International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 2, Pages 313-348

Oscillatory and periodic solutions to a diffusion equation of neutral type

Joseph Wiener and William Heller

Department of Mathematics, University of Texas-Pan American, Edinburg 78539, TX, USA

Received 30 October 1997

Copyright © 1999 Joseph Wiener and William Heller. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We examine a PDE with piecewise constant time delay. The equation is of neutral type since it contains the derivative ut at different values of the t-argument. Furthermore, the argument deviation changes its sign within intervals of unit length, so that the given PDE is alternately of retarded and advanced type. It is shown that the argument deviation generates, under certain conditions, oscillations of the solutions, which is an impossible phenomenon for the corresponding equation without delay. Of special interest is the appearance of periodic solutions as well as solutions asymptotically approaching closed curves which are not solutions of the equation studied.