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- Type of Document: Ph.D. Dissertation
- Language: Farsi
- Document No: 54733 (02)
- University: Sharif University of Technology
- Department: Mathematical Sciences
- Advisor(s): Hesaaraki, Mahmoud; Fotouhi Firoozabad, Morteza
- Abstract:
- Singular differential equations have a wide range of applications. Hardy singularities, which are connected to inequalities of the same name and have various extensions, are the most well-known singularities. The application of Hardy inequalities in quantum physics and also in the linearization of reaction-diffusion equations in thermodynamics and combustion theory motivates researchers to examine them. Singularities on a domain's boundary are another well-known type of singularity. In the study of fluid mechanics and pseudoplastic flows, these singularities emerge.Differential equations with coefficients or functions that are simply functions belonging to $ L^1 $, or bounded Radon measures, are also intriguing and hard from a mathematics standpoint. They're used in mechanics, electromagnetism, and control theory, among other fields. To investigate the uniqueness of solutions to such issues, it is required to first create new ideas of solutions, which is frequently followed by an examination of the equivalence of these various notions of solutions.We will investigate the qualitative features of solutions of singular partial differential equations using rough data, such as $L^1$ data, bounded Radon measure, or, in general, data that is a distribution, in this thesis. Except for the first two chapters, which deal with the introduction to the subject, each chapter of the dissertation will deal with distinct model challenges relating to a range of singularities and irregular data, or both at the same time. The following is a summary of the contents of chapters $ 3 $, $ 4 $, and $ 5 $.In chapter $ 3 $, we'll investigate the boundary value problems with nonlinear singularities of the Lazer and McKenna types and irregular data. To be more specific, we'll look at the existence and uniqueness of positive solutions to a double singularity problem.The crucial threshold power of $ p $ for the solvability of an elliptic semilinear problem with a Hardy potential will be found in Chapter $4$. We'll get some results based on this essential threshold power.The effect of the Hardy potential on the existence or nonexistence of solutions to a stationary fractional problem, as well as its evolutionary counterpart, which is involved with a nonlinear Laser and McKnana singularity term, will be thoroughly studied in Chapter $5$
- Keywords:
- Parabolic Equations ; Semilinear Elliptic Equations ; Elliptic Differential Equations ; Singular Nonlinearity ; Nonregular Data ; Hardy Potential
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