Journal of Inequalities and Applications
Volume 4 (1999), Issue 3, Pages 265-281
doi:10.1155/S1025583499000399
An elementary proof for one-dimensionality of travelling waves in cylinders
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.
Abstract
Let ω be a bounded domain in ℝn−1 with smooth boundary, u+,u−∈ℝ, a>0, and let u∈Wloc2,n((−a,a)×ω)∩C1([−a,a]×ω¯) satisfy −Δu+c(x1)ux1=f(x1,u) and ux1≥0 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 ℝ×ω. The proofs are based on a rearrangement inequality.