On asymptotics of the Green function and Poisson kernels of a mixed parabolic problem in a cone. I
type: Item
instance of: scholarly article
title: On asymptotics of the Green function and Poisson kernels of a mixed parabolic problem in a cone. I
zbMATH Open document ID: 0728.35048
DOI: 10.4171/ZAA/343
author: N. E. Zubov
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1989-01-01T00:00:00Z
review text: The author considers the initial-boundary value problem: \(\sum_{2bk+| \alpha | \leq 2bm}a_{\alpha k}D_ x^{\alpha}\partial_ t^ ku=0\) on \(K\times (0,\infty)\), \(\sum_{2bk+| \alpha | \leq m_ j}b_{j\alpha k}D_ x^{\alpha}\partial_ t^ ku=g_ j\) \((j=1,...,mb)\) on (\(\partial K\setminus 0)\times (0,\infty)\), where K is a cone with a vertex at the origin of \({\mathbb{R}}^ n\), b is a fixed positive integer and \(a_{\alpha k}\), \(b_{j\alpha k}\) are positiv-homogeneous functions. Under some conditions imposed on differential operators he proves existence of the unique solution to the above problem in an appropriate class of functions. Some special solutions of the homogeneous problem are also given. Similar results are obtained for an adjoint parabolic problem. These results are derived by considering an associated parametrized elliptic problem. In the latter case formulas for an asymptotic form of the solutions, and for the coefficients of the expansions are established as well.
Mathematics Subject Classification ID: 35K35
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 35A08
zbMATH DE Number: 4202852
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: uniqueness
zbMATH Keywords: initial-boundary value problem
zbMATH Keywords: existence
reviewed by: S. Burys
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zaa/343
OpenAlex ID: W2563047076
description: scientific article; zbMATH DE number 4202852