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

Galaktionov, Victor A.

doi:10.1088/0951-7715/22/7/012

Cited by 14 publications

(25 citation statements)

References 88 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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: 53%

See 1 more Smart Citation

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

Álvarez-Caudevilla

Galaktionov

2012

Advanced Nonlinear Studies

View full text Add to dashboard Cite

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.

show abstract

Section: 2supporting

confidence: 53%

“…The idea of such Type II(LN) blow-up patterns (according to a classification in [24]) consists of noting that blow-up can occur via some "slow" motion about the stationary solutions (6.20). This formally means the following non-stationary parameter timedependence:…”

Section: 2mentioning

confidence: 99%

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

Álvarez-Caudevilla

Galaktionov

2012

Advanced Nonlinear Studies

View full text Add to dashboard Cite

show abstract

“…This paper continues the study began in [1–4] of blow‐up patterns for the fourth‐order reaction‐diffusion equation (the RDE‐4) For applications of such higher‐diffusion models, see surveys and references in [1, 4]. In general, higher‐order semilinear parabolic equations arise in many physical applications such as thin film, convection‐explosion, and lubrication theory, flame and wave propagation (the Kuramoto–Sivashinsky equation and the extended Fisher–Kolmogorov equation), phase transition at critical Lifshitz points, bi‐stable systems and applications to structural mechanics.…”

Section: Introduction: Self‐similar Blow‐up Patterns Of Higher‐ordsupporting

confidence: 56%

“…It is difficult to prove analytically that actually takes place at some p = p δ > 1, thus numerical methods have been used [3] to support this idea. Namely, the following exponent, for which holds, was detected: In Figure 3, we show such a profile f 0 ( y ) in the case , accompanied by the second one f 1 ( y ), for which C 1 ≠ 0.…”

Section: Self‐similar Blow‐up Patternsmentioning

confidence: 80%

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

Galaktionov¹

2010

Stud Appl Math

View full text Add to dashboard Cite

Abstract. Blow-up behaviour for the 4th-order semilinear reaction-diffusion equation (0.1)is studied. For the classic semilinear heat equation from combustion theoryvarious blow-up patterns were investigated since 1970s, while the case of higher-order diffusion was studied much less. Blow-up self-similar solutions of (0.1),, are shown to admit global extensions for t > T in an analogous similarity form:The continuity at t = T is preserved in the sense thatThis is in a striking difference with blow-up for (0.2), which is known to be always complete in the sense that the minimal (proper) extension beyond blow-up is u(x, t) ≡ +∞ for t > T . Difficult 4th-order dynamical systems for f and F are studied by a combination of various analytic, formal, and careful numerical methods. Other nonsimilarity patterns for (0.1) with non-generic complete blow-up are also discussed.

show abstract

“…It is curious that a negative answer (i.e., similar for the NSEs in Section 5) can be obtained rather convincingly just by a local asymptotic analysis of the elliptic equation (B.34). As happens in practically all blow-up problems for reaction-diffusion and other nonlinear PDEs (see examples in, e.g., [19,23,63]), a "generic" behaviour of its solutions as z = |y| → +∞ is governed by the leading lower-order linear terms, i.e., in the radial representation, this means that, for z ≫ 1,…”

mentioning

confidence: 99%

Boundary characteristic point regularity for Navier–Stokes equations: Blow-up scaling and Petrovskii-type criterion (a formal approach)

Galaktionov

Maz′ya

2012

Nonlinear Analysis: Theory, Methods & Applications

View full text Add to dashboard Cite

Abstract. The three-dimensional (3D) Navier-Stokes equations (0.1) u t + (u · ∇)u = −∇p + ∆u, div u = 0 in Q 0 , where u = [u, v, w] T is the vector field and p is the pressure, are considered. Here,is a smooth domain of a typical backward paraboloid shape, with the vertex (0, 0) being its only characteristic point: the plane {t = 0} is tangent to ∂Q 0 at the origin, and other characteristics for t ∈ [0, −1) intersect ∂Q 0 transversely. Dirichlet boundary conditions on the lateral boundary ∂Q 0 and smooth initial data are prescribed:Existence, uniqueness, and regularity studies of (0.1) in non-cylindrical domains were initiated in the 1960s in pioneering works by J.L. Lions, Sather, Ladyzhenskaya, and Fujita-Sauer. However, the problem of a characteristic vertex regularity remained open.In this paper, the classic problem of regularity (in Wiener's sense) of the vertex (0, 0) for (0.1), (0.2) is considered. Petrovskii's famous "2 √ log log-criterion" of boundary regularity for the heat equation (1934) is shown to apply. Namely, after a blow-up scaling and a special matching with a boundary layer near ∂Q 0 , the regularity problem reduces to a 3D perturbed nonlinear dynamical system for the first Fourier-type coefficients of the solutions expanded using solenoidal Hermite polynomials. Finally, this confirms that the nonlinear convection term gets an exponentially decaying factor and is then negligible. Therefore, the regularity of the vertex is entirely dependent on the linear terms and hence remains the same for Stokes' and purely parabolic problems.Well-posed Burnett equations with the minus bi-Laplacian in (0.1) are also discussed.

show abstract

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

Cited by 14 publications

References 88 publications

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

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

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

Boundary characteristic point regularity for Navier–Stokes equations: Blow-up scaling and Petrovskii-type criterion (a formal approach)

Contact Info

Product

Resources

About