Qualitative analysis of the solutions of some partial differential equations

In this study, the behavior of the solutions of a hyperbolic type partial differential equation with a logarithmic source term is examined. In particular, it has been proven that when the initial conditions are chosen sufficiently large, the solution exhibits exponential growth over time. By using energy methods and appropriate test functions, lower and upper bounds of the solution in certain norms have been obtained. These results show that the logarithmic source plays a decisive role in the dynamics of the equation and, unlike classical polynomial source terms, may cause rapid growth in the solutions. The findings provide a deeper understanding of the mathematical structure of the related equation and open the door to generalizable results for similar types of equations. In the second chapter of the thesis, a literature review is presented. In the third chapter, some important definitions, theorems, lemmas, inequalities, and methods that will be used in the following parts of the thesis are given. In the fourth chapter, the main problem of the thesis is introduced, and the existence and blow-up of the solutions to this problem are studied using the potential well method. The potential well method is a powerful tool to analyze the existence and blow-up of solutions to partial differential equations. This method helps to predict the time- dependent behavior of the solutions by analyzing the energy and potential wells of the system. This analysis is especially important for understanding the stability and blow-up behavior of the solution. Since the existence of solutions to higher-order hyperbolic type equations is examined via the potential well method, this study aims to explore the importance and applications of evolution equations in a broader framework. In the fifth chapter of the thesis, the blow-up of the solutions of a higher-order hyperbolic type equation with a logarithmic source term is investigated through growth estimates.