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

The booklet first stories the actual self-similar singularity options (patterns) of the equations. This technique permits 4 various periods of nonlinear PDEs to be handled concurrently to set up their remarkable universal positive factors. The ebook describes many homes of the equations and examines conventional questions of existence/nonexistence, uniqueness/nonuniqueness, international asymptotics, regularizations, shock-wave concept, and numerous blow-up singularities.

Preparing readers for extra complex mathematical PDE research, the ebook 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 via various kinds of nonlinear PDEs and encourages readers to increase additional this unifying PDE strategy from different viewpoints.

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

Example text

For utt = (un+1 )xx + un+1 in IR × IR+ (u ≥ 0). (11) Note that this PDE, as a hyperbolic 2 × 2 system, admits solutions with shocks, with quite nontrivial properties. But now we describe its similar blow-up (regional) S-regime separate variables solutions, where the resulting ODE takes the form: 2 uS (x, t) = (T − t)− n f˜(x) =⇒ 2 2 n n + 1 f˜ = (f˜n+1 ) + f˜n+1 . (12) Using an extra scaling, f˜(x) = 2(n+2) n 1 n f (x), (13) yields the same ODE (4) and, hence, the exact localized solution (5).

69) The new functional H(r, v) = 1 2 r2 − 1 β rβ |v|β (70) has an absolute minimum point, where Hr (r, v) ≡ r − rβ−1 |v|β = 0 =⇒ r0 (v) = |v|β 1 2−β . (71) . (72) We then obtain the following functional: 2−β 2 ˜ H(v) = H(r0 (v), v) = − 2−β 2β r0 (v) ≡ − 2β |v|β 2 2−β Obviously, the critical points of the functional (72) on the set (69) coincide with those for ˜ H(v) = |v|β , (73) so we arrive at an even, non-negative, convex, and uniformly differentiable functional, to which L–S theory applies [252, § 57]; see also [94, p.

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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