International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 84972, 24 pages
doi: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 Elena I. Kaikina. 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+ℕ(u,ux)+(an∂xn+am∂xm)u=0, (x,t)∈ℝ+×ℝ+,
u(x,0)=u0(x), x∈ℝ+,
∂xj−1u(0,t)=0
for j=1,…,m/2, where the
constants an,am∈ℝ, n, m are integers, the
nonlinear term ℕ(u,ux)
depends on the unknown
function u
and its derivative ux
and satisfies the estimate
|ℕ(u,v)|≤C|u|ρ|v|σ
with
σ≥0, ρ≥1, such that
((n+2)/2n)(σ+ρ−1)=1, ρ≥1, σ∈[0,m).
Also we suppose that ∫ℝ+xn/2ℕdx=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.