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 u(tt) - u(xx) - u(xxt) - u(xxtt) = f(u)(xx), x is an element of Omega , t > 0, u(x, 0) = u(0)(x), u(t)(x, 0) = u(1)(x), x is an element of Omega, u(0, t) = u(1, t) = 0, t >= 0, where Omega = (0,1). First, we obtain the existence of local W-k,W-p solutions. Then, we prove that, if f(s) is an element of Ck+1(R) is nondecreasing, f(0) = 0 and vertical bar f(u)vertical bar <= C-1 vertical bar u vertical bar integral(u)(0) f(s)ds + C-2,C- U-0(x), u(1)(x) is an element of W-k,W-p(Omega) boolean AND W-0(1,p)(Omega), k >= 1, 1 < p <= infinity, then for any T > 0 the problem adrnits a unique solution u(x, t) is an element of W-2,W-infinity(0,T;W-k,W-p(Omega) boolean AND W-0(1,p)(Omega)). Finally, the finite time blow-up of solutions and global W-k,W-p solution of generalized IMBq equations are discussed.