arXiv subjects

V. A. Galaktionov

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

At least 19 recordsLinked to original sources

arXiv2014

Existence of solutions for a system of coupled nonlinear stationary bi-harmonic Schrödinger equations

We obtain existence and multiplicity results for the solutions of a class of coupled semilinear bi-harmonic Schrödinger equations. Actually, using the classical Mountain Pass Theorem and minimization techniques, we prove the existence of critical points of the associated functional constrained on the \emph{Nehari manifold}. Furthermore, we show that using the so-called \emph{fibering method} and the \emph{Lusternik-Schnirel'man theory} there exist infinitely many solutions, actually a countable family of critical points, for such a semiliner bi-hamonic Schrödinger system under study in this work.

math.AP↗

arXiv2012

Boundary Characteristic Point Regularity for Navier-Stokes Equations: Blow-up Scaling and Petrovskii-type Criterion (a Formal Approach)

It is shown that Wiener's regularity of the vertex of a backward paraboloid for 3D Navier-Stokes equations with zero Dirichlet conditions on the paraboloid boundary is given by Petrovskii's criterion for the heat equation (1934), i.e., the nonlinear convection term does not affect the regularity result.

math.AP↗

arXiv2012

Sturmian Multiple Zeros for Stokes and Navier--Stokes Equations in $\re^3$ via Solenoidal Hermite Polynomials

Multiple spatial zero formations for Stokes and Navier-Stokes equations in three dimensions are shown to occur according to nodal sets of solenoidal Hermite polynomials. Extensions to well-posed Burnett equations with the bi-harmonic viscosity operator are also discussed.

math.AP↗

arXiv2012

Classification of global and blow-up sign-changing solutions of a semilinear heat equation in the subcritical Fujita range II. Higher-order diffusion

Two families of sign-changing solutions of higher-order semilinear parabolic equations are studied.

math.AP↗

arXiv2012

Towards the KPP--Problem and Log-Front Shift for Higher-Order Nonlinear PDEs I. Bi-Harmonic and Other Parabolic Equations

Some extensions of the KPP idelogy of 1937 and more recent results to higher-order semilinear parabolic equations are discussed

math.AP↗

arXiv2011

Refined Scattering and Hermitian Spectral Theory for Linear Higher-Order Schrödinger Equations

A classification of large-time and finite-time blow-up asymptotics of solutions of the Cauchy problem for higher-order Schrödinger equations is performed.

math.AP↗

arXiv2011

Variational Approach to Complicated Similarity Solutions of Higher-Order Nonlinear PDEs. II

Some higher-order quasilinear parabolic, hyperbolic, and nonlinear dispersion equations are shown to admit various blow-up, extinction, and travelling wave solutions, which reduce to variational problems admitting countable families of compactly supported solutions.

math.AP↗

arXiv2010

Eigenfunctions and Very Singular Similarity Solutions of Odd-Order Nonlinear Dispersion PDEs

Asymptotic properties of solutions of odd-order nonlinear dispersion equations are studied. The global in time similarity solutions, which lead to eigenfunctions of the rescaled ODEs, are constructed.

math.AP↗

arXiv2010

Local Bifurcation-Branching Analysis of Global and "Blow-up" Patterns for a Fourth-Order Thin Film Equation

The quasilinear fourth-order thin film equation in $R^N$ is shown to admit infinite countable families of global and blow-up similarity solutions, which are originated in the limit from eigenfunctions of the corresponding linear non-self-adjoint operators.

math.AP↗

arXiv2010

On Continuous Branches of Very Singular Similarity Solutions of the Stable Thin Film Equation

Continuous branches of similarity solutions of the fourth-order stable thin film equation are studied.

math.AP↗

arXiv2009

Schock waves and compactons for fifth-order nonlinear dispersion equations. II

It is shown that fifth-order nonlinear dispersion equations from compacton theory admit shock and rarefaction waves. A self-similar gradient blow-up is shown to admit infinitely many similarity extensions beyond blow-up time, meaning principal nonuniqueness of solutions of the Cauchy problem after occurring such singularities.

math.AP↗

arXiv2009

Branching analysis of a countable family of global similarity solutions of a fourth-order thin film equation

We show that various asymptotic properties of global solutions of a fourth-order quasilinear thin film equation can be described by branching from corresponding solutions of the linear bi-harmonic equation. This includes a countable family of self-similar solutions and other aspects.

math.AP↗

arXiv2009

On source-type solutions and the Cauchy problem for a doubly degenerate sixth-order thin film equation I. Local oscillatory properties

Local oscillatory and other properties of source-type solutions of doubly nonlinear sixth-order parabolic thin film equations are studied.

math.AP↗

arXiv2009

Very Singular Similarity Solutions and Hermitian Spectral Theory for Semilinear Odd-Order PDEs

Very singular self-similar solutions of semilinear odd-order PDEs are studied on the basis of a Hermitian-type spectral theory for linear rescaled odd-order operators.

math.AP↗

arXiv2009

Self-similar blow-up in parabolic equations of Monge--Ampère type

We propose nonlinear parabolic equations of Monge--Ampére (M--A) type that admit regional, single poin, and global blow-up of similarity type. A similar model is derived for a fourth-order M--A flow.

math.AP↗

arXiv2009

Three tupes of self-similar blow-up for the fourth-order p-Laplacian equaiton with source: variational and branching approaches

The fourth-order quasilinear reaction-diffusion equation with a p-Laplacian operator is shown to admit three types of blow-up. Self-similar patterns are first constructed for the regional blow-up case, where the rescaled problem admits a variational setting. Next, these were extended via a branching approach to non-variational problems.

math.AP↗

arXiv2009

On blow-up shock waves for a nonlinear PDE associated with Euler equations

A second-order PDE is derived from Euler's equaitons under certain assumptions. It is shown that this PDE admits shock and rarefaction waves, and that a single point gradient blow-up admits a unique similarity extension after blow-up that settles uniqueness/entropy issues for such equations.

math.AP↗

arXiv2009

Shock waves and compactons for fifth-order nonlinear dispersion equaitons

Fifth-order 1D nonlinear dispersion equations are shown to admit blow-up formation of shock waves as well as rarefaction waves. The concepts of smooth deformations are applied to distinguish "entropy" shocks from smooth rarefaction waves. Single point gradient catastrophe is shown to lead to nonuniqueness after blow-up.

math.AP↗