Cauchy problem for semilinear parabolic equations with initial data in \(H_p^s(\mathbb{R}^n)\) spaces
type: Item
instance of: scholarly article
title: Cauchy problem for semilinear parabolic equations with initial data in \(H_p^s(\mathbb{R}^n)\) spaces
zbMATH Open document ID: 0910.35065
DOI: 10.4171/RMI/232
author: Francis Ribaud
published in: Revista Matemática Iberoamericana
publication date: 1998-09-28T00:00:00Z
full work available at URL: https://eudml.org/doc/39552
review text: Summary: We study local and global Cauchy problems for the semilinear parabolic equations \(\partial_t U-\Delta U=P(D) F(U)\) with initial data in fractional Sobolev spaces \(H_p^s (\mathbb{R}^n)\). In most of the studies on this subject, the initial data \(U_0(x)\) belongs to Lebesgue spaces \(L^p(\mathbb{R}^n)\) or to supercritical fractional Sobolev spaces \(H^s_p (\mathbb{R}^n)\) \((s>n/p)\). Our purpose is to study the intermediate cases (namely for \(0<s <n/p)\). We give some mapping properties for functions with polynomial growth on subcritical \(H_p^s (\mathbb{R}^n)\) spaces, and we show how to use them to solve the local Cauchy problem for data with low regularity. We also give some results about the global Cauchy problem for small initial data.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1203672
zbMATH Keywords: subcritical fractional Sobolev spaces
zbMATH Keywords: small initial data
zbMATH Keywords: initial data with low regularity
zbMATH Keywords: local and global Cauchy problems
zbMATH Keywords: polynomial growth
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1203672