{"id":{"repo_id":"tdl","oai_identifier":"oai:tdl-ir.tdl.org:1969.1/3312"},"canonical_url":"https://search.dev.ndltd.org/etd/tdl/oai:tdl-ir.tdl.org:1969.1/3312","repository":{"repo_id":"tdl","name":"Texas Digital Library","base_url":"https://tdl-ir.tdl.org/server/oai/request"},"display":{"title":"Global existence of reaction-diffusion equations over multiple domains","abstract":"Systems of semilinear parabolic differential equations arise in the modelling of many chemical and biological systems. We consider m component systems of the form ut = DΔu + f (t, x, u) ∂uk/∂η =0 k =1, ...m where u(t, x)=(uk(t, x))mk=1 is an unknown vector valued function and each u0k is zero outside Ωσ(k), D = diag(dk)is an m ?? m positive definite diagonal matrix, f : R ?? Rn?? Rm → Rm, u0 is a componentwise nonnegative function, and each Ωi is a bounded domain in Rn where ∂Ωi is a C2+αmanifold such that Ωi lies locally on one side of ∂Ωi and has unit outward normal η. Most physical processes give rise to systems for which f =(fk) is locally Lipschitz in u uniformly for (x, t) ∈ Ω ?? [0,T ] and f (??, ??, ??) ∈ L∞(Ω ?? [0,T ) ?? U ) for bounded U and the initial data u0 is continuous and nonnegative on Ω. The primary results of this dissertation are three-fold. The work began with a proof of the well posedness for the system . Then we obtained a global existence result if f is polynomially bounded, quaipositive and satisfies a linearly intermediate sums condition. Finally, we show that systems of reaction-diffusion equations with large diffusion coeffcients exist globally with relatively weak assumptions on the vector field f.","abstract_html":"Systems of semilinear parabolic differential equations arise in the modelling of many chemical and biological systems. We consider m component systems of the form ut = DΔu + f (t, x, u) ∂uk/∂η =0 k =1, ...m where u(t, x)=(uk(t, x))mk=1 is an unknown vector valued function and each u0k is zero outside Ωσ(k), D = diag(dk)is an m ?? m positive definite diagonal matrix, f : R ?? Rn?? Rm → Rm, u0 is a componentwise nonnegative function, and each Ωi is a bounded domain in Rn where ∂Ωi is a C2+αmanifold such that Ωi lies locally on one side of ∂Ωi and has unit outward normal η. Most physical processes give rise to systems for which f =(fk) is locally Lipschitz in u uniformly for (x, t) ∈ Ω ?? [0,T ] and f (??, ??, ??) ∈ L∞(Ω ?? [0,T ) ?? U ) for bounded U and the initial data u0 is continuous and nonnegative on Ω. The primary results of this dissertation are three-fold. The work began with a proof of the well posedness for the system . Then we obtained a global existence result if f is polynomially bounded, quaipositive and satisfies a linearly intermediate sums condition. Finally, we show that systems of reaction-diffusion equations with large diffusion coeffcients exist globally with relatively weak assumptions on the vector field f.","abstract_has_math":false,"creators":["Ryan, John Maurice-Car"],"institution":"Texas A&M University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Walton, Jay"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2006,"date_issued":"2006-04-12","date_published":"2006-04-12","updated_at":"2026-08-21T22:21:56Z","subjects":["Existence","Differential Equations","Reaction-Diffusion"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1969.1/3312","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"source_record":{"url":"https://tdl-ir.tdl.org/server/oai/request?verb=GetRecord&metadataPrefix=dim&identifier=oai%3Atdl-ir.tdl.org%3A1969.1%2F3312","prefix":"dim"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Walton, Jay"]},{"key":"dc:creator","label":"Author","values":["Ryan, John