Regularization for a class of ill-posed evolution problems in Banach space
type: Item
instance of: scholarly article
title: Regularization for a class of ill-posed evolution problems in Banach space
zbMATH Open document ID: 1267.47014
author: Matthew A. Fury
author: Rhonda J. Hughes
published in: Semigroup Forum
publication date: 2013-02-15T00:00:00Z
review text: In this attractive paper, the authors deal with the issue of Banach space regularization in a very specific sense. They consider ill-posed versions of the evolution problem \[ \frac{du(t)}{dt}= A(t,D) u(t), \quad 0 \leq s \leq t <T, \] \[ u(s)=\chi, \] where the solution \(u(t)\) takes values in a Banach space \(X\), \(i D\) denotes a bounded strongly continuous group on \(X\), and the operator \(A\) attains the form \[ A(t,D)= \sum_{j=1}^k a_j(t) D^j, \] for \(t \in [0,T]\) with \(a_j \in C([0,T]: \mathbb{C})\) and \(1 \leq j \leq k\). To overcome the ill-posedness, a regularization approach with a one-parametric family \(R_\beta(t)\) of regularization operators is exploited. Precisely, for regularization parameters \(0<\beta<1\), the regularized solutions \(v_\beta(t)=R_{\beta(\delta)}(t)\chi\) are defined by solving the evolution problem \[ \frac{dv_\beta(t)}{dt}= f_\beta(t,D)v_\beta(t), \quad 0 \leq s \leq t <T, \] \[ v_\beta(s)=\chi. \] There are given sufficient conditions such that this auxiliary problem is well-posed for all \(\beta\) under consideration and, moreover, as common in all facets of regularization theory, convergence \(\|u(t)-R_{\beta(\delta)}(t)\chi_\delta\| \to 0\) with \(\beta(\delta) \to 0\) as \(\delta \to 0\) takes place whenever \(\|\chi-\chi_\delta\| \leq \delta\), here for all \(0 \leq s \leq t \leq T\). For \(X=L^p(\mathbb{R}),\;p \not=2,\) the situation is made explicit when \(f_\beta(t,D)=A(t,D)-\beta D^{2k}\). Examples of this kind include the linearized Korteweg-de Vries equation as well as the Schrödinger equation. Then, even a Hölder type conditional stability estimate \[ \|u(t)-v_\beta(t)\| \leq C \beta^\frac{T-t}{T-s} M^\frac{t-s}{T-s} \] with two positive constants \(C\) and \(M\) independent of \(\beta\) occurs under additional requirements, which leads to convergence rates \(\|u(t)-R_{\beta(\delta)}(t)\chi_\delta\|=O\left(\delta^{\frac{T-1}{2(T-s)}}\right)\) as \(\delta \to 0\) for an a priori parameter choice \(\beta = \sqr…
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35R25
zbMATH DE Number: 6136735
zbMATH Keywords: evolution equations
zbMATH Keywords: ill-posed problems
zbMATH Keywords: Cauchy problems
zbMATH Keywords: regularization
zbMATH Keywords: semigroup theory
zbMATH Keywords: Banach space
zbMATH Keywords: Hölder conditional stability
reviewed by: Bernd Hofmann
MaRDI profile type: Publication
full work available at URL: s00233-011-9353-3
OpenAlex ID: W1978644850
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DOI: 10.1007/S00233-011-9353-3
description: scientific article; zbMATH DE number 6136735