Interior differentiability of weak solutions to parabolic systems with quadratic growth nonlinearities
type: Item
instance of: scholarly article
title: Interior differentiability of weak solutions to parabolic systems with quadratic growth nonlinearities
zbMATH Open document ID: 0894.35050
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1998-09-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108447
review text: Let \(\Omega\subset \mathbb{R}^n\) \((n\geq 2)\) be a bounded open set, and \(0<T<\infty\). In the cylinder \(Q= \Omega\times (0,T)\), the authors consider the system of nonlinear PDE's: \[ u^i_t- \sum^n_{\alpha= 1} D_\alpha A^\alpha_i(\nabla u)= B_i(\nabla u)\quad (i= 1,\dots, N).\tag{1} \] The authors prove the theorem on interior differentiability of weak solutions of (1): Let \(u=(u^1,\dots, u^N)\), \(u^j\in W^{1,0}_2(Q)\cap C^\gamma(\overline Q)\) \((0<\gamma< 1; j=1,\dots, N)\) be a weak solution of (1). Then for any \(\Omega'\subset\subset\Omega\) and \(0< t_0< t_1<T: \nabla u^j\in (L^\sigma(\Omega'\times (t_0,t_1))^n\) \((\sigma\in[4,4(1+ {\gamma\over n})[, j=1,\dots, N)\); \(\nabla u^j\in (L^4(t_0, t_1; (L^s(\Omega))^N\) \((s\in [4,{4n\over n-2\gamma}[, j=1,\dots, N)\), where \(L^4(t_0, t_1; L^s(\Omega))\) denotes the vector space of all Bochner measurable functions \(v:(t_0, t_1)\to L^s(\Omega)\) such that \(\| v\|_{L^4(t_0, t_1; L^s(\Omega))}= \left(\int^{t_1}_{t_0}\| v(t)\|^4_{L^s(\Omega)}dt\right)^{1/4}< \infty\).
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35K40
zbMATH DE Number: 1115664
author: Joachim Naumann
author: Jörg Wolf
MaRDI profile type: Publication
cites work: Q4073917
cites work: Q3930113
cites work: Q3803530
cites work: A result of differentiability of nonlinear parabolic systems under monotonicity hypothesis
cites work: On the interior differentiability of weak solutions of parabolic systems with quadratic growth nonlinearities
cites work: Q4836063
description: scientific article; zbMATH DE number 1115664