{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/101465"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/101465","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Fourth order spectral theory and diffusion-driven instability","abstract":"In Part I, we study the spectrum of the one-dimensional vibrating free rod equation u′′′′ − τ u′′ = μu under tension (τ > 0) or compression (τ < 0). The eigenvalues μ as functions of the tension/compression parameter τ exhibit three distinct types of behavior. In particular, eigenvalue branches in the lower half-plane exhibit a cascading pattern of barely-avoided crossings. We provide a complete description of the eigenfunctions and eigenvalues by implicitly parameterizing the eigenvalue curves. We also establish properties of the eigenvalue curves such as monotonicity, crossings, asymptotic growth, cascading and phantom spectral lines. In Part II, we analyze diffusion-driven (Turing) instability of a reaction-diffusion system. The innovation is that we replace the traditional Laplacian diffusion operator with a combination of the fourth order bi-Laplacian operator and the second order Laplacian. We find new phenomena when the fourth order and second order terms are competing, meaning one of them stabilizes the system whereas the other destabilizes it. We characterize Turing space in terms of parameter values in the system, and also find criteria for instability in terms of the domain size and tension parameter.","abstract_html":"In Part I, we study the spectrum of the one-dimensional vibrating free rod equation u′′′′ − τ u′′ = μu under tension (τ &gt; 0) or compression (τ &lt; 0). The eigenvalues μ as functions of the tension/compression parameter τ exhibit three distinct types of behavior. In particular, eigenvalue branches in the lower half-plane exhibit a cascading pattern of barely-avoided crossings. We provide a complete description of the eigenfunctions and eigenvalues by implicitly parameterizing the eigenvalue curves. We also establish properties of the eigenvalue curves such as monotonicity, crossings, asymptotic growth, cascading and phantom spectral lines. In Part II, we analyze diffusion-driven (Turing) instability of a reaction-diffusion system. The innovation is that we replace the traditional Laplacian diffusion operator with a combination of the fourth order bi-Laplacian operator and the second order Laplacian. We find new phenomena when the fourth order and second order terms are competing, meaning one of them stabilizes the system whereas the other destabilizes it. We characterize Turing space in terms of parameter values in the system, and also find criteria for instability in terms of the domain size and tension parameter.","abstract_has_math":false,"creators":["Chung, Jooyeon"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Laugesen, Richard S.","DeVille, Lee","Bronski, Jared","Rapti, Zoi"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-09-27T16:17:18Z","date_published":"2018-09-27T16:17:18Z","updated_at":"2026-07-22T22:24:40Z","subjects":["Bi-Laplacian","cascading","avoided crossings","Turing diffusion-driven instability","reaction-diffusion system","fourth order"],"languages":["en"],"rights":["Copyright 2018 Jooyeon Chung"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/101465","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Laugesen, Richard S.","DeVille, Lee","Bronski, Jared","Rapti, Zoi"]},{"key":"dc:creator","label":"Author","values":["Chung, Jooyeon"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-09-27T16:17:18Z","2018-05-29","2018-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Bi-Laplacian","cascading","avoided crossings","Turing diffusion-driven instability","reaction-diffusion system","fourth order"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2018 Jooyeon Chung"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/101465"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In Part I, we study the spectrum of the one-dimensional vibrating free rod equation u′′′′ − τ u′′ = μu under tension (τ > 0) or compression (τ < 0). The eigenvalues μ as functions of the tension/compression parameter τ exhibit three distinct types of behavior. In particular, eigenvalue branches in the lower half-plane exhibit a cascading pattern of barely-avoided crossings. We provide a complete description of the eigenfunctions and eigenvalues by implicitly parameterizing the eigenvalue curves. We also establish properties of the eigenvalue curves such as monotonicity, crossings, asymptotic growth, cascading and phantom spectral lines. In Part II, we analyze diffusion-driven (Turing) instability of a reaction-diffusion system. The innovation is that we replace the traditional Laplacian diffusion operator with a combination of the fourth order bi-Laplacian operator and the second order Laplacian. We find new phenomena when the fourth order and second order terms are competing, meaning one of them stabilizes the system whereas the other destabilizes it. We characterize Turing space in terms of parameter values in the system, and also find criteria for instability in terms of the domain size and tension parameter.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-09-27 without embargo terms","The student, Jooyeon Chung, accepted the attached license on 2018-05-23 at 08:22.","The student, Jooyeon Chung, submitted this Dissertation for approval on 2018-05-23 at 09:38.","This Dissertation was approved for publication on 2018-05-29 at 11:40.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12572 on 2018-09-27 at 10:44:03","Made available in DSpace on 2018-09-27T16:17:18Z (GMT). No. of bitstreams: 4 CHUNG-DISSERTATION-2018.pdf: 1824398 bytes, checksum: c78278c5058fe49005c0f0ac86f07c44 (MD5) Thesis(revision).tex: 205502 bytes, checksum: 38a33f0ce16ff428e440d27f22869aaa (MD5) LICENSE.txt: 4210 bytes, checksum: 4f7860799c261a7657aff95945001c56 (MD5) PROQUEST_LICENSE.txt: 4556 bytes, checksum: edc804f7387d4ada8e396df37c350ef5 (MD5) Previous issue date: 2018-05-29"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Fourth order spectral theory and diffusion-driven instability"]}]}],"canonical_facts":{"dc:contributor":["Laugesen, Richard S.","DeVille, Lee","Bronski, Jared","Rapti, Zoi"],"dc:creator":["Chung, Jooyeon"],"dc:date":["2018-09-27T16:17:18Z","2018-05-29","2018-08"],"dc:description":["In Part I, we study the spectrum of the one-dimensional vibrating free rod equation u′′′′ − τ u′′ = μu under tension (τ > 0) or compression (τ < 0). The eigenvalues μ as functions of the tension/compression parameter τ exhibit three distinct types of behavior. In particular, eigenvalue branches in the lower half-plane exhibit a cascading pattern of barely-avoided crossings. We provide a complete description of the eigenfunctions and eigenvalues by implicitly parameterizing the eigenvalue curves. We also establish properties of the eigenvalue curves such as monotonicity, crossings, asymptotic growth, cascading and phantom spectral lines. In Part II, we analyze diffusion-driven (Turing) instability of a reaction-diffusion system. The innovation is that we replace the traditional Laplacian diffusion operator with a combination of the fourth order bi-Laplacian operator and the second order Laplacian. We find new phenomena when the fourth order and second order terms are competing, meaning one of them stabilizes the system whereas the other destabilizes it. We characterize Turing space in terms of parameter values in the system, and also find criteria for instability in terms of the domain size and tension parameter.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-09-27 without embargo terms","The student, Jooyeon Chung, accepted the attached license on 2018-05-23 at 08:22.","The student, Jooyeon Chung, submitted this Dissertation for approval on 2018-05-23 at 09:38.","This Dissertation was approved for publication on 2018-05-29 at 11:40.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12572 on 2018-09-27 at 10:44:03","Made available in DSpace on 2018-09-27T16:17:18Z (GMT). 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