A mollification method for a noncharacteristic Cauchy problem for a parabolic equation
type: Item
instance of: scholarly article
title: A mollification method for a noncharacteristic Cauchy problem for a parabolic equation
zbMATH Open document ID: 0861.35033
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-05-04T00:00:00Z
review text: The object of this paper is the study of the following noncharactristic Cauchy problem \[ u_t-a(x)u_{xx}-b(x)u_x-c(x)u=0,\quad u(0,t)=\varphi(t),\quad u_x(0,t)=0. \] The analysis of this problem is clear and was studied by the author in previous papers, too. It is proved, that under certain assumptions for the coefficients the above problem has a \(L_p\)-solution if and only if the datum \(\varphi=\varphi(t)\) belongs to a Gevrey space of order 2 basing on the \(L_p(\mathbb{R}^n)\)-norm. Moreover, the problem is ill-posed. For this reason, the author is interested in error estimates if one prescribes instead of \(\varphi\) perturbed data \(\varphi_\varepsilon\). A mollification method is proposed. The main idea is to mollify \(\varphi_\varepsilon\) in such a way that the new data belong to spaces of entire functions of exponential type. These are classes of well-posedness for noncharacteristic Cauchy problems. Some numerical examples complete the paper.
Mathematics Subject Classification ID: 35K10
Mathematics Subject Classification ID: 35R25
zbMATH DE Number: 902580
zbMATH Keywords: Gevrey space
zbMATH Keywords: error estimates
zbMATH Keywords: classes of well-posedness
author: Dinh Nho Hào
reviewed by: Michael Reissig
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0181
OpenAlex ID: W2044078015
DOI: 10.1006/JMAA.1996.0181
description: scientific article; zbMATH DE number 902580