The development of modern science and technology requires accurate and comprehensive mathematical modeling of physical and hydrodynamic processes of continuous media. For example, experimental studies show that the presence of even small amounts of polymeric substances in a viscous medium can significantly affect to motion of fluid. A comparison of the physical characteristics of water and weak aqueous solutions of polymers showed that at almost the same density and viscosity, these fluids sharply differ in their relaxation properties. In recent years, the investigation of solvability and qualitative properties such as blow up and extinction of a solution in a finite time, large time behavior of a solution of the problems in various statements for pseudo-parabolic equations has been rapidly developing. This is confirmed by publications have been publishing in world ranking scientific journals. This paper is devoted to the study of a mathematical model of the one-dimensional motion of a Kelvin-Voigt fluid with complex rheological properties, which describes by a nonlinear pseudo-parabolic equation (Sobolev type equation) with p-Laplacian. The study of pseudo-parabolic equations was first started by Sobolev in [1] for the linear version. In this work the asymptotic behavior of solutions of pseudo-parabolic equations at large times, in clearly, the properties of exponential and power-law decay of a solution are proved.