Acta mathematica scientia, Series B >
GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS
Received date: 2006-09-06
Online 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.
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|u| ∫0uf(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.
XU Run-Zhang , LIU YA-Cheng . GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS[J]. Acta mathematica scientia, Series B, 2010 , 30(5) : 1793 -1807 . DOI: 10.1016/S0252-9602(10)60173-7
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