Steady state solutions of the Satsuma-Mimura diffusion equation
type: Item
instance of: scholarly article
title: Steady state solutions of the Satsuma-Mimura diffusion equation
zbMATH Open document ID: 0661.35046
DOI: 10.4171/ZAA/293
author: Lothar von Wolfersdorf
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1988-01-01T00:00:00Z
review text: \textit{J. Satsuma} and \textit{M. Mimura} [Exact treatment of nonlinear diffusion equations with singular integral terms, J. Phys. Soc. Japan 54, 894-900 (1985); see also \textit{M. Mimura}, Contemp. Math. 17, 343-351 (1983; Zbl 0547.35055)] investigated a class of nonlinear nonlocal diffusion equations involving singular integral terms. They developed an exact linearization method for these equations and derived some interesting particular solutions in explicit form by this method. In the present paper we rederive the steady state solutions of Satsuma and Mimura in systematic way by reducing the steady equations to complex differential equations of first order. We find that apart from an obvious invariant shifting transformation the steady state solutions found by Satsuma and Mimura are the only ones with a prescribed asymptotic behaviour at infinity. Besides we obtain new periodic steady state solutions for some corresponding equations with variable diffusion coefficient.
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 4080024
zbMATH Keywords: nonlocal diffusion equations
zbMATH Keywords: singular integral terms
zbMATH Keywords: linearization
zbMATH Keywords: solutions in explicit form
zbMATH Keywords: steady state
zbMATH Keywords: invariant shifting transformation
zbMATH Keywords: periodic steady state solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4080024