Lyapunov function of semi-groups generated by a class of Sobolev type equations

H. A. Mamikonyan

Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2008), pp. 3-9

Résumé

In this paper Lyapunov function the following initial boundary value problem for a class of Sobolev type equations is considered $$\left\{\begin{array}{l} A\left(\frac{\partial u}{\partial t}\right)+Bu=0,\\ u\Big|_{t=0}=u_0,\\ u\Big|_{\Sigma}=0, \end{array}\right.$$ where $A$ and $B$ are nonlinear operators of the following form: $$Au=-\sum_{i=1}^n\frac{\partial}{\partial x_i}a_i(x,\nabla u), \quad Bu=-\sum_{i=1}^n\frac{\partial}{\partial x_i}b_i(x,\nabla u).$$ 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.