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...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 Wkp solutions. Then, we prove that, if f(s) ∈ Ck+1(R) is nondecreasing, f(0) = 0 and |f(u)| ≤ C1|u|∫0uf(s)ds + C2, u0(x),u1(x) ∈ WkP(Ω) ∩ W01,P(Ω), k≥ 1, 1 〈 p≤ ∞, then for any T 〉 0 the problem admits a unique solution u(x, t) ∈ W2,∞(0, T; WkP(Ω) ∩ W01,P(Ω)). Finally, the finite time blow-up of solutions and global Wkp solution of generalized IMBo equations are discussed.展开更多
基金supported by National Natural Science Foundation of China(10871055,10926149)Natural Science Foundation of Heilongjiang Province (A2007-02+2 种基金A200810)Science and Technology Foundation of Education Office of Heilongjiang Province(11541276)Foundational Science Founda-tion of Harbin Engineering University
文摘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 Wkp solutions. Then, we prove that, if f(s) ∈ Ck+1(R) is nondecreasing, f(0) = 0 and |f(u)| ≤ C1|u|∫0uf(s)ds + C2, u0(x),u1(x) ∈ WkP(Ω) ∩ W01,P(Ω), k≥ 1, 1 〈 p≤ ∞, then for any T 〉 0 the problem admits a unique solution u(x, t) ∈ W2,∞(0, T; WkP(Ω) ∩ W01,P(Ω)). Finally, the finite time blow-up of solutions and global Wkp solution of generalized IMBo equations are discussed.