Stability of the evolutionary linear Sobolev type equation

Sobolev type equations are a part of extensive area of non-classical equations of mathematical physics. These are equations that are not solved respective to the highest derivative with respect to time. Research of different problems for equations of the given type nowadays are very relevant, as such equations appear during modeling of different processes in natural and engineering sciences. In this article, stability of stationary solution of an evolutionary equation, which appeared in the filter theory and which describes development of form of the filterable liquid’s free surface, is researched.
For this equation, an initial boundary-value problem in limited area is considered. The article consists of an introduction, a list of references and two parts. In the first part, general concepts and theory assertions concerning $p$-sectorial operators are given. After that, reduction of the considered problem to the Cauchy problem for a Sobolev type abstract linear equation, by the means of selecting the corresponding Banach spaces and linear operators, is carried out. Then the phase space of our problem is described.
In the second part, general concepts of the stability theory such as flow, stationary point of the flow, and Lyapunov functional, are given. Theorems of stability and asymptotical stability of a stationary point of the flow are given. In this article, the method of Lyapunov functional, modified for the case of complete normalized spaces, is used. It should be noted, that modification of the method lies in transition from incomplete normalized spaces to complete normalized spaces. As a result, the uniformity of stability and asymptotic stability is lost, but the class of problems being solved gets considerably expanded. The main result of the article are conditions formulated as a theorem of stability and asymptotic stability of zero solution of the considered problem.