ATTRACTORS OF SEMIGROUPS GENERATED BY AN EQUATION OF SOBOLEV TYPE
DOI:
https://doi.org/10.46991/PYSU:A/2008.42.1.018Keywords:
Sobolev type equations, attractor, semigroupAbstract
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 $^0 W^1_ 2 (\Omega)$.
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