Behavior of solutions of variable coefficient evoluation equations

Partial differential equations are essential tools for understanding phenomena in the universe. For instance, many events such as the vibration of a violin string, sound waves, or heat conduction in a rod can be modeled using partial differential equations. When one of the independent variables in a partial differential equation is time (ttt), such equations are called evolution equations. Evolution equations constitute a class of differential equations used to model physical systems that change over time. These equations are of great importance not only in mathematics but also in other disciplines such as physics, mechanics, and materials science. Moreover, in real-world scenarios, many physical processes cannot be accurately represented by constant-coefficient mathematical models due to the fact that environmental conditions often vary with time or space. In such cases, variable-coefficient differential equations provide more realistic modeling. For example, in areas like heat conduction, wave propagation, diffusion, and fluid dynamics, properties of the medium—such as temperature, density, or elasticity—may vary in time or space. Analyzing such systems inevitably requires the use of variable-coefficient equations. In this context, the first part of the thesis briefly introduces the theory of differential equations, the evolution equations arising in applied sciences such as engineering and physics, and the behavior of their solutions. In the second chapter, the physical interpretations of the equations considered in this thesis are discussed, along with a historical overview of the related studies conducted up to the present day. In the third chapter, the fundamental definitions, lemmas, theorems, inequalities, and equalities that will be used throughout the thesis are introduced. In addition, several lemmas regarding the existence, decay, and blow-up of solutions are provided. In the fourth chapter, the blow-up behavior of solutions to a variable coefficient strongly damped wave equation is investigated. In the fifth chapter, the existence, decay, and blow-up of solutions for a weighted m-biharmonic equation with nonlinear damping and source terms are studied. In the sixth chapter, the upper and lower bounds for the global existence and blow-up time of solutions to a higher-order reaction-diffusion equation with singular potential are examined. In the seventh chapter, the decay of solutions to a variable coefficient Petrovsky equation with delay term is analyzed.