Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Qualitative Properties of Solutions of Some Keller-Segel Type Systems(01. Izmir Institute of Technology, 2022) Batal, Ahmet; Özsarı, Türker; Batal, Ahmet; Özsarı, Türker; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThe main objective of this thesis is to summarize results related with solutions of some Keller - Segel type systems, which model chemotaxis. This work surveys mathematical studies starting with the work that first presented these systems in 1970. This study emphasizes the local and global existence of solutions of Keller - Segel type systems, in particular the boundedness and blow-up of solutions.Master Thesis Semigroup Theory and Some Applications(Izmir Institute of Technology, 2020) Batal, Ahmet; Batal, Ahmet; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technologyn the present thesis, we consider the evolution equation (Cauchy problem) which is the basis for our study. We show how various linear partial differential equations can be transformed into the Cauchy problem form. Solving the Cauchy problem is equivalent to find a family of evolution operators T(t) which sends the initial state of the system to the solution state at a later time t. It turns out that this family of operators T(t) must satisfy some properties which we call semigroup properties. We state the Hille-Yosida and Lumer-Phillips theorems to characterize contraction semigroups. Moreover, we apply these theorems to the heat and wave equations as examples. We also consider strongly continuous operator groups and Stone's theorem. Finally, we give some essential conditions to obtain wellposed evaluation equation and introduce an inhomogeneous Cauchy problem.