The behavior of the generalized solution of the initial-boundary value problem for the nonlinear parabolic equation

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Within the framework of this work, we study the behavior of the generalized solution (or the so-called energy solution) of an interesting initial-boundary value problem (namely, the Cauchy-Dirichlet problem is considered) for nonlinear parabolic equation. The research is carried out in a cylindrical area. A structural condition is imposed on the parameters of the equation corresponding to the slow diffusion process. So, in the article we are dealing with the distribution of the substance concentration in space and time, taking into account the initial and boundary conditions. This process has a practical aspect and is used in physics and engineering, for example, to study the diffusion of matter in environments with variable concentration or chemical influence. Solving such problems allows obtaining important data on the evolution of the system and predicting its behavior in various conditions. In the work, as a result of our research, several integral ratios, various estimates and inequalities were established, which lead to the need to analyze the behavior of the differential system, which in turn makes it possible to establish the presence of the solution localization property. So, relying on the well-known results regarding the behavior of the solution of the resulting differential system, it is possible to find a condition that guarantees the localization of the solution carrier for the Cauchy-Dirichlet problem under study. The main result of the work is a theorem that is proved for an arbitrary finite initial function and under the condition of fulfilling a certain restriction on the limit mode. The article has a fairly standard structure and, in addition to annotations and literature, contains the following structural elements: introduction; Formulation of the problem; basic definitions; formulation of the main result; auxiliary inequalities for proving the main result; auxiliary statements for proving the theorem; proving the main result; conclusions.