LYAPUNOV FUNCTION OF SEMI-GROUPS GENERATED BY A CLASS OF SOBOLEV TYPE EQUATIONS

Abstract

In this paper Lyapunov function 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: $$Au=-\displaystyle\sum^n_{i,j=1}\dfrac{\partial}{\partial x_i}a_j(x,u, \triangledown u), ~~Bu= -\displaystyle\sum^{n}_{i,j=1}\dfrac{\partial}{\partial x_i}b_j (x,u,\triangledown u).$$

The existence of Lyapunov function on the attractor of the semi-group generated by this equation is proved. It is given the construction of attractor by the fixed points of that semi-group.