Unstable sixth-order thin film equation: I. Blow-up similarity solutions

Evans, Jonathan D.; Galaktionov, Victor A.; King, John R.

doi:10.1088/0951-7715/20/8/002

Cited by 51 publications

(79 citation statements)

References 40 publications

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“…For instance, the pure sixth-order parabolic thin film equation (TFE) was first introduced in [1,2] to describe the spreading of a thin viscous fluid (with possible slip at the solid interface) under the driving force of an elastica (or light plate). There are many related works on blow-up solutions to these kinds of parabolic equations and systems (see [3][4][5][6][7][8][9] and the references therein).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The method used here is also related to the so-called Fibering method introduced by Pohozev in the 1970s [16][17][18][19]. It is easy to verify that g is increasing for 0 < α < α 1 , decreasing for α > α 1 ; g(α) → −∞ as α → +∞ and g(α 1 ) = E 1 , where α 1 is given in (5). As E(0) < E 1 , there exists α 2 > α 1 such that g(α 2 ) = E(0).…”

Section: Proof Of Theoremmentioning

confidence: 99%

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Blow-up of solutions for the sixth-order thin film equation with positive initial energy

Liu

Chen

2015

Pramana - J Phys

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

Blow-up of solutions for the sixth-order thin film equation with positive initial energy

Liu

Chen

2015

Pramana - J Phys

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“…Analysis of finite-time blow-up and rupture in many higher order PDEs starts along similar lines [35,11,76,77,79,34,22,25].…”

Section: Self-similar Finite-time Blow-up In a Higher-order Pdementioning

confidence: 99%

Stability and dynamics of self-similarity in evolution equations

Bernoff

Witelski

2009

J Eng Math

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Abstract. We discuss a methodology for studying the linear stability of self-similar solutions. We will illustrate these fundamental ideas on three prototype problems: a simple ODE with finite-time blowup, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, we will discuss the use of dimensional analysis to derive scaling invariant similarity variables, and the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions we will demonstrate that the symmetries give rise to positive eigenvalues associated with the symmetries, and show how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions.

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“…The equation (1.1) is a typical higher order equation, which is obtained for power-law fluids spreading on a horizontal substrate [5,6,8]. We refer also the following relevant equation 2) For any ϕ ∈ C ∞ 0 (Q T ), the following integral equality holds:…”

Section: Introductionmentioning

confidence: 99%

A sixth order degenerate equation with the higher order p-laplacian operator

Liu

2010

Mathematica Slovaca

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ABSTRACT. We consider a initial-boundary value problem for a sixth order degenerate parabolic equation. Under some assumptions on the initial value, we establish the existence of weak solutions by the time-discrete method. The uniqueness, asymptotic behavior and the finite speed of propagation of perturbations of solutions are also discussed. IntroductionThis paper is concerned with a sixth order degenerate parabolic equation of the form ∂u ∂twhere Ω ⊂ R N is a bounded domain with smooth boundary. On the basis of physical consideration, as usual the equation (1.1) is supplemented with the natural boundary value conditions 2) and the initial value conditionThe equation (1.1) is a typical higher order equation, which is obtained for power-law fluids spreading on a horizontal substrate [5,6,8]. We refer also the following relevant equation M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary

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Unstable sixth-order thin film equation: I. Blow-up similarity solutions

Abstract: We study blow-up behaviour of solutions of the sixth-order thin film equationwhere n > 0, p > 1, containing an unstable (backward parabolic) second-order term. By a formal matched expansion technique, we show that, for the first critical exponent

Cited by 51 publications

References 40 publications

Blow-up of solutions for the sixth-order thin film equation with positive initial energy

Blow-up of solutions for the sixth-order thin film equation with positive initial energy

Stability and dynamics of self-similarity in evolution equations

A sixth order degenerate equation with the higher order p-laplacian operator

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