数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (5): 1793-1807.doi: 10.1016/S0252-9602(10)60173-7

• 论文 • 上一篇    下一篇

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

徐润章, 刘亚成   

  1. College of Science, Harbin Engineering University, Harbin 150001, China
  • 收稿日期:2006-09-06 出版日期:2010-09-20 发布日期:2010-09-20
  • 基金资助:

    This work is supported by National Natural Science Foundation of China (10871055, 10926149); Natural Science Foundation of Heilongjiang
    Province (A2007-02; A200810); Science and Technology Foundation of Education Office of Heilongjiang Province (11541276); Foundational
    Science Foundation of Harbin Engineering University.

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

 XU Run-Zhang, LIU YA-Cheng   

  1. College of Science, Harbin Engineering University, Harbin 150001, China
  • Received:2006-09-06 Online:2010-09-20 Published:2010-09-20
  • Supported by:

    This work is supported by National Natural Science Foundation of China (10871055, 10926149); Natural Science Foundation of Heilongjiang
    Province (A2007-02; A200810); Science and Technology Foundation of Education Office of Heilongjiang Province (11541276); Foundational
    Science Foundation of Harbin Engineering University.

摘要:

In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation
utt-uxx-uxxt-uxxtt=f(u)xx, x ∈Ω, t >0,
u(x, 0)=u0(x), ut(x, 0)=u1(x), x ∈Ω,
u(0, t)=u(1, t)=0, t ≥0,
where Ω =(0, 1). First, we obtain the existence of local Wk, p solutions. Then, we prove that, if f(s) ∈ in Ck+1(R) is ondecreasing, f(0)=0 and |f(u)| ≤ C1|u0uf(s)ds + C2, u0(x), u1(x) ∈Wk, p(Ω) ∩ W0{1, p}(Ω), k ≥1, 1< p ≤∞, then for any T>0 the problem admits a unique solution u(x, t) ∈ W2, ∞ (0, T; Wk, p(Ω)∩W01, p(Ω) ). Finally, the finite time blow-up of solutions and global Wk, p solution of generalized IMBq equations are discussed.

关键词: Pochhammer-Chree equations, initial boundary value problem, Wk, p solution, global existence, blow-up

Abstract:

In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation
utt-uxx-uxxt-uxxtt=f(u)xx, x ∈Ω, t >0,
u(x, 0)=u0(x), ut(x, 0)=u1(x), x ∈Ω,
u(0, t)=u(1, t)=0, t ≥0,
where Ω =(0, 1). First, we obtain the existence of local Wk, p solutions. Then, we prove that, if f(s) ∈ in Ck+1(R) is ondecreasing, f(0)=0 and |f(u)| ≤ C1|u0uf(s)ds + C2, u0(x), u1(x) ∈Wk, p(Ω) ∩ W0{1, p}(Ω), k ≥1, 1< p ≤∞, then for any T>0 the problem admits a unique solution u(x, t) ∈ W2, ∞ (0, T; Wk, p(Ω)∩W01, p(Ω) ). Finally, the finite time blow-up of solutions and global Wk, p solution of generalized IMBq equations are discussed.

Key words: Pochhammer-Chree equations, initial boundary value problem, Wk, p solution, global existence, blow-up

中图分类号: 

  • 35L75