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JOURNALS // Proceedings of the Yerevan State University, series Physical and Mathematical Sciences // Archive

Proceedings of the YSU, Physical and Mathematical Sciences, 2008 Issue 1, Pages 18–23 (Mi uzeru283)

This article is cited in 1 paper

Mathematics

Attractors of semigroups generated by an equation of Sobolev type

H. A. Mamikonyan

Chair of the theory of optimal control and approximate methods YSU, Armenia

Abstract: In this paper the behavior of solutions of the following initial boundary value problem for a class of sobolev type equations is considered.
$$A\left(\frac{\partial u}{\partial t}\right)+Bu=0,~u\Big|_{t=0}=u_0,~u\Big|_{\Sigma}=0 $$
where $A$ and $B$ are nonlinear operators of the following form:
$$Au=-\sum_{i,j=1}^n\frac{\partial}{\partial x_i}a_j(x, u, \nabla u),~~Bu=-\sum_{i,j=1}^n\frac{\partial}{\partial x_i}b_j(x, u, \nabla u)$$
It’s proved that when functions $a_j(x, u, \nabla u)$ and $b_j(x, u, \nabla u)$ specify some conditions, the semigroup generated by this equation has attractor $\{S_t,~t \geq0\}$,, which is bounded in $W_2^1(\Omega)$.

UDC: 517.9

Received: 19.04.2007
Accepted: 30.08.2007



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