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

doi:10.1016/S0252-9602(10)60173-7

Abstract

Abstract:

In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation
ut_t-u_xx-u_xxt-u_xxtt=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 W_{k, p} solutions. Then, we prove that, if f(s) ∈ in C^k⁺¹(R) is ondecreasing, f(0)=0 and |f(u)| ≤ C₁|u| ∫₀^uf(s)ds + C₂, u₀(x), u₁(x) ∈W^{k, p}(Ω) ∩ W₀{^1, ^p}(Ω), k ≥1, 1< p ≤∞, then for any T>0 the problem admits a unique solution u(x, t) ∈ W^2, ∞ (0, T; W^k, ^p(Ω)∩W₀^1, ^p(Ω) ). Finally, the finite time blow-up of solutions and global W^{k, p} solution of generalized IMBq equations are discussed.

Key words: Pochhammer-Chree equations, initial boundary value problem, W^{k, p} solution, global existence, blow-up

CLC Number:

35L75

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.

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