Abstract: The existence and uniqueness of weak solutions for the Cauchy problem of an evolutionary Laplacian systems were investigated. The exsitence was obtained by virtue of the parabolic regularized method. Owing to the difficulty of the comparison between the two solutions of the system consided, wecombined the enery method with the decay method to reduce our problem to the scalar case, so as to establish the uniqueness of solution.

Key words: nonlinear diffusion system, existence, uniqueness, absorption, p-Laplacian system