Maurice-Car"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2006-04-12T16:06:22Z","2017-04-07T19:51:19Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2006-04-12T16:06:22Z","2017-04-07T19:51:19Z"]},{"key":"dc:date.issued","label":"Date","values":["2006-04-12"]},{"key":"dc:publisher","label":"Institution","values":["Texas A&M University"]},{"key":"dc:type","label":"Dc Type","values":["Book","Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Existence","Differential Equations","Reaction-Diffusion"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1969.1/3312"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Systems of semilinear parabolic differential equations arise in the modelling of many chemical and biological systems. We consider m component systems of the form ut = DΔu + f (t, x, u) ∂uk/∂η =0 k =1, ...m where u(t, x)=(uk(t, x))mk=1 is an unknown vector valued function and each u0k is zero outside Ωσ(k), D = diag(dk)is an m ?? m positive definite diagonal matrix, f : R ?? Rn?? Rm → Rm, u0 is a componentwise nonnegative function, and each Ωi is a bounded domain in Rn where ∂Ωi is a C2+αmanifold such that Ωi lies locally on one side of ∂Ωi and has unit outward normal η. Most physical processes give rise to systems for which f =(fk) is locally Lipschitz in u uniformly for (x, t) ∈ Ω ?? [0,T ] and f (??, ??, ??) ∈ L∞(Ω ?? [0,T ) ?? U ) for bounded U and the initial data u0 is continuous and nonnegative on Ω. The primary results of this dissertation are three-fold. The work began with a proof of the well posedness for the system . Then we obtained a global existence result if f is polynomially bounded, quaipositive and satisfies a linearly intermediate sums condition. Finally, we show that systems of reaction-diffusion equations with large diffusion coeffcients exist globally with relatively weak assumptions on the vector field f."]},{"key":"dc:title","label":"Title","values":["Global existence of reaction-diffusion equations over multiple domains"]}]}],"canonical_facts":{"dc:contributor":["Walton, Jay"],"dc:creator":["Ryan, John Maurice-Car"],"dc:date.accessioned":["2006-04-12T16:06:22Z","2017-04-07T19:51:19Z"],"dc:date.available":["2006-04-12T16:06:22Z","2017-04-07T19:51:19Z"],"dc:date.issued":["2006-04-12"],"dc:description.abstract":["Systems of semilinear parabolic differential equations arise in the modelling of many chemical and biological systems. We consider m component systems of the form ut = DΔu + f (t, x, u) ∂uk/∂η =0 k =1, ...m where u(t, x)=(uk(t, x))mk=1 is an unknown vector valued function and each u0k is zero outside Ωσ(k), D = diag(dk)is an m ?? m positive definite diagonal matrix, f : R ?? Rn?? Rm → Rm, u0 is a componentwise nonnegative function, and each Ωi is a bounded domain in Rn where ∂Ωi is a C2+αmanifold such that Ωi lies locally on one side of ∂Ωi and has unit outward normal η. Most physical processes give rise to systems for which f =(fk) is locally Lipschitz in u uniformly for (x, t) ∈ Ω ?? [0,T ] and f (??, ??, ??) ∈ L∞(Ω ?? [0,T ) ?? U ) for bounded U and the initial data u0 is continuous and nonnegative on Ω. The primary results of this dissertation are three-fold. The work began with a proof of the well posedness for the system . Then we obtained a global existence result if f is polynomially bounded, quaipositive and satisfies a linearly intermediate sums condition. Finally, we show that systems of reaction-diffusion equations with large diffusion coeffcients exist globally with relatively weak assumptions on the vector field f."],"dc:identifier.uri":["http://hdl.handle.net/1969.1/3312"],"dc:language.iso":["en_US"],"dc:publisher":["Texas A&M University"],"dc:subject":["Existence","Differential Equations","Reaction-Diffusion"],"dc:title":["Global existence of reaction-diffusion equations over multiple domains"],"dc:type":["Book","Thesis"]},"updated_at":"2026-08-21T22:21:56Z"}