Incomplete Self‐Similar Blow‐Up in a Semilinear Fourth‐Order Reaction‐Diffusion Equation

Galaktionov, Victor A.

doi:10.1111/j.1467-9590.2009.00474.x

Cited by 14 publications

(7 citation statements)

References 38 publications

(178 reference statements)

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“…It is worth mentioning that self-similar blow-up for (1.1) is incomplete, i.e., blow-up solutions, in general, admit global extensions for t > T . Such principal questions are studied in [29] and will not be treated here.…”

Section: Type I(ss): Self-similar Blow-upmentioning

confidence: 99%

Five types of blow-up in a semilinear fourth-order reaction–diffusion equation: an analytic–numerical approach

Galaktionov

2009

Nonlinearity

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Abstract. Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion typeare discussed. For the semilinear heat equation u t = ∆u + u p , various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application. The types of blow-up include: 1. From second-order to higher-order blow-up R-D models: a PDE route from XXth to XXIst century 1.1. The RDE-4 and applications. This paper is devoted to a description of blow-up patterns for the fourth-order reaction-diffusion equation (the RDE-4 in short)(1.1)where ∆ stands for the Laplacian in R N . This has the bi-harmonic diffusion −∆ 2 and is a higher-order counterpart of classic second-order PDEs, which we begin our discussion with. For applications of such higher-diffusion models, see short surveys and references in

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Section: Type I(ss): Self-similar Blow-upmentioning

confidence: 99%

Five types of blow-up in a semilinear fourth-order reaction–diffusion equation: an analytic–numerical approach

Galaktionov

2009

Nonlinearity

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“…The scenario of incomplete blow-up and self-similar global "peaking solutions", that is solutions which shrink self-similarly, blow up, and then expand self-similarly for a while (with this scenario possibly repeating a number of times) has been studied in the past for the harmonic map flow [4] and other parabolic equations: the semilinear heat equations [5,6], the mean curvature flow [4,[7][8][9], the Yang-Mills flow [10], the Ricci flow [11], and more recently for a fourth-order reaction-diffusion equation [12]. Most of these studies emphasized non-uniqueness of continuation beyond blow-up.…”

Section: Introductionmentioning

confidence: 99%

Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

Biernat

Bizoń

2011

Nonlinearity

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Using mixed analytical and numerical methods we investigate the development of singularities in the heat flow for corotational harmonic maps from the d-dimensional sphere to itself for 3 ≤ d ≤ 6.By gluing together shrinking and expanding asymptotically self-similar solutions we construct global weak solutions which are smooth everywhere except for a sequence of times T 1 < T 2 < · · · < T k < ∞ at which there occurs the type I blow-up at one of the poles of the sphere. We give evidence that in the generic case the continuation beyond blow-up is unique, the topological degree of the map changes by one at each blow-up time T i , and eventually the solution comes to rest at the zero energy constant map.

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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: 2mentioning

confidence: 52%

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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Incomplete Self‐Similar Blow‐Up in a Semilinear Fourth‐Order Reaction‐Diffusion Equation

Cited by 14 publications

References 38 publications

Five types of blow-up in a semilinear fourth-order reaction–diffusion equation: an analytic–numerical approach

Five types of blow-up in a semilinear fourth-order reaction–diffusion equation: an analytic–numerical approach

Shrinkers, expanders, and the unique continuation beyond generic blowup in the heat flow for harmonic maps between spheres

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

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