Stability of solutions of some partial differential equations with the komornik inequality

Research on partial differential equations in the 19th and 20th centuries aimed to reveal the relationship of these equations with other branches of mathematics and applied sciences such as engineering. Partial differential equations not only provide solutions to theoretical problems, but also provide the basis for mathematical modeling of events encountered in fields such as engineering, chemistry and physics. Events in nature appear as mathematical problems, and creating mathematical models to solve these problems contributes to the theoretical development of science. These models are often based on nonlinear partial differential equations, and such equations may not always have an explicit solution. In nonlinear partial differential equations for which no solution can be found, the equations must be limited to certain conditions in order to find approximate solutions or understand the behavior of the solution. In this context, investigating the conditions under which equations have solutions has become an important field of study in mathematics. In the first part of this thesis, the definition of differential equations and their methods of obtaining them; The definition of partial differential equations, the studies on partial differential equations in the 19th and 20th centuries, the relationship of partial differential equations with other applied sciences, the solution of a given problem for partial differential equations, and finally the evolution equations are summarized. In the second part of this thesis, studies on higher order parabolic Kirchhoff type equations and the historical development of these studies are discussed. In the third chapter, basic definitions, lemmas, theorems and inequalities that will be used throughout the thesis are presented. In the fourth chapter, the global existence and energy decay of solutions of a higher order parabolic equation are examined. These studies demonstrate the applicability of mathematical theory and its contributions to solving practical problems in engineering and science. In particular, such studies carried out to understand the solution behavior of nonlinear partial differential equations are of great theoretical and practical importance. Keywords: Kirchhoff type reaction-diffusion equation, Global existence, Energy decay, Komornik's inequality