Journal of Inequalities and Applications
Volume 4 (1999), Issue 3, Pages 265-281

An elementary proof for one-dimensionality of travelling waves in cylinders

Friedemann Brock

Fakultät für Mathematik und Informatik, Universität Leipzig, Augustusplatz 10, Leipzig D-04109, Germany

Received 5 September 1998; Revised 16 December 1998

Copyright © 1999 Friedemann Brock. 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.


Let ω be a bounded domain in n1 with smooth boundary, u+,u, a>0, and let uWloc2,n((a,a)×ω)C1([a,a]×ω¯) satisfy Δu+c(x1)ux1=f(x1,u) and ux10 in (a,a)×ω, u=u± on {±a}×ω and u/v=0 on (a,a)×ω, where c is bounded and nonincreasing and f is continuous and nondecreasing in x1. We prove that u is a function of x1 only. The same result is shown for a related problem in the infinite cylinder ×ω. The proofs are based on a rearrangement inequality.