Unique solvability of parabolic equations with almost periodic coefficients in Hölder spaces.
type: Item
instance of: scholarly article
title: Unique solvability of parabolic equations with almost periodic coefficients in Hölder spaces.
zbMATH Open document ID: 1063.35067
DOI: 10.1023/A:1024001930334
author: V. B. Levenshtam
published in: Mathematical Notes
publication date: 2004-12-02T00:00:00Z
review text: Let \(P = {\partial \over \partial t} + {\sum_{| \alpha| \leq 2m}} a_\alpha(x,t)D^\alpha\) be a linear parabolic operator. Assume that the coefficients \(a_\alpha\) are Hölder continuous and almost periodic in the variables \((x,t)\). Consider the problem \(Pu = f\). Indicate with \(Q(P)\) the class of operators of the form \(Q = {\partial \over \partial t} + {\sum_{| \alpha| \leq 2m}} b_\alpha(x,t)D^\alpha\), such that the coefficients \(b_\alpha(x,t)\) are uniform limits of sequences of the form \(a_\alpha(x+\xi_k,t+\tau_k)\) (same sequence \((\xi_k,\tau_k)\) for every \(\alpha\)). The main result of the paper states that this problem has a unique solution \(u\) which is bounded in \(\mathbb R^{n+1}\) for every \(f \in C^{\gamma,\gamma/(2m)}\) (\(0 < \gamma < 1\)), if and only if, for every \(Q \in H(P)\), the problem \(Qv = 0\) has the trivial solution as unique bounded solution in the whole \(\mathbb R^{n+1}\). In this case, \(u\) belongs to the class \(C^{2m,1,\gamma,\gamma/(2m)}\) and is almost periodic if \(f\) is almost periodic. This result is connected to the notion of exponential dichotomy and extensions to other classes of data are considered. Finally, the author examines some quasilinear perturbations of the basic operator.
reviewed by: Davide Guidetti
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 35B15
zbMATH DE Number: 2120996
zbMATH Keywords: quasilinear perturbations
zbMATH Keywords: exponential dichotomy
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2120996