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

By Victor A. Galaktionov

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

The ebook first reports the actual self-similar singularity suggestions (patterns) of the equations. This procedure permits 4 assorted periods of nonlinear PDEs to be handled at the same time to set up their awesome universal positive aspects. The e-book describes many homes of the equations and examines conventional questions of existence/nonexistence, uniqueness/nonuniqueness, worldwide asymptotics, regularizations, shock-wave concept, and diverse blow-up singularities.

Preparing readers for extra complicated mathematical PDE research, the ebook demonstrates that quasilinear degenerate higher-order PDEs, even unique and awkward ones, should not as daunting as they first look. It additionally illustrates the deep positive factors shared through different types of nonlinear PDEs and encourages readers to improve additional this unifying PDE method from different viewpoints.

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

Example text

5) This explicit integration of ODE (4) was amazing and rather surprising in the middle of the 1970s and led then to the foundation of blow-up and heat localization theory. In dimension N ≥ 2, the blow-up solution (3) does indeed exist [359, p. 183], but not in an explicit form (so that, it seems, (5) is the only available elegant form of such a localized solution). Note that (5) is a compactly supported in space weak solution, due to a strong degeneracy at u = 0 of the diffusion operator, meaning finite propagation of perturbations.

Though, even for m = 1, this is not that univalent: there are other structures that do not obey the Sturmian order. , ±F0 (y + ak )} for k ≥ 2 of y-shifted first patterns F0 (y + ai ) without sign changes. This is easy, since each F0 is compactly supported, so, in such an easy construction via y-shifting, one needs to avoid overlapping of supports of all the components; see more comments below. For m ≥ 2, though such a “shifting construction” makes the same sense, in general, this question is more difficult, and seem does not to admit a clear rigorous treatment, since, unlike the case m = 1, there exist infinitely many other “gluing connections” between such patterns.

We will show how this affects the oscillatory properties of solutions for odd and even m’s. Periodic oscillatory components We now look for periodic solutions of (102), which are the simplest nontrivial bounded solutions that can be continued up to the interface at s = −∞. 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.

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