Abstract: It is the purpose of this paper to consider some semilinear evolution systems of partial differential equations, which are found in physics, chemical reactions and biology. The existence, uniqueness and regularity of solutions for periodic boundary problems in global are proved for following degenerate parabolic and hyperbolic systems of higher order
∂u/∂t=Σ^m-1M(-1)^m+1A_m(x)∂^2mu/∂x²_m+Σ_τ=0^RB_τ(t)∂_2r+1u/∂x^2r+1+f(u),
where the matrices A(t), m=1,…, M are nonnegative definite and the matrices B(t), r=1.., M are symmetric. One can show that the nonlinear Schrödinger equation or system, which may be treated as a real degenerate parabolic system, the Sine-Gordon equation, Klein-Gordon equation, some simultaneous equations of Schrödinger equation and wave equation etc. may be changed into the above type.