Asymptotic Solutions of the Dirichlet Problem for the Heat Equation at a Characteristic Point

Abstract

Bibliografische Information der Deutschen NationalbibliothekDie Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.de abrufbar. Abstract. The Dirichlet problem for the heat equation in a bounded domain G ⊂ R n+1 is characteristic, for there are boundary points at which the boundary touches a characteristic hyperplane t = c, c being a constant. It was I.G. Petrovskii (1934) who first found necessary and sufficient conditions on the boundary which guarantee that the solution is continuous up to the characteristic point, provided that the Dirichlet data are continuous. This paper initiated standing interest in studying general boundary value problems for parabolic equations in bounded domains. We contribute to the study by constructing a formal solution of the Dirichlet problem for the heat equation in a neighbourhood of a characteristic boundary point and showing its asymptotic character. Universitätsverlag

Asymptotic Expansions at Nonsymmetric Cuspidal Points

“…If λ 2 is an eigenvalue of A(0) subject to B, the question of solvability of equations (3.3) requires a careful examination, cf. [AKT14].…”

“…To do this we first localise the problem to a neighbourhood of the singular point O to see that it degenerates at O to a nonelliptic problem. Hence it follows that the techniques of constructing formal solutions to the boundary value problem developed in [AKT14] do not apply to suggest any crude solution. Although the problem reduces to an ordinary differential equationsU = A(t)U + F with operator-valued coefficients independent of t up to a separate interfering factor e 2t , the long-standing results of [Paz67] do not lead to a satisfactory solution, for the limit problem related to A(−∞) is quite sophisticated.…”