The system ut = uxx - (uvx)x, vt = u - Av is considered where A is a non-negative, self-adjoint operator which commutes with the Laplacian. The operator is considered to have eigenvalues lambda n = nrholambda1, and the system is considered on [0,1] x [0,T] with homogeneous Neumann boundary conditions. The operators which lead to global solutions and those that lead to solutions which blow up in finite time are considered as a function of rho, using an application of the methods of Hillen and Potapov [Math. Methods Appl. Sci., 27 (2004), pp. 1783-1801] to analyze the global case and those of Halverson, Levine, and Renclawowicz [Siam J. Appl. Math., 65 (2004), pp. 336--360; 66 (2005), pp. 361--364] to analyze the finite time blowup case. Some numerical results are provided to back up the analysis. Some questions and directions for future study are posed.
","abstract_html":"<p>The system ut = uxx - (uvx)x, vt = u - Av is considered where A is a non-negative, self-adjoint operator which commutes with the Laplacian. The operator is considered to have eigenvalues lambda n = nrholambda1, and the system is considered on [0,1] x [0,T] with homogeneous Neumann boundary conditions. The operators which lead to global solutions and those that lead to solutions which blow up in finite time are considered as a function of rho, using an application of the methods of Hillen and Potapov [Math. Methods Appl. Sci., 27 (2004), pp. 1783-1801] to analyze the global case and those of Halverson, Levine, and Renclawowicz [Siam J. Appl. Math., 65 (2004), pp. 336--360; 66 (2005), pp. 361--364] to analyze the finite time blowup case. Some numerical results are provided to back up the analysis. Some questions and directions for future study are posed.</p>","abstract_has_math":false,"creators":["Halverson, Matthew"],"institution":null,"degree_name":"Doctor of Philosophy","degree_level":"dissertation","degree_discipline":null,"degree_department":"Department of Mathematics","school":null,"contributors":[],"advisors":["Howard Levine","Elgin Johnston","Leslie Hogben"],"committee_chairs":[],"committee_members":[],"year":2008,"date_issued":"2008-01-01","date_published":"2008-01-01","updated_at":"2026-07-24T02:39:53Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.31274/rtd-180813-16867"],"render_values":[{"text":"https://doi.org/10.31274/rtd-180813-16867","href":"https://doi.org/10.31274/rtd-180813-16867","code":true}]},{"key":"dc:identifier","label":"Identifier","values":["archive/lib.dr.iastate.edu/rtd/15654/"],"render_values":[{"text":"archive/lib.dr.iastate.edu/rtd/15654/","href":null,"code":true}]}]},"links":{"outbound_url":"https://dr.lib.iastate.edu/handle/20.500.12876/69308","outbound_label":"Repository record","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Howard Levine","Elgin Johnston","Leslie Hogben"]},{"key":"dc:contributor.department","label":"Department","values":["Department of Mathematics"]},{"key":"dc:creator","label":"Author","values":["Halverson, Matthew"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-08-22T20:26:39.000"]},{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2020-06-30T07:46:12Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2020-06-30T07:46:12Z"]},{"key":"dc:date.issued","label":"Date","values":["2008-01-01"]},{"key":"dc:type","label":"Dc Type","values":["dissertation"]},{"key":"thesis:degree_level","label":"Degree Level","values":["dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["archive/lib.dr.iastate.edu/rtd/15654/"]},{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.31274/rtd-180813-16867"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://dr.lib.iastate.edu/handle/20.500.12876/69308"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The system ut = uxx - (uvx)x, vt = u - Av is considered where A is a non-negative, self-adjoint operator which commutes with the Laplacian. The operator is considered to have eigenvalues lambda n = nrholambda1, and the system is considered on [0,1] x [0,T] with homogeneous Neumann boundary conditions. The operators which lead to global solutions and those that lead to solutions which blow up in finite time are considered as a function of rho, using an application of the methods of Hillen and Potapov [Math. Methods Appl. Sci., 27 (2004), pp. 1783-1801] to analyze the global case and those of Halverson, Levine, and Renclawowicz [Siam J. Appl. Math., 65 (2004), pp. 336--360; 66 (2005), pp. 361--364] to analyze the finite time blowup case. Some numerical results are provided to back up the analysis. Some questions and directions for future study are posed.
"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Asymptotic behavior of the solutions to a family of PDE's arising from the chemotaxis equations of Keller and Segal"]}]}],"canonical_facts":{"dc:contributor.advisor":["Howard Levine","Elgin Johnston","Leslie Hogben"],"dc:contributor.department":["Department of Mathematics"],"dc:creator":["Halverson, Matthew"],"dc:date":["2018-08-22T20:26:39.000"],"dc:date.accessioned":["2020-06-30T07:46:12Z"],"dc:date.available":["2020-06-30T07:46:12Z"],"dc:date.issued":["2008-01-01"],"dc:description.abstract":["The system ut = uxx - (uvx)x, vt = u - Av is considered where A is a non-negative, self-adjoint operator which commutes with the Laplacian. The operator is considered to have eigenvalues lambda n = nrholambda1, and the system is considered on [0,1] x [0,T] with homogeneous Neumann boundary conditions. The operators which lead to global solutions and those that lead to solutions which blow up in finite time are considered as a function of rho, using an application of the methods of Hillen and Potapov [Math. Methods Appl. Sci., 27 (2004), pp. 1783-1801] to analyze the global case and those of Halverson, Levine, and Renclawowicz [Siam J. Appl. Math., 65 (2004), pp. 336--360; 66 (2005), pp. 361--364] to analyze the finite time blowup case. Some numerical results are provided to back up the analysis. Some questions and directions for future study are posed.
"],"dc:format.mimetype":["application/pdf"],"dc:identifier":["archive/lib.dr.iastate.edu/rtd/15654/"],"dc:identifier.doi":["https://doi.org/10.31274/rtd-180813-16867"],"dc:identifier.uri":["https://dr.lib.iastate.edu/handle/20.500.12876/69308"],"dc:language.iso":["en"],"dc:title":["Asymptotic behavior of the solutions to a family of PDE's arising from the chemotaxis equations of Keller and Segal"],"dc:type":["dissertation"],"thesis:degree_level":["dissertation"],"thesis:degree_name":["Doctor of Philosophy"]},"updated_at":"2026-07-24T02:39:53Z"}