Geodesic
The stabilization of the solutions of certain parabolic equations and systems
Matematičeskie zametki, Tome 3 (1968) no. 1, pp. 85-92

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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. 
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     author = {M. I. Freidlin},
     title = {The stabilization of the solutions of certain parabolic equations and systems},
     journal = {Matemati\v{c}eskie zametki},
     pages = {85--92},
     publisher = {mathdoc},
     volume = {3},
     number = {1},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1968_3_1_a10/}
}
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M. I. Freidlin. The stabilization of the solutions of certain parabolic equations and systems. Matematičeskie zametki, Tome 3 (1968) no. 1, pp. 85-92. http://geodesic.mathdoc.fr/item/MZM_1968_3_1_a10/