有外力干擾的二階非線性微分方程

Abstract

在這一篇論文中，我們討論的是常微分方程u\" =u^P(C₁+C2(u`)^q\")我們發現一些現象，爆破率、爆破常數、爆破時間。而且我們還發現爆破時問與係數之間的關係，我們將在之後討論。

In this paper we work with the ordinary differential equation u\" = u^P(C₁+C2(u`)^q\"). We have found some phenomena, blow-up, blow-up rate, blow-up constant, blow-up time are obtained in this work. Further, we have also found the relationship between blow-up time and blow-up coefficients, we shall detail illustrate it later.

Abstract-----i\r\n中文摘要-----ii\r\n1 Introduction-----1\r\n1.1 The Calligraphy Equation (Li,1999)-----1\r\n1.2 The Existence of Solutions-----2\r\n\r\n2 Blow-up Phenomena for 2 > q ≧1-----6\r\n2.1 Blow-up Rate and Blow-up Constant of u(t)-----10\r\n2.2 Blow-up Rate and Blow-up Constant of u`t)-----11\r\n2.3 Blow-up Rate and Blow-up Constant of u\"\"(t)-----12\r\n\r\n3 Blow-up Phenomena for q = 2-----13\r\n3.1 Blow-up Rate and Blow-up Constant of u(t)-----13\r\n3.2 Blow-up Rate and Blow-up Constant of u`(t)-----14\r\n3.3 Blow-up Rate and Blow-up Constant of u\"\"(t)-----15\r\n\r\n4 Blow-up Phenomena for q > 2-----16\r\n4.1 Blow-up Rate and Blow-up Constant of u`(t)-----17\r\n4.2 Blow-up Rate and Blow-up Constant of u\"\"{t)-----18\r\n\r\n5 Conclusions-----19\r\n5.1 Tables-----19\r\n5.1.1 Blows up Phenomena for u under uo,u1,c2 > 0-----19\r\n5.1.2 Blows up Phenomena for u` under uo,u1,c2 > 0-----19\r\n5.1.3 Blows up Phenomena for u\"\" under uo,u1,c1,c2 > 0-----19\r\n5.2 Properties of Blow-up Constant and Coefficients-----19\r\n5.2.1 The Case of 1 5.2.2 The Case of q= 2-----22\r\n5.3 Properties of Blow-up Time and Coefficients-----23\r\n5.3.1 The Case of 1 5.3.2 The Case of q=2-----25\r\n\r\nReferences-----26