Global smooth solutions to the spatially periodic Cauchy problem for dissipative nonlinear evolution equations
type: Item
instance of: scholarly article
title: Global smooth solutions to the spatially periodic Cauchy problem for dissipative nonlinear evolution equations
zbMATH Open document ID: 0895.35052
author: Ling Hsiao
author: Huaiyu Jian
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-09-20T00:00:00Z
review text: The authors study the problem \[ \psi_t= -(\sigma- \alpha) \psi- \sigma \theta_x+ \alpha\psi_{xx}, \] \[ \theta_t= -(1-\beta) \theta+ \nu\psi_x+ (\psi\theta)_x +\beta \theta_{xx}, \quad 0\leq x\leq 1 \] with the conditions \[ \begin{aligned} (\psi,\theta) (0,t) & =(\psi, \theta) (1,t),\;(\psi_x, \theta_x) (0,t)= (\psi_x, \theta_x) (1,t),\;0\leq t\leq T,\\ (\psi,\theta) (x,0) & =\bigl( \psi_0 (x), \theta_0(x) \bigr),\;0\leq x\leq 1,\end{aligned} \] \((\alpha,\beta, \sigma\) and \(\nu\) are constants with \(\alpha,\beta\) being positive). Using the energy method, \(L^1\)-estimates and the Leray-Schauder fixed point theorem, they prove the existence and uniqueness of global classical solutions \(\psi, \theta\in C^{2,\delta} ([0,1] \times [0,T]\), where \(C^{2,\delta}\) are parabolic Hölder spaces.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 1117606
zbMATH Keywords: \(L^1\)-estimates
zbMATH Keywords: energy method
zbMATH Keywords: Leray-Schauder fixed point theorem
reviewed by: Wolfgang Watzlawek
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5535
OpenAlex ID: W1967194015
cites work: On partial differential equations related to Lorenz system
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DOI: 10.1006/JMAA.1997.5535
description: scientific article; zbMATH DE number 1117606