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 sorts of higher-order nonlinear evolution partial differential equations (PDEs) have many commonalities via their specific quasilinear degenerate representations. The authors current a unified method of take care of those quasilinear PDEs.

The booklet first experiences the actual self-similar singularity suggestions (patterns) of the equations. This strategy permits 4 varied periods of nonlinear PDEs to be handled at the same time to set up their notable universal positive factors. The ebook 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, aren't as daunting as they first seem. It additionally illustrates the deep good points shared by means of various kinds of nonlinear PDEs and encourages readers to strengthen extra this unifying PDE technique from different viewpoints.

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Extra resources for Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schroedinger Equations

Sample text

It seems that, with such a huge, at least, countable variety of similar patterns, we first distinguish the profile that delivers the critical value c2 given by (74) by comparing the values (83) for each pattern. 1 for n = 1, for which the critical values (83) are ˜ cF = H(CF )= |F |3/2 (− ˜ m F |2 + |D F 2 )3/4 β= 3 2 . 2. This and many other graphical representations of such patterns were obtained by using the bvp4c solver in MATLAB. 5 for details). 855... is delivered by F1 . (90) 24 Blow-up Singularities and Global Solutions Notice that the critical values cF for F1 and F+2,2,+2 are close by just two percent.

Periodic solutions, together with their stable manifolds, are simple connections with the interface, as a singular point of ODE (9). Note that (102) does not admit variational setting, so we cannot apply well-developed potential theory [303, Ch. 8] (see a large amount of related existence–nonexistence results and further references therein), or a degree one 1 Self-Similar Blow-up and Compacton Patterns 31 [251, 252]. 1], which can be extended to m = 3 as well. Nevertheless, uniqueness of a periodic orbit is still open, so we conjecture the following result supported by various numerical and analytical evidence (cf.

Here, the structure of the stationary rescaled set S becomes key for understanding the blow-up behavior of general solutions of the higher-order local parabolic flow (1). Thus, the above analysis shows again that the “stationary” elliptic problems (8) and (56) are crucial for revealing various local and global evolution properties of all four classes of PDEs involved. We begin this study with an application of the classic variational techniques. 3 Problem “existence”: variational approach to countable families of solutions by the Lusternik–Schnirel’man category and Pohozaev’s fibering theory Variational setting and compactly supported solutions Thus, we study, in a general multi-dimensional geometry, the existence and a multiplicity of compactly supported solutions of the elliptic problem in (8).

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Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schroedinger Equations by Victor A. Galaktionov


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