A simple description of blow-up solutions through dynamics at infinity in nonautonomous ODEs

Kaname Matsue

doi:10.3934/dcds.2026020

Article Contents

2026, Volume 51: 1-45. Doi: 10.3934/dcds.2026020

This volume Previous Article Next Article Dynamics of the energy-critical nonlinear Schrödinger system in $ \mathbb R^{4} $

A simple description of blow-up solutions through dynamics at infinity in nonautonomous ODEs

Kaname Matsue^{1, 2, ,}

1.
Institute of Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan
2.
International Institute for Carbon-Neutral Energy Research (WPI-I²CNER), Kyushu University, Fukuoka 819-0395, Japan

Received: April 19, 2025

Revised: November 24, 2025

Early access: December 31, 2025

Published online: December 31, 2025

Abstract / Introduction Full Text(HTML) Figure(1) / Table(4) Related Papers Cited by

Abstract

A simple criterion of the existence of (type-Ⅰ) blow-up solutions for nonautonomous ODEs is provided. In a previous study [24], geometric criteria for characterizing blow-up solutions for nonautonomous ODEs are provided by means of dynamics at infinity. The basic idea towards the present aim is to correspond such criteria to leading-term equations associated with blow-up ansatz characterizing multiple-order asymptotic expansions, which originated from the corresponding study developed in the framework of autonomous ODEs. Restricting our attention to constant coefficients of leading terms of blow-ups, results involving the simple criterion of blow-up characterizations in autonomous ODEs can be mimicked to nonautonomous ODEs.

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Mathematics Subject Classification: 34A26, 34C08, 34D05, 34E10, 34C41, 37C25, 58K55.

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Figure 1. Trajectory for (4.19)

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Table 1. Correspondence of eigenstructures from $ A^{\rm ext} $ to $ Dg^{\rm ext}_\ast $

	$ A^{\rm ext} $	$ Dg^{\rm ext}_\ast $
Transversal eigenvalue	$ 1 $	$ -C_\ast $
Transversal eigenvector ($ k>1 $)	$ {\bf v}_{0, \alpha}^{\rm ext} $ (Proposition 3.12)	$ {\bf v}_{\ast, \alpha}^{\rm ext} $ (Proposition 3.13)
Transversal eigenvector ($ k=1 $)	$ {\bf v}_{0, \alpha}^{\rm ext} $ (Proposition 3.12)	$ \tilde {\bf v}_{\ast, \alpha}^{\rm ext} $ (Proposition 3.21)
Tangential eigenvalue (nonautonomous)	$ 0 $	$ 0 $
Tangential eigenvector (nonautonomous)	$ \begin{pmatrix} 1 \\-A^{-1}D_t f_{\alpha, k}(t_\ast, {\bf Y}_0) \end{pmatrix} $	$ r_{{\bf Y}_0}^{-\Lambda_\alpha^{\rm ext}}\begin{pmatrix} 1\\-A^{-1}D_t f_{\alpha, k}(t_\ast, {\bf Y}_0) \end{pmatrix} $
Tangential eigenvalue	$ \tilde \lambda $	$ \lambda = r_{{\bf Y}_0}^{-k}\tilde \lambda $
Tangential (generalized) eigenvector	$ \begin{pmatrix}0\\\tilde {\bf U} \end{pmatrix} $	$ (I-P_\ast) r_{{\bf Y}_0}^{-\Lambda_\alpha^{\rm ext}} \begin{pmatrix}0\\\tilde {\bf U} \end{pmatrix} $
The constant $ r_{{\bf Y}_0} $ is $ p_\alpha({\bf Y}_0) $. Once a nonzero root $ {\bf Y}_0 $ of the balance law and eigenpairs of $ A $ are given, corresponding equilibrium on the horizon $ (t_\ast, {\bf x}_\ast) $ and all eigenpairs of $ Dg^{\rm ext}_\ast $ are constructed by the rule on the table.

