Existence and uniqueness of solutions of some Cauchy problems for the Emden-Fowler equation
type: Item
instance of: scholarly article
title: Existence and uniqueness of solutions of some Cauchy problems for the Emden-Fowler equation
zbMATH Open document ID: 1484.34092
DOI: 10.1134/S0012266121080024
author: Yanyan Li
published in: Differential Equations
publication date: 2021-09-08T00:00:00Z
review text: The article is devoted to the following the nonlinear Emden-Fowler differential equation \[ y''-x^a y^\sigma=0,\tag{1} \] where \(a, \sigma\) are real parameters and \(\sigma \neq1\). The article consists of 3 sections and the list of references. The first section is devoted to the general introduction into the problem. In Section 2, the authors consider the equation (1) with the following initial conditions \[ y(0)=c, \quad y'(0)=\lambda , \quad c\in(0;+\infty), \lambda \in \mathbb{R}\cup\{-\infty\}. \tag{2} \] There are obtained necessary and sufficient conditions on the parameters \(a, \sigma\) of the equation (1) for the Cauchy problem (1), (2) to have (or not to have) a solution and about uniqueness of such solutions in case its existence. An example shows that the solution of the Cauchy problem (1), (2) is not necessarily unique in case \(\lambda=-\infty\). Section 3 of the article is devoted to the equation (1) with the following initial conditions \[ y(x_0)=c, \quad y'(x_0)=\lambda , \quad x_0>0, \quad a, \lambda \in \mathbb{R}\cup\{0\}, \sigma <0.\tag{3} \] Theorem 3.1 gives the existence of a solution of the Cauchy problem (1.1), (1.3). The question arises about the uniqueness of such a solution. For \(\lambda > 0 (\lambda < 0)\), the answer was obtained in a Theorem 3.2., the case \(\lambda = 0\) was considered in the theorem 3.3. The results presented in the article are very important for the further study of solutions of Emden-Fowler differential equation.
reviewed by: Olga Chepok
Mathematics Subject Classification ID: 34B30
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 7392228
Wikidata QID: Q115249827
MaRDI profile type: Publication
cites work: Q5822220
cites work: Q4692282
cites work: Vertical asymptotes of solutions of the Emden-Fowler equation
cites work: On the Cauchy problem for an Emden-Fowler equation
cites work: Note on asymptotical behavior of solutions of Emden-Fowler equation and the existence and uniqueness of solution of some Cauchy problem
cites work: Q5279936
full work available at URL: s0012266121080024
OpenAlex ID: W3196574600
description: scientific article; zbMATH DE number 7392228