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V. A. Galaktionov

Publications and source records attributed to V. A. Galaktionov.

35 records · Page 2Linked to original sources
arXiv2009

Formation of shocks in higher-order nonlinear dispersion PDEs: nonuniqueness and nonexistence of entropy

It is shown that third-order 1D nonlinear dispersion equations admit single point gradient catastrophe, described by blow-up-type similarity solutions. After blow-up, the solutions admit shock wave-type self-similar extensions. Snce such extensions are not unique, this implies the principle nonuniqueness of shock-type solutions and also nonexistence of any entropy-type description of proper unique solutions. A difficult free-boundary setting, with extra conditions specified on shocks, are necessary to restore uniqueness in such problems.

math.AP↗
arXiv2009

Variational approach to complicated similarity solutions pf higher-order nonlinear PDEs. I

Three classes of higher-order nonlinear parabolic hyperbolic, and nonlinear dispersion equations are shown to admit exact blow-up or compacton solutions, which are induced by elliptic equations with non-Lipschitz nonlinearities. Variational techniques give countable families of various compactly supported solutions with oscllatory behaviour cloase to the interfaces. Overall, the whole set of such compact patterns reveals certain chaotic properties and contains solutions of arbitrary complexity.

math.AP↗
arXiv2009

On global solutions and blow-up for Kuramoto-Sivashinsky-type models, and well-posed Burnett equations

The initial boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto--Sivashinsky equation and other related $2m$th-order semilinear parabolic partial differential equations in one and N dimensions are considered. Global existence and blow-up as well as uniform bounds are reviewed by using: (i) classic tools of interpolation theory and Galerkin methods, (ii) eigenfunction and nonlinear capacity methods, (iii) Henry's version of weighted Gronwall's inequalities, and (vi) two types of scaling (blow-up) arguments. For the IBVPs, existence of global solutions is proved for both Dirichlet and "Navier" boundary conditions. For some related higher-order PDEs in N dimensions uniform boundedness of global solutions of the Cauchy problem are established. As another related application, the well-posed Burnett-type equations, which are a higher-order extension of the classic Navier-Stokes equations, are studied. As a simple illustration, a generalization of the famous Leray-Prodi-Serrin-Ladyzhenskaya regularity results is obtained.

math.AP↗
arXiv2009

Regional, single point, and global blow-up for the fourth-order porous medium type equation with source

Three types of blow-up for a fourth-order degenerate reaction-diffusion equation are studied by a combination of analytic and numerical methods. At the critical values of parameters, there occurs a variational problem with a countable set of solutions obtained by Lysternik--Shnirelman category theory, which then are extended to neighbouring values of parameters by a homotopy-like approach.

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arXiv2009

On blow-up "twistors" for the Navier--Stokes equations in $R^3$: a view from reaction-diffusion theory

Various formal blow-up scenarious for the Navier--Stokes equaitons in 3D are discussed. A particular interest is payed to "twistor mechanisms" based on angular logarithmic blow-up travelling waves, and to a formation of "blow-up tornado" about the singular stationary Slezkin--Landau solutions (1934--44) of a submerged jet. A survey on various related aspects of singularity analysis for nonlinear parabolic PDEs is enclosed.

math.AP↗
arXiv2009

Vast multiplicity of very singular self-similar solutions of a semilinear higher-order diffusion equation with time-dependent absorption

It is shown that a fourth-order semilinear parabolic equation with time-dependent absorption admit a vast multiplicity of the so-called very singular self-similar solutions (VSSs), which can bifurcate from some eigenfunctions of the linearized operators and also are generated by branching from other critical point. For the second-order semilinear heat equations, such VSSs have been known since the beginning of the 1980s and were constructed in papers by Brezis, Friedman, Galaktionov, Kyrdyumov, Samarskii, Peletier, Kamin, Veron, and many others.

math.AP↗
arXiv2009

Blow-up in higher-order reaction-diffusion and wave equations: how $\sqrt{log log}$ factor occurs

The origin of non self-similar blow-up in higher-order reaction-diffusion (parabolic), wave (hyperbolic) and nonlinear dispersion equations is explained by a combination of various methods. Some links and similarities with double-log blow-up terms occurring in earlier studies of plasma physics second-order parabolic equations and the nonlinear critical Schrödinger equation are discussed. On the other hand, the log-log factor obtained in Petrovskii's boundary regularity study of a paraboloid vertex for the heat equation in 1934 was the first its appearance in PDE theory.

math.AP↗
arXiv2009

Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type $$ u_t= -Δ^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1, \quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, $$ are discussed. For the semilinear heat equation $u_t= Δu+ u^p$, various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide range of its application.

math.AP↗
arXiv2009

On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach

It is shown that, for the bi-harmonic equation, an optimal regularity criterion of the vertex of typical paraboloids can be expressed in terms of Osgood-Dini integral conditions of Petrovskii's type for the heat equation derived in 1934. Some extensions of the first Fourier coefficent method to other PDEs are discussed. A survey on boundary point regularity for elliptic and parabolic PDEs is enclosed.

math.AP↗
arXiv2009

On centre subspace behaviour in thin flm equations

It is shown that the fourth-order parabolic thin film equation with critical absorption admits the large time behaviour, where special logarithmic perturbations enter the standard source-type asymptotics. Such asymptpotic phenomena on existence of logarithmic factors are known for semilinear heat equation with absorption since the 1980s.

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