Solvability of nonlinear elliptic boundary value problems
- UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
- Shalmali Bandyopadhyay (Creator)
- Institution
- The University of North Carolina at Greensboro (UNCG )
- Web Site: http://library.uncg.edu/
- Advisor
- Maya Chhetri
Abstract: This dissertation focuses on the study of steady states of reaction diffusion problems that are motivated by applications. In particular, we focus on elliptic boundary value problems where the nonlinear reaction may appear in the interior or on the boundary of a domain in the Euclidean space. First, we study linear elliptic problems with nonlinear reaction on the boundary. In this case, we establish the existence of maximal and minimal solutions for both monotone and non monotone cases. We then extend these results to the systems case. Next, we prove the existence, nonexistence, multiplicity and global bifurcation results of positive solutions of superlinear problems. To support our analytical results we numerically approximate solutions using finite difference methods including existence and stability analysis. Second, we study problems that are nonlinear inside the domain and linear on the boundary in the context of a model arising in mathematical ecology. To begin with we perform computational simulations for the problem in the one dimensional setting. Then, motivated by the bifurcation diagrams that are obtained, we prove several analytical results such as existence, uniqueness and nonexistence.
Solvability of nonlinear elliptic boundary value problems
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Created on 8/1/2023
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Additional Information
- Publication
- Dissertation
- Language: English
- Date: 2023
- Keywords
- Elliptic problem, Finite difference approximation, Mathematical ecology, Maximal and minimal solution, Nonlinear boundary condition, Superlinear problem
- Subjects
- Reaction-diffusion equations
- Differential equations, Elliptic
- Nonlinear boundary value problems
- Ecology $x Mathematical models