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The existence and uniqueness of the global generalized solution and the global classical solution to the initial boundary value problem for a system of generalized IMBq equations $$\eqalign{u_{tt}-Du_{xx}-u_{xxtt} =\,f(v)_{xx},\quad v_{tt}{-}v_{xxtt}\,=\,g(v)_{xx}\cr u(0,t)\,=\,u(\ell,t)\,=\,0,\quad v(0,t)\,=\,v(\ell,t)\,{=}\,0 \cr u(x,0)\,=\,u_{0}(x),\quad u_{t}(x,0)\,=\,u_{1}(x),\quad v(x,0)\,=\,v_{0}(x), \enspace v_{t}(x,0)\,{=}\,v_{1}(x)}$$are proved. This paper also arrives at some sufficient conditions of blow up of the solution in finite time. Copyright © 2004 John Wiley & Sons, Ltd.

The existence and uniqueness of the global generalized solution and the global classical solution to the initial boundary value problem for a system of generalized IMBq equations $$\eqalign{u_{tt}-Du_{xx}-u_{xxtt} =\,f(v)_{xx},\quad v_{tt}{-}v_{xxtt}\,=\,g(v)_{xx}\cr u(0,t)\,=\,u(\ell,t)\,=\,0,\quad v(0,t)\,=\,v(\ell,t)\,{=}\,0 \cr u(x,0)\,=\,u_{0}(x),\quad u_{t}(x,0)\,=\,u_{1}(x),\quad v(x,0)\,=\,v_{0}(x), \enspace v_{t}(x,0)\,{=}\,v_{1}(x)}$$ are proved. This paper also arrives at some sufficient conditions of blow up of the solution in finite time. Copyright © 2004 John Wiley & Sons, Ltd.

The initial value problems and the first boundary problems for the quasilinear wave equation $$u_{tt} - \left[ {a_0 + na_1 \left( {u_x } \right)^{n - 1} } \right]u_{xx} - a_2 u_{xxtt} = 0$$ are considered, wherea0,a2 > 0 are constants,a1 is an arbitrary real number,n is a natural number. The existence and uniqueness of the classical solutions for the initial value problems and the first boundary problems of the equation (1) are proved by the Galerkin method.

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.

In this paper, the periodic boundary problem and the initial value problem for the nonlinear system of parabolic type ut=A(x, t)ux4+B(x, t)ux2+(g(u))x2+(grad h(u))x+f(u) are studied, where u(x, t)=(u1(x, t). …, uJ(x, t)is a J-dimensional unknown vector valued function, f(u)and g(u)are the J-dimensional vector valued function of u(x, t),h(u)is a scalar function of u, A(x, t)and B(x, t)are JJmatrices of functions. The existence, uniqueness and regularities of the generalized global solution and classical global solution of the problems are proved. When J= 1, h(u) =0, g(u)=au3, A=a1B=a2, where a1, a2, aare constants, the system is a generalized diffusion model equation in population problem.