Global existence and blow-up of solutions for generalized Pochhammer-Chree equations

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)∈Ck+1(R)f(s)∈Ck+1(R) is nondecreasing, f  (0) = 0 and |f(u)|≤C1|u|∫0uf(s)ds+C2,u0(x),u1(x)∈Wk,p(Ω)∩W01,p(Ω), k≥1, 1 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.