Positive solutions of the generalized Emden-Fowler equation in Hölder spaces

A singular sublinear BVP related to the Emden-Fowler equation is considered. Existence, nonexistence, and regularity of positive solutions in Hölder spaces is obtained.     

On the structure of generalized solutions of the one-dimensional equations of a polytropic viscous gas

A model system of equations that defines the unsteady one-dimensional flow of a visoucs gas is considered on the assumption that the pressure is determined by the adiabatic Poisson law. Generalized solutions are investigated in the class of discontinuous functions, a class of correctness is separated, and the structure of solutions of this class is clarified. It is shown that the initial velocity discontinuities are instantly smoothed out, and from the discontinuity points of the initial density,lines of contact discontinuity are formed. These lines exist for an infinite time, and the pressure jumps on them vanish exponentially.     

Entire positive solutions of the singular Emden-Fowler equation with nonlinear gradient term

Let p and q be locally Hölder functions in ?^N, p > 0 and q ≥ 0. We study the Emden-Fowler equation $-\triangle u+q(x)\mid \nabla u\mid^\alpha=p(x)u^{-\gamma}$ in ?^N, where α and γ are positive numbers. Our main result establishes that the above equation has a unique positive solutions decaying to zero at infinity. Our proof is elementary and it combines the maximum principle for elliptic equations with a theorem of Crandall, Rabinowitz and Tartar.     

On singularities of continuity equation solutions

Generalized solutions to the continuity equation in one-dimensional and multidimensional cases are constructed in the case of a discontinuous velocity field.     

Integration ofD-dimensional cosmological models with two factor spaces by reduction to the generalized Emden-Fowler equation

The D-dimensional cosmological model on the manifold M=R×M₁×...×M_n, describing the evolution of Einstein factor spaces M_i in the presence of a multicomponent perfect fluid source, is considered. The barotropic equation of state for the mass-energy densities and pressures of the components is assumed in each space. Where the number of non-Ricci-flat factor spaces and the number of perfect fluid components are both equal to two, the Einstein equations for the model are reduced to the generalized Emden-Fowler (second-order ordinary differential) equation, which has been recently investigated by Zaitsev and Polyanin using discrete-group analysis. We generate new integrable cosmological models using the integrable classes of this equation and present the corresponding metrics. The method is demonstrated for the special model with Ricci-flat spaces M₁ and M₂ and a two-component perfect fluid source. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 114, No. 3, pp. 454–469, March, 1998.     

Multiple positive solutions to a singular boundary value problem for a superlinear Emden-Fowler equation

A multiplicity result for the singular ordinary differential equation y^″+λx⁻²yσ=0, posed in the interval (0,1), with the boundary conditions y(0)=0 and y(1)=γ, where σ>1, λ>0 and γ?0 are real parameters, is presented. Using a logarithmic transformation and an integral equation method, we show that there exists Σ^?∈(0,σ/2] such that a solution to the above problem is possible if and only if λγσ−1?Σ^?. For 0<λγσ−1<Σ^?, there are multiple positive solutions, while if γ=(λ⁻¹Σ^?)^1/(σ−1) the problem has a unique positive solution which is monotonic increasing. The asymptotic behavior of y(x) as x→⁰⁺ is also given, which allows us to establish the absence of positive solution to the singular Dirichlet elliptic problem −Δu=d⁻²(x)uσ in Ω, where Ω⊂RN, N?2, is a smooth bounded domain and d(x)=dist(x,∂Ω).     

On the continuity of the generalized spectral radius in max algebra

Given a bounded set Ψ of n×n non-negative matrices, let ρ(Ψ) and μ(Ψ) denote the generalized spectral radius of Ψ and its max version, respectively. We show that

     

Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation

Positive solutions of the nonlinear second-order differential equation $(p(t)|x'|^{\alpha - 1} x')' + q(t)|x|^{\beta - 1} x = 0,\alpha > \beta > 0,$ are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.     

On the solutions of generalized discrete Poisson equation

The set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete Poisson equation. It is shown that its solutions with square-summable discrete derivatives are unique up to a constant. The proof uses the Fourier transform as the main tool. The necessary condition for the existence of the solution is provided.     

On positive solutions concentrating on spheres for the Gierer-Meinhardt system

We consider the stationary Gierer-Meinhardt system in a ball of R^N:

     

On the stability of some exact solutions to the generalized convection-reaction-diffusion equation

Stability of a set of traveling wave solutions to the convection-reaction-diffusion equation taking into account the effects of memory is studied by means of the qualitative methods and numerical simulation.     

On positive solutions of a semilinear equation

A semilinear elliptic equation is considered in a domain with smooth boundary. The authors prove the existence and uniqueness of positive solutions of different types, singular at an inner point, subject to the Dirichlet boundary conditions. Bibliography: 13 titles. Translated from Trudy Seminara imeni I., G. Petrovskogo. No. 18, pp. 157–169, 1995.     

Random generalized solutions to the heat equation

A solution for the heat conduction problem with random source term and random initial and boundary conditions is defined. Existence, uniqueness, properties, and asymptotic behavior of such a solution are investigated. Applications to one-dimensional problems are presented.     

On solutions of the generalized Stein quaternion matrix equation

In this paper, some solution expressions to the quaternion matrix equation X?AXF=BY are obtained. They are the further conclusions of the paper (Song et al. in Int. J. Comput. Math. 89:890–900, 2012). By applying of Kronecker map and complex representation of a quaternion matrix, the sufficient conditions for finding the solution have been established. At the end of the article, numerical examples show the applications of the proposed explicit solution.     

On the support of solutions to the Ostrovsky equation with positive dispersion

In this paper we prove that sufficiently smooth solutions of the Ostrovsky equation with positive dispersion,