International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 62601, 12 pages
doi:10.1155/IJMMS/2006/62601
A nonexistence result for a nonlinear PDE with Robin condition
Department of Mathematics, Faculty of Sciences, University of Annaba, P.O. Box 12, Annaba, Algeria
Received 16 May 2005; Revised 24 April 2006; Accepted 4 May 2006
Copyright © 2006 Brahim Khodja. 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
Under the assumption λ>0
and f
verifying f(x,y,0)=0 in D, 2F(x,y,u)−uf(x,y,u)≥0, u≠0, and if Ω=R×D, we show the convexity of function E(t)=∬D|u(t,x,y)|2dxdy, where u:Ω→ℝ is a solution of problem λ(∂2u/∂t2)−(∂/∂x)(p(x,y)(∂u/∂x))−(∂/∂y)(q(x,y)(∂u/∂y))+f(x,y,u)=0 in Ω, u+ε(∂u/∂n)=0 on ∂Ω,
considered in H2(Ω)∩L∞(Ω),
p,q:D¯→ℝ are two nonnull functions on D, ε is a positive real number, and D=]a1,b1[×]a2,b2[,
(F(x,y,s)=∫0sf(x,y,t)dt).