The profiles u^
are defined for b G BQ and a C&. It is convenient to extend the
definition to all b and a £U.
1.8. There is a C°° function W on O x [0, oo[ such that
i) W(6, a,z)=0 when b e Bo and z = 0,
ii) for all compact set JC C xU, there are 8 0 and C such that
V 6 G S , VaG/C,Vz0 : \W(b,a,z)-a\ Ce~6z
Hi) when b G BQ and a G Cb then z i— W(6, a, z) zs a solution of (1.6)
One can parametrize a neighborhood of 9ft using normal coordinates
x y + xnv(y). Near (6, a) G C, one can use coordinates a = (a!,a") such that C
is given by the equations a" = h(b,af). Then one can extend locally the function
w as
W(b,a,z) =wh
,Mbbiaf)(z) + (a / / -/i(6 b ,a / )) tanhz.
where 6b (t,y,bo) if b (£,#, 6Q) and a: i/ + xnv{y). When x is outside a
neighborhood of dft, one can take W = a. Gluing the pieces by a partition of unity
yields the result. D
Taking p G C°°(ft) such that / ? = 0 and dip = v on 9ft, introduce
(1.23) ix§(t,a) = W(6(£,x),u0(t,x),p(a;)/e).
By construction, it satisfies
( 1 ' 2 4 ) \ t i 5 - u o = 0 ( c - ^ )
Thus I^Q is a perturbation of ito in the interior and the general idea is that UQ is
close to a solution of (1.4). In this direction, the main step is to prove that the
linearized equations from (1.4) around UQ are stable. More precisely, consider a
family of perturbations

x ft) and the linearized equation from
(1.4) around
: UQ +
(1-25) VueQ+£v(t,x,dudx)u = / , U|an=0
TV+ev is a differential operator, first order in t and second order in x, whose coef-
ficients depend on 6, UQ and v and their derivatives. Its explicit form is computed
in section 4 below.
The main new result of this paper is that, under Assumptions 1.1 and 1.4, the
equations (1.25) are well posed in L2. With To 0 given, we assume that
( u0 G
x ft), b G
x ft)
I ^p (KlUoc + lleVt^llLco + lkX^IU-)
o c
stability). There are C 0 and £o s^c/i that for all e
and / G
x ft) vanishing for t 0, the equation (1.25) has a unique
solution which vanishes for t 0. Moreover
(1.27) H |
L 2
+ /ER«||L2 + e3/2\\d2xu\\Li C\\f\\L,.
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