Global smooth solutions for a quasilinear fractional evolution equation
type: Item
instance of: scholarly article
title: Global smooth solutions for a quasilinear fractional evolution equation
zbMATH Open document ID: 1035.45005
author: Emilia Bazhlekova
author: Philippe Clément
published in: Journal of Evolution Equations
publication date: 2003-08-06T00:00:00Z
review text: The authors prove the global existence of smooth solutions for quasilinear fractional evolution equations of the form \[ u_t(t, x)= \int^t_0 g_\beta(t- s)[\sigma(u_x(s, x))_x+ f(s, x)]\,ds,\tag{1} \] where \(\sigma\) is smooth and strictly increasing, and the convolution kernel is given by \[ g_\beta(t):= t^{\beta- 1}/\Gamma(\beta)\quad t> 0,\quad \beta\in (0,1). \] The proof is based on rewriting (1) as a differential equation of fractional order \(\alpha:=\beta+1\) and resorting to \(L^p (L^q)\) maximal regularity results for the corresponding linear version of (1). The paper generalizes earlier results by \textit{G. Gripenberg} [J. Differ. Equations 102, 382--390 (1993; Zbl 0780.45012)] and \textit{H. Engler} [Trans. Am. Math. Soc. 348, 267--290 (1996; Zbl 0848.45002)].
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 1960720
zbMATH Keywords: quasilinear fractional evolution equation
zbMATH Keywords: smooth solutions
zbMATH Keywords: global existence
reviewed by: H. Brunner
MaRDI profile type: Publication
full work available at URL: s00028-003-0092-0
OpenAlex ID: W4252512853
DOI: 10.1007/S00028-003-0092-0
description: scientific article; zbMATH DE number 1960720