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 exhibits how 4 kinds of higher-order nonlinear evolution partial differential equations (PDEs) have many commonalities via their detailed quasilinear degenerate representations. The authors current a unified method of take care of those quasilinear PDEs.

The e-book first stories the actual self-similar singularity ideas (patterns) of the equations. This technique permits 4 various sessions of nonlinear PDEs to be taken care of concurrently to set up their impressive universal positive factors. The ebook describes many homes of the equations and examines conventional questions of existence/nonexistence, uniqueness/nonuniqueness, worldwide asymptotics, regularizations, shock-wave conception, and numerous 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 beneficial properties shared through different types of nonlinear PDEs and encourages readers to improve additional this unifying PDE strategy from different viewpoints.

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

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4, while the F -term, giving F 2 , is negligible in the tail. , eliminating the tail in between F0 ’s, will decrease the value cF , since, in (83), the numerator decreases and the denominator increases. , making the minimal number of internal transversal zeros between single structures. Regardless of the simple variational-oscillatory meaning (93) of this FRPC, we do not know how to make this rule sound rigorous. 1. , (−1)k−1 F0 }, (95) where each neighboring pair {F0 , −F0 } or {−F0 , F0 } has a single transversal zero in between the structures.

This agrees with the obviously correct similar result for n = 0, namely, for the linear equation (98) for α = 0 (or, more correctly, for very small values of α > 0). Namely, taking into account the nonlinear term only and passing to the limit α → +0 (n → +0) yields F (4) = −|F |−α F =⇒ F (4) = −F as y → −∞. (109) Here the interface is infinite, so its position corresponds to y = −∞, so that the interface satisfies y0 (n) → −∞ as n → +0. Setting, as usual, F (y) = eμy gives the characteristic equation and a unique exponentially decaying pattern: as y → −∞, μ4 = −1 =⇒ F (y) ∼ e y √ 2 A cos √y 2 + B sin √y 2 , (110) where A, B ∈ IR are constants.

6. Numerical construction of periodic orbits for m = 3 Consider now the second equation in (103), which admits constant equilibria (104) existing for all n > 0. It is easy to check that the equilibria ±ϕ0 are asymptotically stable as s → +∞. Then, the necessary periodic orbit is situated between these stable equilibria, so it is unstable as s → +∞. 7 for n = 15, obtained by shooting from s = 0 with prescribed Cauchy data. 6 Convergence to the stable periodic solution of ODE (114) (the limit value n = +∞).

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