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Asymptotic behavior of solutions of x''=e^{\alpha\lambda t} x^{1+\alpha} where -1 < \alpha lpha < 0

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  • Physics

Abstract

Tsukamoto, I. Osaka J. Math. 40 (2003), 595–620 ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF x ′′ = e tx1+ WHERE −1 0 ICHIRO TSUKAMOTO (Received February 5, 2002) 1. Introduction Let us consider second order nonlinear differential equations ′′ = ± β 1+α(1 1)± ′′ = ± σ 1+α(1 2)± where ′ = / , the double signs correspond in the same order in every equation and α, β, σ are parameters. Using Chapter 7 of [1], we can state value of solving these. First these can be derived from an important second order nonlinear differential equa- tion (1.3) ( ρ ) ± σ = 0 ρ, σ, being parameters, which contains the Emden equation of astrophysics and the Fermi-Thomas equation of atomic physics and so has several interesting physical ap- plications. Second (1.3) is mathematically interesting, because (1.3) is nontrivial, non- linear and has a large class of solutions whose behavior can be ascertained with as- tonishing accuracy nevertheless these cannot be generally obtained explicitly. In addi- tion (1 1)±, (1 2)± are examples of differential equations positive radial solutions of a nonlinear elliptic partial differential equation satisfy (cf. [17]). Actually many authors have considered (1 1)±, (1 2)± and (1.3) in more general form in [2], [5] through [9], [13], [20] and so on. In these papers they mainly dis- cussed asymptotic behavior of the solution continuable to ∞. On the other hand, ini- tial value problems of (1 1)+, (1 2)+, (1 2)− and (1 1)− were considered in [10], [11], in [14], [16], in [15], [16] and in [17] respectively in case of α > 0 and asymptotic behavior of all the solutions was studied. In the case α < 0, the initial value problems of (1 1)±, (1 2)± are not considered yet, while in [5], [8], [20] etc. this case was already considered for differential equa- tions with more general form than (1 1)±, (1 2)± and for the solutions continuable to ∞. So in this paper, we shall consider (1 2)+ where −1 < α < 0 as a first step. Since 596 I. TSUKAMOTO it is convenient to put σ = αλ

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