In this work, we investigate a system of doubly nonlinear degenerate parabolic equations. We consider it with zero Dirichlet boundary conditions. The peculiarity of the problem is that nonlinear source and nonlinear absorption terms are involved in the given problem, which complicates the learning process. We show global solvability and blow-up of the solution, which are important qualitative properties of the solution of problem, through self-similar analysis, comparison principle and nonlinear splitting methods. It is shown that nonlinear diffusion, source and absorption terms affect the qualitative properties of the solution. The obtained results are useful in studying the qualitative properties of the solutions of similar problems.