Premium
On the Linearization of the Satsuma‐Mimura Diffusion Equation
Author(s) -
von Wolfersdorf L.
Publication year - 1990
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19901450119
Subject(s) - mathematics , traveling wave , linearization , mathematical physics , nonlinear system , calculus (dental) , mathematical economics , humanities , mathematical analysis , physics , philosophy , quantum mechanics , medicine , dentistry
Some years ago J. SATSUMA and M. MIMURA [2-4] introduced a class of non-linear diffusion equations involving singular integral terms for describing some diffusion phenomena with non-local aggregation. They developed an exact linearization method for these equations and derived some interesting particular solutions in explicit form by this method. Recently in their paper [5] they investigated the stability behaviour of the steady state solutions in the spatially periodic case. The steady state solutions of Satsuma and Mimura are rederived in systematic way in our paper [6] by reducing the stationary equations to complex differential equations of first order. Methods of complex analysis are also used in our joint paper with W.GERLACH [1] to reduce the general initial-value problem for the equation of Satsuma-Mimura in the spatially periodic case to a Hammerstein integral equation for which by Schauders and Banachs fixed point theorems the existence of a solution in some time interval is proved. In our paper [7] this method is extended to the general equations of Satsuma-Mimura. Besides in [81 we use the complex analogue of the Hopf-Cole transformation for linearizing the equations of Satsuma-Mimura and studying the general Cauchy problem on the real axis. In the present paper this method is employed to the equations of Satsuma-Mimura in the spatially periodic case. We derive representation formulas in integral and series form for the solution of the general periodic Cauchy problem, study the asymptotic behaviour of the solution, and give some simple criteria for existence and non-existence (i.e., blow up) of the solution for all times. The somewhat involved general picture of the existence of such a solution is demonstrated by examples.