The Stabilization Theorems For Parabolic Systems With Analytic Nonlinearity And Ljapunov Functional

• Main Author: Mikhail Vishnevskii
• Format: Article
• Language: English
• Published: Sociedade Brasileira de Matemática 2004-07-01
• Series: Boletim da Sociedade Paranaense de Matemática
• Subjects: parabolic system; stabilization; Ljapunov Functional.
• Online Access: http://www.periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/7496/4315

Let u(x,t) denote the solution of a boundary value problem forparabolic 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 ofu(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 con-sists of stationary solution and connected orbits. The °ow on global attractor is agradient-like °ow. The similar result obtained also for the Canh - Hilliard equation.