Let u(x; t) denote the solution of a boundary value problem for parabolic system . We say the solution u(x; t) stabilizes as t tends to plus infinity (minus infinity) if the set of all partial limits as t tends to plus infinity (menus infinity) of the solution u(x; t) consists of a single stationary solution. In this communication we consider the nonlinear parabolic system with analytic dependence of u(x; t) and gradient of u(x; t) on the space variable and with Liapunov functional. It is shown that any solution of the problem uniformly bounded for positive t (or fornegative t) stabilizes. In particular the global attractor of this kind of problem consists of stationary solution and connected orbits. The flow on global attractor is a gradient-like flow. The similar result obtained also for the Canh - Hilliard equation.
@article{7496,
title = {The Stabilization Theorems For Parabolic Systems With Analytic Nonlinearity And Ljapunov Functional - doi: 10.5269/bspm.v22i1.7496},
journal = {Boletim da Sociedade Paranaense de Matem\'atica},
volume = {23},
year = {2009},
doi = {10.5269/bspm.v22i1.7496},
language = {EN},
url = {http://dml.mathdoc.fr/item/7496}
}
Vishnevskii, Mikhail. The Stabilization Theorems For Parabolic Systems With Analytic Nonlinearity And Ljapunov Functional - doi: 10.5269/bspm.v22i1.7496. Boletim da Sociedade Paranaense de Matemática, Tome 23 (2009) . doi : 10.5269/bspm.v22i1.7496. http://gdmltest.u-ga.fr/item/7496/