1 research outputs found
Critical convective-type equations on a half-line
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+β(u,ux)+(anβxn+amβxm)u=0, (x,t)ββ+Γβ+,
u(x,0)=u0(x), xββ+,
βxjβ1u(0,t)=0
for j=1,β¦,m/2, where the
constants an,amββ, n, m are integers, the
nonlinear term β(u,ux)
depends on the unknown
function u
and its derivative ux
and satisfies the estimate
|β(u,v)|β€C|u|Ο|v|Ο
with
Οβ₯0, Οβ₯1, such that
((n+2)/2n)(Ο+Οβ1)=1, Οβ₯1, Οβ[0,m).
Also we suppose that β«β+xn/2βdx=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