Abstract
The authors consider the first boundary value problem for a second order parabolic equation with variable coefficients in a domain with conical points or edges. In the first part of the paper, they study the Green function for this problem in the domain K× R n − m , where K is an infinite cone in R m , 2≤m≤n. They obtain the asymptotics of the Green function near the vertex (n=m) and edge (n>m), respectively. This result is applied in the second part of the paper, which deals with the initial-boundary value problem in this domain. Here, the right-hand side f of the differential equation belongs to a weighted L p space. At the end of the paper, the initial-boundary value problem in a bounded domain with conical points or edges is studied.
Highlights
1 Introduction The present paper is concerned with an initial-boundary value problem for a second order parabolic equation in a n-dimensional domain with a (n – m)-dimensional edge M, n > m ≥
Initial-boundary value problems for parabolic equations in domains with angular or conical points and edges were studied in a number of papers
Concerning the heat equation in domains with angular or conical points, we mention the papers by Grisvard [ ], Kozlov and Maz’ya [ ], de Coster and Nicaise [ ], where the asymptotics of the solutions near the singular boundary points was studied
Summary
The present paper is concerned with an initial-boundary value problem for a second order parabolic equation in a n-dimensional domain with a (n – m)-dimensional edge M, n > m ≥. As was proved in [ ], the Green function G(x, y, t, τ ) of the problem ( ) satisfies the estimate Under some smoothness conditions on the coefficients of the differential operator, we obtain the same decomposition of the weak solution near an edge point as in the case of the previously considered domain D
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