Qualitative analysis of the solutions of a high-order partial differential equation

The higher-order partial differential equations play a significant role in the development of mathematical physics and engineering. These equations are prevalent in various fields such as materials science, fluid dynamics, and wave propagation. The study of these equations began in the 18th century with the works of the Bernoulli brothers and Euler on vibrating strings and beams. Subsequent mathematicians have made significant contributions to the applications of these equations in fluid mechanics and wave theory. Modern research focuses on the applications of high-order partial differential equations in areas such as viscoelastic and smart materials. Understanding and solving these equations is critically important both from a theoretical mathematics perspective and for engineering applications. The primary aim of this thesis is to examine the existence and blow-up of solutions for a high-order partial differential equation with appropriate initial and boundary conditions. Previous studies have investigated similar problems, focusing on the global existence, uniqueness, and stability of solutions. This work addresses a high-order partial differential equation and generalizes blow-up conditions using the method of generalized concavity. In the first chapter of the thesis, the definition of high-order partial differential equations emerging in applied sciences and engineering fields is briefly introduced, along with the concepts of global existence and blow-up of solutions, illustrated using a simple ordinary differential equation example. The second chapter reviews the historical development of research conducted on high-order partial differential equations up to the present day. The third chapter provides information on fundamental definitions, lemmas, theorems, necessary function spaces, and inequalities that will be used throughout the thesis. The fourth chapter forms the core of the thesis, where the global existence of weak solutions to the given initial-boundary value problem is investigated, along with the conditions leading to blow-up. The fifth chapter concludes the thesis with results and offers suggestions for future studies.