Geodesic
Lyapunov function of semi-groups generated by a class of Sobolev type equations
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2008), pp. 3-9 
H. A. Mamikonyan. Lyapunov function of semi-groups generated by a class of Sobolev type equations. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2008), pp. 3-9. http://geodesic.mathdoc.fr/item/UZERU_2008_3_a0/
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     author = {H. A. Mamikonyan},
     title = {Lyapunov function of semi-groups generated by a class of {Sobolev} type equations},
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     year = {2008},
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     url = {http://geodesic.mathdoc.fr/item/UZERU_2008_3_a0/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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