By Victor A. Galaktionov
Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrödinger Equations exhibits how 4 forms 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 ebook first stories the actual self-similar singularity suggestions (patterns) of the equations. This method permits 4 assorted periods of nonlinear PDEs to be taken care of concurrently to set up their outstanding universal good points. The ebook describes many houses of the equations and examines conventional questions of existence/nonexistence, uniqueness/nonuniqueness, worldwide asymptotics, regularizations, shock-wave idea, and diverse blow-up singularities.
Preparing readers for extra complex mathematical PDE research, the publication 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 different types of nonlinear PDEs and encourages readers to increase extra this unifying PDE technique from different viewpoints.
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Extra resources for Blow-up for higher-order parabolic, hyperbolic, dispersion and Schrödinger equations
For the N -dimensional PDE (10), looking for the same solution (12), after scaling, leads to the same elliptic equation (8). Indeed, the existence of such blow-up compactly supported solutions means that the above quasilinear wave equation can describe processes with a ﬁnite propagation of perturbations, a phenomenon that was much less studied for 1 Self-Similar Blow-up and Compacton Patterns 7 higher-order hyperbolic equations than for parabolic ones. It seems that, in general, it is an open mathematical question.
Indeed, such nonstationary instant compactiﬁcation phenomena for quasilinear absorption-diﬀusion equations with singular absorption −|u|p−1 u, with p < 1, have been known since the 1970s 18 Blow-up Singularities and Global Solutions and are also called the instantaneous shrinking of the support of solutions. These phenomena have been proved for quasilinear higher-order parabolic equations with non-Lipschitz absorption terms; see . Thus, to reveal the compactly supported patterns F (y), we can pose the problem in bounded balls that are suﬃciently large.
Evidently, the 2mth-order counterpart (1) admits the regional blow-up solution of the same form (3), but the proﬁle f = f (y) (here y = x, but, for a future convenience, we almost always denote the independent similarity variable by y; later on it 1 Self-Similar Blow-up and Compacton Patterns 5 will be scaled by a power of t) then solves a more complicated quasilinear elliptic equation (−1)m+1 Δm (|f |n f ) + |f |n f = 1 n in IRN . f (6) After natural change, this gives the following semilinear elliptic equation with a non-Lipschitz nonlinearity in the last reaction-absorption term: F = |f |n f =⇒ (−1)m+1 Δm F + F − 1 n F n − n+1 F = 0 in IRN .