Analysis of solutions for hyperbolic type equations in variable exponent Sobolev spaces

Differential Differential equations are mathematical tools used to model the changes in physical systems over time and play a significant role in many fields (physics, chemistry, engineering, etc.). Evolution equations, on the other hand, are used to model time-dependent processes and help us understand how systems evolve over time. Variable exponent differential equations, as more complex forms of these equations, differ from classical differential equations by allowing the exponents to depend on space (𝑥 ) or time (𝑡), offering more flexible models. These models enable the study of more heterogeneous structures and are utilized in solving many modern engineering and scientific problems. In this context, the first chapter of the thesis discusses variable exponent equations that arise in applied sciences such as engineering and physics. The second chapter provides a detailed review of previous studies on variable exponent hyperbolic-type equations. In the third chapter, the fundamental definitions, lemmas, and theorems to be used in the thesis are presented. This chapter establishes the mathematical groundwork necessary for understanding subsequent theoretical results. In the fourth chapter, the existence and blow-up of solutions to variable exponent nonlinear Petrovsky equations are examined. The fifth chapter focuses on the blow-up of solutions to a variable exponent higher-order Kirchhoff-type equation with a logarithmic source term. In the sixth chapter, the global existence and blow-up of solutions for the variable exponent viscoelastic equation is addressed. The seventh chapter examines the global existence and decay of solutions for the variable exponent m-Laplacian equation with a logarithmic source term. Finally, the eighth chapter investigates the variable exponent Klein-Gordon system, presenting its blow-up and numerical results. Two-dimensional numerical examples are provided, and graphs are illustrated.