This article is cited in 1 paper

Solution of parabolic equations by means of Lyapunov functionals

M. M. Lavrent'ev (Jn.)

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We propose a new approach to defining the notion of a solution to linear and nonlinear parabolic equations. The main idea consists in studying connections between solutions to dynamic problems in the variational shape and the properties of the corresponding Lyapunov functionals which are strictly decreasing along the trajectories of the above-mentioned dynamic equations except for the equilibrium points. It turns out that the families of Lyapunov functionals constructed by T. I. Zelenyak enable us to propose a new approach to defining solutions to both linear and nonlinear parabolic problems. All results are given in the case of smooth solutions. Note that the Lyapunov functionals can be used for studying solutions with unbounded gradients.

Keywords: parabolic equation, Lyapunov functional.

UDC: 517.95

Received: 19.04.2005

 English version: Siberian Mathematical Journal, 2005, 46:5, 867–878

Bibliographic databases:

• 
• 
• 
•