Shock waves and compactons for fifth-order non-linear dispersion equations

Advanced Nonlinear Studies

2012

Self 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.

Section: 4mentioning

confidence: 99%

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

Álvarez-Caudevilla

Advanced Nonlinear Studies

2012

Self Cite

“…Actually, this ODE is not that difficult for application of standard shooting methods, which in greater detail are explained in [4, § 3, 4]. Moreover, even analogous fifth‐order ODEs associated with the shocks for the NDE–5 also admit similar shooting analysis [3], though, in view of the essential growth of the dimension of the phase space (5D), some more delicate issues become more difficult. Therefore, we will widely use numerical methods for illustrating and even justifying of some of our conclusions.…”

Section: Gradient Blow‐up Similarity Solutionsmentioning

confidence: 99%

“…A compacton solution of is a compactly supported travelling wave [10] given by The physical motivation, references, and results for the NDEs such as , and others, which appear in many areas of application, with a large number of key papers, are available in surveys in [3, § 1] or in [4, § 1].…”

Section: Introduction: Nonlinear Dispersion Pdes and Main Directiomentioning

confidence: 99%

“…Increasing the order of the PDE under consideration, we then enlarge the dimension of the parametric space, and this may imply even stronger nonuniqueness conclusions. For example, nonuniqueness and nonentropy features are available for the fifth‐order nonlinear dispersion equation (the NDE–5) and others; see models and results in [2, 3]; nonuniqueness is discussed in [3, § 5].…”

Section: Introduction: Nonlinear Dispersion Pdes and Main Directiomentioning

confidence: 99%

“…This and the whole series of papers on NDEs [ 1–4 ] and a monograph to follow, as a first systematic study of shocks, rarefaction, blow‐up singularities, and principal nonuniqueness‐nonentropy in such nonlinear PDEs, are dedicated to the memory of author's son , Oleg Galaktionov (The Bloomberg, The City, London, UK) and his fiance Natalia Karavaeva (Moscow, Russia), who were brutally killed, with 25 other victims and many severely injured, in a terroristic (? – still unknown and seems will never be known in this Godless KGB‐ruled country) attack on the 27th November 2009, 9:30 p.m., on the train “Nevskii Express” 300 km from Moscow and St.‐Petersburg.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Single Point Gradient Blow‐Up and Nonuniqueness for a Third‐Order Nonlinear Dispersion Equation

2010

Stud Appl Math

Self Cite

A basic mechanism of a formation of shocks via gradient blow‐up from analytic solutions for the third‐order nonlinear dispersion PDE from compacton theory is studied. Various self‐similar solutions exhibiting single point gradient blow‐up in finite time, as t → T− < ∞, with locally bounded final time profiles u(x, T−), are constructed. These are shown to admit infinitely many discontinuous shock‐type similarity extensions for t > T, all of them satisfying generalized Rankine–Hugoniot's condition at shocks. As a consequence, the nonuniqueness of solutions of the Cauchy problem after blow‐up is detected. This is in striking difference with general uniqueness‐entropy theory for the 1D conservation laws such as (a partial differential equation, PDE, Euler's equation from gas dynamics) created by Oleinik in the middle of the 1950s. Contrary to and not surprisingly, self‐similar gradient blow‐up for is shown to admit a unique continuation. Bearing in mind the classic form , the NDE can be written as with the standard linear integral operator (−D2x)−1 > 0. However, because is a nonlocal equation, no standard entropy and/or BV‐approaches apply (moreover, the x‐variations of solutions of is increasing for BV data u0(x)).

On source-type solutions and the Cauchy problem for a doubly degenerate sixth-order thin film equation, I: Local oscillatory properties

Chaves

Nonlinear Analysis: Theory, Methods & Applications

2010

As a key example, the sixth-order doubly degenerate parabolic equation from thin film theory,with two parameters, n ≥ 0 and m ∈ (−n, n + 2), is considered. In this first part of the research, various local properties of its particular travelling wave and source type solutions are studied. Most complete analytic results on oscillatory structures of these solutions of changing sign are obtained for m = 1 by an algebraic-geometric approach, with extension by continuity for m ≈ 1.