TY - JOUR
T1 - Geometric treatments and a common mechanism in finite-time singularities for autonomous ODEs
AU - Matsue, Kaname
N1 - Funding Information:
The author was partially supported by Program for Promoting the reform of national universities (Kyushu University), Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, World Premier International Research Center Initiative (WPI), MEXT, Japan, and JSPS Grant-in-Aid for Young Scientists (B) (No. JP17K14235 ). He would like also to thank Professors Koichi Anada, Tetsuya Ishiwata and Takeo Ushijima for giving him very essential suggestions to the present study. Finally, he would like to thank to reviewers of the journal for providing him with helpful suggestions about contents, organizations of this paper.
Publisher Copyright:
© 2019
PY - 2019/12/5
Y1 - 2019/12/5
N2 - Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and dynamics on their center-stable manifolds. In particular, we show that dynamics on center-stable manifolds of invariant sets at infinity with appropriate time-scale desingularizations as well as blowing-up of singularities characterize dynamics of blow-up solutions as well as their rigorous blow-up rates. Similarities for characterizing finite-time extinction and asymptotic behavior of compacton traveling waves to blow-up solutions are also shown.
AB - Geometric treatments of blow-up solutions for autonomous ordinary differential equations and their blow-up rates are concerned. Our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces, and dynamics on their center-stable manifolds. In particular, we show that dynamics on center-stable manifolds of invariant sets at infinity with appropriate time-scale desingularizations as well as blowing-up of singularities characterize dynamics of blow-up solutions as well as their rigorous blow-up rates. Similarities for characterizing finite-time extinction and asymptotic behavior of compacton traveling waves to blow-up solutions are also shown.
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U2 - 10.1016/j.jde.2019.07.022
DO - 10.1016/j.jde.2019.07.022
M3 - Article
AN - SCOPUS:85070231187
SN - 0022-0396
VL - 267
SP - 7313
EP - 7368
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 12
ER -