Solutions of evolution equations with logarithmic source term

In the first part of this thesis, the definition and historical process of the differential equation that emerged in applied sciences such as science and engineering are briefly mentioned. It is also mentioned that the developments in differential equations provide the emergence of new fields in mathematics and the advances in many fields in mathematics offer new perspectives for differential equations. In the second part, the purpose of the thesis is stated and the areas where the equations with logarithmic source terms are used in general are given. Then, basic studies on problems with logarithmic source terms are given. Here, first the problem that Birula and Mycielski worked on in 1976 is given, and then the work of Cazenave and Haraux in 1980 is given. After the basic problems are given, many studies carried out until today are given in this section. The third part consists of three parts. Firstly, the definition of Lebesgue space is given. In the second part, the definition of Sobolev space and the Sobolev Embedding theorem are given. In the third part, the inequalities and lemmas required throughout the thesis are included. In the fourth chapter, the Klein-Gordon equation with one-dimensional logarithmic source terms is studied. The local existence of the solutions of the logarithmic nonlinear Klein-Gordon problem is discussed. Then, the proof of the weak solution of the problem is given, and the proof consists of three steps: Approach Finding the Solution, Predictive Estimation and Passing the Limit. Here, Galerkin method, Logarithmic Sobolev Inequality and Compactness theorems are used to prove the weak solution. In addition, the Aubin Lions lemma is used in the step of Passing the Limit, which is the third step of the proof. The fifth chapter is the original part of the thesis, and the higher order Klein-Gordon equation with logarithmic source terms has been studied. First, the energy equation of our problem is found and shown, and necessary lemmas and definitions are given. Then, the theorems about the Global Existence and Exponential Decay of our problem are given and the lemmas to be used in the proof of the theorems are given together with their proofs. Finally, the Blow up of the Solution of our problem has been attempted. The blow up of our problem has been shown using the concavity method. In the sixth chapter, the conclusion and suggestions of the thesis are given.