Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations

Budd, Chris; Williams, J. F.; Галактионов, В. А.

doi:10.1137/s003613990241552x

Cited by 55 publications

(87 citation statements)

References 34 publications

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“…for such a blow-up/extension study, see [11,24,26]. This construction can be connected with Leray's scenario of self-similar blow-up/extension proposed in 1934 for the Navier-Stokes equations in R 3 , [43, p. 245].…”

Section: 2supporting

confidence: 54%

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

Álvarez-Caudevilla

Galaktionov

2012

Advanced Nonlinear Studies

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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.

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Section: 2supporting

confidence: 54%

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

Álvarez-Caudevilla

Galaktionov

2012

Advanced Nonlinear Studies

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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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“…[12][13][14][15][16][17]) and more recently for their higherorder generalizations (cf. [18][19][20]). The applications of singularity formation are diverse, including fourth-order lubrication theory models from fluids (cf.…”

Section: Similaritymentioning

confidence: 99%

“…In contrast, for high-order PDE models exhibiting singularity formation, there can be a countably infinite set of local self-similar solutions whose stability properties must be studied on a case-by-case basis (cf. [18][19][20]27]); -the role of quasi-self-similar solutions (which satisfy the governing equations only asymptotically, rather than exactly) in describing ultimate (long-time or close to finite-time singularity) evolutions (cf. [17,[30][31][32]).…”

Section: Similaritymentioning

confidence: 99%