The Global Solution For A Coupled Nolinear Partial Differential Equations

In this paper, we studied an abstract coupled nonlinear initial problem, that is,the following coupled equations nder the initial conditionsWe give the proof of the existence of the global weak solution ,whereΩ=(0, l),0 <α≤1, t∈[0, T ](0 < T<∞),A is a positive self-adjoint operater defined in H ,functions M and N satisfy certain conditions.In particular, studies are as follows:1. We made simple sum-up and comment on the developing and actuality study on partial differential equations relevant with this paper;2. We give some important definitions and lemmas, and the rignt part of the text symbol is explained;3. Using Galerkin method to prove the existence of a global solution for the problems(1)-(4) in a Hilbert space framework;4. By improving the smoothness of the initial data (u 0 , v0 ), (u 1 , v1 ) and functions f , g ,we get a more smoothly global solution.