Initial value problem for a class of nonlinear pseudo-hyperbolic equations

The existence of global weak solutions to the periodic boundary problem or the initial value problem for the nonlinear Pseudo-hyperbolic equation $$u_{tt} - \left[ {a_1 + a_2 \left( {u_x } \right)^{2m} } \right]u_{xx} - a_3 u_{xxt} = f\left( {x,t,u,u_x } \right)$$ is proved by the method of the vanishing of the additional diffusion terms, Leray-Schauder's fixed-point argument and Sobolev's estimates, wherem≥1 is a natural number andai>0 (i=1, 2, 3) are constants.