ATTRACTORS OF SEMIGROUPS GENERATED BY AN EQUATION OF SOBOLEV TYPE

Abstract

In this paper the behavior of solutions of the following initial boundary value problem for a class of Sobolev type equations is considered. $$\begin{cases}  A\left(\dfrac{\partial u}{\partial t}\right)+Bu=0, \\ u|_{t=0}=u_0, \\u|_{\Sigma}=0, \end{cases}$$ where $A$ and $B$ are nonlinear operators of the following form: $$Au=-\displaystyle\sum^n_{i,j=1}\dfrac{\partial}{\partial x_i}a_j(x,u, \triangledown u), ~~Bu= -\displaystyle\sum^{n}_{i,j=1}\dfrac{\partial}{\partial x_i}b_j (x,u,\triangledown u).$$

It’s proved that when functions $a_j (x, u,\triangledown u)$ and $b_j (x, u,\triangledown u)$ specify some conditions, the semigroup generated by this equation has attractor, which is bounded in $^0 W^1_ 2 (\Omega)$.