{"id":{"repo_id":"aachen","oai_identifier":"oai:publications.rwth-aachen.de:59933"},"canonical_url":"https://search.dev.ndltd.org/etd/aachen/oai:publications.rwth-aachen.de:59933","repository":{"repo_id":"aachen","name":"RWTH Aachen University","base_url":"https://publications.rwth-aachen.de/oai2d"},"display":{"title":"Travelling wave solutions of the heat equation in an unbounded cylinder with a non-linear boundary condition","abstract":"Let $Omega$ be a bounded domain in $R^2$. The existence of travelling wave solutions for the heat equation in the unbounded cylinder $RimesOmega$ subject to the nonlinear boundary condition $partial u over partial n = f(u)$ is investigated. Finding such a solution amounts to solving the elliptic PDE $Delta u - c partial_x u = 0$ for $(x,y)inRimesOmega$ with $partial u over partial n = f(u)$ on $RimespartialOmega$. The main result is the existence of a non-trivial solution of this equation for a large class of non-linearities $f$. Additionally, asymptotic behavior at $x=pminfty$ and regularity properties are established. A variational approach is used. More specifically, a weak solution is found as the minimizer of a constrained minimization problem posed in a weighted Sobolev space. Due to underlying domain $RimesOmega$ being unbounded, this problem suffers from a lack of compactness. The problem is solved using a suitable approximation.","abstract_html":"Let $Omega$ be a bounded domain in <span class=\"etd-inline-math\">R<sup>2</sup></span>. The existence of travelling wave solutions for the heat equation in the unbounded cylinder $RimesOmega$ subject to the nonlinear boundary condition $partial u over partial n = f(u)$ is investigated. Finding such a solution amounts to solving the elliptic PDE <span class=\"etd-inline-math\">Delta u - c partial<sub>x</sub> u = 0</span> for $(x,y)inRimesOmega$ with $partial u over partial n = f(u)$ on $RimespartialOmega$. The main result is the existence of a non-trivial solution of this equation for a large class of non-linearities $f$. Additionally, asymptotic behavior at $x=pminfty$ and regularity properties are established. A variational approach is used. More specifically, a weak solution is found as the minimizer of a constrained minimization problem posed in a weighted Sobolev space. Due to underlying domain $RimesOmega$ being unbounded, this problem suffers from a lack of compactness. The problem is solved using a suitable approximation.","abstract_has_math":true,"creators":["Kyed, Mads"],"institution":"Publikationsserver der RWTH Aachen University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Lugscheider, Erich"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2005,"date_issued":"2005","date_published":"2005","updated_at":"2026-07-30T19:42:48Z","subjects":["info:eu-repo/classification/ddc/510","Wärmeleitungsgleichung","nichtlineares Randwertproblem","fortschreitende Wellenlösung","Mathematik","semilinear parabolic equations","semilinear elliptic equations","cylindrical domains","nonlinear boundary condition","travelling wave"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121673%22"],"render_values":[{"text":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121673%22","href":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121673%22","code":true}]}]},"links":{"outbound_url":"https://publications.rwth-aachen.de/record/59933","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Lugscheider, Erich"]},{"key":"dc:creator","label":"Author","values":["Kyed, Mads"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:coverage","label":"Dc Coverage","values":["DE"]},{"key":"dc:date","label":"Dc Date","values":["2005"]},{"key":"dc:publisher","label":"Institution","values":["Publikationsserver der RWTH Aachen University"]},{"key":"dc:relation","label":"Dc Relation","values":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-11066"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["info:eu-repo/classification/ddc/510","Wärmeleitungsgleichung","nichtlineares Randwertproblem","fortschreitende Wellenlösung","Mathematik","semilinear parabolic equations","semilinear elliptic equations","cylindrical domains","nonlinear boundary condition","travelling wave"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/record/59933","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121673%22"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let $Omega$ be a bounded domain in $R^2$. The existence of travelling wave solutions for the heat equation in the unbounded cylinder $RimesOmega$ subject to the nonlinear boundary condition $partial u over partial n = f(u)$ is investigated. Finding such a solution amounts to solving the elliptic PDE $Delta u - c partial_x u = 0$ for $(x,y)inRimesOmega$ with $partial u over partial n = f(u)$ on $RimespartialOmega$. The main result is the existence of a non-trivial solution of this equation for a large class of non-linearities $f$. Additionally, asymptotic behavior at $x=pminfty$ and regularity properties are established. A variational approach is used. More specifically, a weak solution is found as the minimizer of a constrained minimization problem posed in a weighted Sobolev space. Due to underlying domain $RimesOmega$ being unbounded, this problem suffers from a lack of compactness. The problem is solved using a suitable approximation."]},{"key":"dc:source","label":"Dc Source","values":["Aachen : Publikationsserver der RWTH Aachen University VI, 75 S. : graph. Darst. (2005). = Aachen, Techn. Hochsch., Diss., 2005"]},{"key":"dc:title","label":"Title","values":["Travelling wave solutions of the heat equation in an unbounded cylinder with a non-linear boundary condition"]}]}],"canonical_facts":{"dc:contributor":["Lugscheider, Erich"],"dc:coverage":["DE"],"dc:creator":["Kyed, Mads"],"dc:date":["2005"],"dc:description":["Let $Omega$ be a bounded domain in $R^2$. The existence of travelling wave solutions for the heat equation in the unbounded cylinder $RimesOmega$ subject to the nonlinear boundary condition $partial u over partial n = f(u)$ is investigated. Finding such a solution amounts to solving the elliptic PDE $Delta u - c partial_x u = 0$ for $(x,y)inRimesOmega$ with $partial u over partial n = f(u)$ on $RimespartialOmega$. The main result is the existence of a non-trivial solution of this equation for a large class of non-linearities $f$. Additionally, asymptotic behavior at $x=pminfty$ and regularity properties are established. A variational approach is used. More specifically, a weak solution is found as the minimizer of a constrained minimization problem posed in a weighted Sobolev space. Due to underlying domain $RimesOmega$ being unbounded, this problem suffers from a lack of compactness. The problem is solved using a suitable approximation."],"dc:identifier":["https://publications.rwth-aachen.de/record/59933","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121673%22"],"dc:language":["eng"],"dc:publisher":["Publikationsserver der RWTH Aachen University"],"dc:relation":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-11066"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:source":["Aachen : Publikationsserver der RWTH Aachen University VI, 75 S. : graph. Darst. (2005). = Aachen, Techn. Hochsch., Diss., 2005"],"dc:subject":["info:eu-repo/classification/ddc/510","Wärmeleitungsgleichung","nichtlineares Randwertproblem","fortschreitende Wellenlösung","Mathematik","semilinear parabolic equations","semilinear elliptic equations","cylindrical domains","nonlinear boundary condition","travelling wave"],"dc:title":["Travelling wave solutions of the heat equation in an unbounded cylinder with a non-linear boundary condition"],"dc:type":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-30T19:42:48Z"}