Attractors of semigroups generated by an equation of Sobolev type

H. A. Mamikonyan

Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2008), pp. 18-23

Résumé

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 $W_2^1(\Omega)$.