ◽  
Vol 18 (3) ◽  
pp. 273-321 ◽  
Author(s):  
J. D. EVANS ◽  
V. A. GALAKTIONOV ◽  
J. R. KING
Keyword(s):  
Thin Film ◽  
Cauchy Problem ◽  
Fourth Order ◽  
Similarity Solutions ◽  
Diffusion Term ◽  
Thin Film Equation ◽  
The Cauchy Problem ◽  
Self Similar ◽  
Cahn Hilliard Equation ◽  
Image Position

We consider the fourth-order thin film equation (TFE) with the unstable second-order diffusion term. We show that, for the first critical exponent where N ≥ 1 is the space dimension, the free-boundary problem the with zero contact angle and zero-flux conditions admits continuous sets (branches) of self-similar similarity solutions of the form For the Cauchy problem, we describe families of self-similar patterns, which admit a regular limit as n → 0+ and converge to the similarity solutions of the semilinear unstable limit Cahn-Hilliard equation studied earlier in [12]. Using both analytic and numerical evidence, we show that such solutions of the TFE are oscillatory and of changing sign near interfaces for all n ∈ (0,nh), where the value characterizes a heteroclinic bifurcation of periodic solutions in a certain rescaled ODE. We also discuss the cases p ⧧ = p0, the interface equation, and regular analytic approximations for such TFEs as an approach to the Cauchy problem.

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