Download Blow-up for higher-order parabolic, hyperbolic, dispersion by Victor A. Galaktionov PDF

By Victor A. Galaktionov

ISBN-10: 1482251728

ISBN-13: 9781482251722

ISBN-10: 1482251736

ISBN-13: 9781482251739

Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations indicates how 4 different types of higher-order nonlinear evolution partial differential equations (PDEs) have many commonalities via their particular quasilinear degenerate representations. The authors current a unified method of take care of those quasilinear PDEs.

The ebook first stories the actual self-similar singularity options (patterns) of the equations. This method permits 4 diversified periods of nonlinear PDEs to be taken care of concurrently to set up their awesome universal beneficial properties. The booklet describes many houses of the equations and examines conventional questions of existence/nonexistence, uniqueness/nonuniqueness, international asymptotics, regularizations, shock-wave conception, and diverse blow-up singularities.

Preparing readers for extra complex mathematical PDE research, the e-book demonstrates that quasilinear degenerate higher-order PDEs, even unique and awkward ones, will not be as daunting as they first look. It additionally illustrates the deep gains shared via various kinds of nonlinear PDEs and encourages readers to strengthen additional this unifying PDE procedure from different viewpoints.

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Additional info for Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations

Example text

For instance, quasilinear second-order Schr¨ odinger-type models, some of which can be written as iut + Δu + β|u|p−1 u + θ(Δ|u|2 )u = 0 in IRN × IR+ , (26) are not a novelty in several physical situations, such as superfluid theory, dissipative quantum mechanics, and in turbulence and Bose–Einstein condensation theory. We refer to some papers from the 1970s and 1980s [198, 335, 374], to [190] for further references, and to [265] for a more mathematical knowledge, as well as to [281] and to the papers of Zakharov’s et al [412, 413], as a sufficient source of other reference and deep physical and mathematical results.

For further discussion of geometric shapes of patterns, it is convenient to recall that, utilizing Berger’s version [28, p. 368] of this min-max analysis of L– S category theory [252, p. 387], the critical values {ck } and the corresponding critical points {vk } are given by ˜ ck = inf F ∈Mk supv∈F H(v), (74) where F ⊂ H0 are closed sets, and Mk denotes the set of all subsets of the form BS k−1 ⊂ H0 , where S k−1 is a suitable sufficiently smooth (k − 1)dimensional manifold (say, sphere) in H0 , and B is an odd continuous map.

Given a solution F of (8) (a critical point of (64)), let us calculate the corresponding critical value cF of (73) on the set (69), by taking v = CF ∈ H0 so that =⇒ ˜ = cF ≡ H(v) C= 1 − ˜ m F |2 + |D ˜ m F |2 + |D 1/2 , (83) |F |β (− F2 F 2 )β/2 β= n+2 n+1 . This formula is important in what follows. Genus one. 9. Indeed, this simple shape, with a single dominant maximum at y = 0, is associated with the variational formulation for F0 : F0 = r(v0 )v0 , ˜ with v0 : inf H(v) ≡ inf |v|β , v ∈ H0 . (84) This is (74) with the simplest choice of closed sets as points, F = {v}.

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Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations by Victor A. Galaktionov


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