Stability of the solutions for a some partial differential equations with nakao's inequality

Partial differential equations are of indispensable importance, especially in fields related to science and engineering. In studies on partial differential equations since the 19th century, on the one hand, the relationship of partial differential equations with other branches of mathematics was investigated, and on the other hand, their connection with engineering and other applied sciences was tried to be revealed. For this reason, studying equations of this type not only provides theoretical solutions to problems, but also provides mathematical modeling for many engineering, chemistry, physics, etc. It also paves the way for the study of problems arising in these fields. Obtaining mathematical models to approach such problems will benefit the advancement of science in theoretical terms. The models put forward to mathematically express, understand and interpret the events in nature are generally based on non-linear partial differential equations. The explicit solution of nonlinear partial differential equations may not always be found. In order to find an approximate solution to non-linear partial differential equations whose solutions cannot be obtained at the moment or to have an idea about the behavior of the solution, it has become necessary to limit the equation with some conditions. Even though these problems cannot be fully solved, investigating under what conditions and at what time a solution exists constitutes an important field of study in mathematics. In the first part of this thesis, partial differential equations and their types, engineering, medicine, physics, chemistry, etc. are discussed. Evolution equations that we encounter in sciences are mentioned. In the second chapter, previous studies on hyperbolic type p-Laplacian and p-Biharmonic equations are discussed. In the third chapter, basic definitions, lemmas, theorems and inequalities that will be used throughout the thesis are included as preliminary information. In the fourth chapter, the global existence of solutions of the nonlinear hyperbolic type p-Biharmonic equation and the energy reduction of the solution are studied. In the fifth chapter, results and recommendations are given.