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International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 84972, 24 pages
http://dx.doi.org/10.1155/IJMMS/2006/84972

Critical convective-type equations on a half-line

Instituto de Matemáticas Unidad Morelia, Universidad Nacional Autónoma de México (UNAM), Campus Morelia, AP 61-3 (Xangari), Morelia Michoacán CP 58180, Mexico

Received 16 June 2005; Revised 6 October 2005; Accepted 4 January 2006

Copyright © 2006 Hindawi Publishing Corporation. 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

We are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for critical convective-type dissipative equations ut+(u,ux)+(anxn+amxm)u=0, (x,t)+×+, u(x,0)=u0(x), x+, xj1u(0,t)=0 for j=1,,m/2, where the constants an,am, n, m are integers, the nonlinear term (u,ux) depends on the unknown function u and its derivative ux and satisfies the estimate |(u,v)|C|u|ρ|v|σ with σ0, ρ1, such that ((n+2)/2n)(σ+ρ1)=1, ρ1, σ[0,m). Also we suppose that +xn/2dx=0. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem above-mentioned. We find the main term of the asymptotic representation of solutions in critical case. Also we give some general approach to obtain global existence of solution of initial-boundary value problem in critical convective case and elaborate general sufficient conditions to obtain asymptotic expansion of solution.