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Table 2. Correspondence of eigenstructures from $ Dg^{\rm ext}_\ast $ to $ A^{\rm ext} $

	$ Dg^{\rm ext}_\ast $	$ A^{\rm ext} $
Transversal eigenvalue	$ -C_\ast $	$ 1 $
Transversal eigenvector ($ k>1 $)	$ {\bf v}_{\ast, \alpha}^{\rm ext} $ (Proposition 3.13)	$ {\bf v}_{0, \alpha}^{\rm ext} $ (Proposition 3.12)
Transversal eigenvector ($ k=1 $)	$ \tilde {\bf v}_{\ast, \alpha}^{\rm ext} $ (Proposition 3.21)	$ {\bf v}_{0, \alpha}^{\rm ext} $ (Proposition 3.12)
Tangential eigenvalue (nonautonomous)	$ 0 $	$ 0 $
Tangential eigenvector (nonautonomous)	$ \begin{pmatrix} 1\\ - ((D_{\bf x} g)_\ast)^{-1}(D_t g)_\ast \end{pmatrix} $	$ r_{{\bf x}_\ast}^{\Lambda_\alpha^{\rm ext}}\begin{pmatrix} 1\\ - ((D_{\bf x} g)_\ast)^{-1}(D_t g)_\ast \end{pmatrix} $
Tangential eigenvalue	$ \lambda $	$ \tilde \lambda = r_{{\bf x}_\ast}^k\lambda $
Tangential (generalized) eigenvector	$ (I-P_\ast) \begin{pmatrix}0\\ \tilde {\bf u} \end{pmatrix} $	$ r_{{\bf x}_\ast}^{\Lambda_\alpha^{\rm ext}} (A_g^{\rm ext} - kC_\ast I) \begin{pmatrix}0\\ \tilde {\bf u} \end{pmatrix} $
The constant $ r_{{\bf x}_\ast} $ is $ (kC_\ast)^{-1/k} $, which is positive by Theorem 3.7. Once an equilibrium on the horizon $ (t_\ast, {\bf x}_\ast) $ and eigenpairs of $ Dg^{\rm ext}_\ast $ are given, corresponding (nonzero) root of the balance law $ (t_\ast, {\bf Y}_0) $ and all eigenpairs of $ A^{\rm ext} $ are constructed by the rule on the table.

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Table 3. $ t_0 $, $ t_{\max} $, $ x_1x_3 $ and $ \sin (t_{\max}) $ for (4.14)

$ t_0 $	$ t_{\max} $	$ x_1x_3 $	$ \sin (t_{\max}) $
$ 0.0 $	$ 32.8825988 $	$ - $	$ 0.994583983 $
$ 0.02 $	$ 27.2058393 $	$ - $	$ 0.876476743 $
$ 0.04 $	$ 26.8604357 $	$ - $	$ 0.987716715 $
$ 0.06 $	$ 32.3622666 $	$ - $	$ 0.811281139 $
$ 0.08 $	$ 58.0551668 $	$ - $	$ 0.997933643 $
$ 0.10 $	$ 64.9226436 $	$ - $	$ 0.867822074 $
$ 0.12 $	$ 167.810591 $	$ + $	$ -0.965193131 $
$ 0.14 $	$ 64.9566839 $	$ - $	$ 0.850408826 $
$ 0.16 $	$ 101.590941 $	$ - $	$ 0.872343878 $
$ 0.18 $	$ 27.2550679 $	$ - $	$ 0.851723643 $
$ 0.20 $	$ 27.3562087 $	$ - $	$ 0.794464435 $

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Table 4. $ t_0 $, $ t_{\max} $, $ x_1x_3 $ and $ \sin (t_{\max}) $ for (4.19)

$ t_0 $	$ t_{\max} $	$ x_1x_3 $	$ \sin (t_{\max}) $
$ 0.0 $	$ 16.8997936 $	$ - $	$ -0.929047680 $
$ 0.02 $	$ 23.1796559 $	$ + $	$ -0.927813165 $
$ 0.04 $	$ 17.0537921 $	$ - $	$ -0.97480137 $
$ 0.06 $	$ 17.0394695 $	$ - $	$ -0.971506471 $
$ 0.08 $	$ 16.8828818 $	$ - $	$ -0.922658414 $
$ 0.10 $	$ 16.6462128 $	$ - $	$ -0.806524463 $
$ 0.12 $	$ 16.7658949 $	$ - $	$ -0.871342465 $
$ 0.14 $	$ 16.7562709 $	$ - $	$ -0.866579900 $
$ 0.16 $	$ 16.9183831 $	$ - $	$ -0.935764124 $
$ 0.18 $	$ 17.0743196 $	$ - $	$ -0.979174827 $
$ 0.20 $	$ 17.3945135 $	$ - $	$ -0.993307996 $

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