Travelling wave solutions of the heat equation in an unbounded cylinder with a non-Linear boundary condition [Elektronische Ressource] / vorgelegt von Mads Kyed

Travelling Wave Solutions of the Heat Equation in an UnboundedCylinder with a Non-Linear Boundary ConditionVon der Fakult˜ at fur˜ Mathematik, Informatik und Naturwissenschaften der Rheinisch-Westf˜ alischen Technischen Hochschule Aachen zur Erlangung des akademischen Gradeseines Doktors der Naturwissenschaften genehmigte Dissertationvorgelegt vonCand. Scient.Mads Kyedaus Kolding, D˜ anemarkBerichter: Prof. Dr. Josef BemelmansProf. Dr. Heiko von der MoselTag der mundlic˜ hen Prfung: 10.05.2005Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.˜Abstract2Let › ‰ R be a bounded domain. The existence of travelling wave solutions for theheat equation @u¡ ¢u = 0 in the unbounded cylinder R£ › subject to the nonlineart@uboundary condition =f(u) is investigated. Finding such a solution amounts to solving@nthe semi-linear elliptic PDE8< ¢u¡c@ u = 0 for (x;y)2 R£ ›x(⁄) @u: =f(u) for (x;y)2 R£@› :@nThe main result is the existence of a non-trivial solution of (⁄) for a large class of non-linearities f. Additionally, asymptotic behavior at x = §1 and regularity properties areestablished.A variational approach is used. More speciflcally, a weak solution of (⁄) is found asthe minimizer of a constrained minimization problem posed in the weighted Sobolev space1 ¡xH (R£ ›; e ). Due to underlying domain R£ › being unbounded, this problem sufiers2from a lack of compactness.