The stabilization of the solutions of certain parabolic equations and systems

This paper concerns the investigation of the stabilization of solutions of the Cauchy problem for a system of equations of the form $\frac{\partial u}{\partial t}=\Delta u+F_1(u,v)$. It is proved that under certain assumptions the behavior of solutions as $t\to\infty$ is determined by mutual arrangement of the set of initial conditions $\{(u,v):u=f_1(x),\ v=f_2(x),\ x\in R^n\}$ and the trajectories of the system of ordinary differential equations $\frac{du}{dt}=F_1(u,v)$. The question of stabilization of the solutions of a single quasilinear parabolic equation is also considered.