116 T. Gramchev
for all β, γ ∈ Z
n +
, β 6= 0.
We point out that, as a corollary of our theorem, we obtain for σ = 1 and f [v] ≡ 0 a
seemingly new result, namely that every eigenfunction φ
j
x of −1+V x is extended
to a holomorphic function in {z ∈ C
n
: |ℑz| T
} for some T 0. Next, we have
a corollary from our abstract result on uniform G
1
regularity for the H
2
R solitary
wave solutions r x + ct, c 0 in [29] satisfying
49 Pu
= r
′′′′
+ µr
′′
+ cr = f r, r
′
, r
′′
= f r, r
′
+ f
1
r, r
′
r
′′
, µ ∈ R
with f
j
being homogeneous polynomial of degree d − j, j = 0, 1, d ≥ 3 and |µ|
2 √
−c. Actually, by Lemma 6, we find that every solution r to 49 belonging to H
s
R , s 32, is extended to a holomorphic function in
{z ∈ C : |I m z| T } for some T 0.
4. Uniform Gevrey regularity of L
∞
stationary solutions
It is well known that the traveling waves to dissipative equations like Burgers, Fisher– Kolmogorov, Kuramoto–Sivashinsky equations have typically two different nonzero
limits for x → ±∞ see the example 6. Now we investigate the G
σ un
R
n
regularity of such type of solutions for semilinear elliptic equations.
We shall generalize Theorem 4.1 in [8] for G
θ
nonlinear terms f . We restrict our attention to 30 for n
= 1, m = 0, P = PD being a constant coefficients elliptic
p.d.o. or Fourier multiplier of order m. T
HEOREM
2. Let θ ≥ 1, σ ≥ θ, m − m
≥ 1, f ∈ G
θ
C
L
, f 0 = 0, w ∈
G
σ ∞
T ; H
s
for some T 0. Suppose that v
∈ L
∞
R is a weak solution of 30
satisfying ∇v ∈ H
s
R . Then there exists T
′
, depending on T , PD, f ,
kvk
∞
and k∇vk
s
such that v ∈ G
σ ∞
H
s
R ; T
′
. In particular, if σ = θ = 1 then v can be
extended to a holomorphic function in {z ∈ C : |Im z| T
′
}. Without loss of generality we suppose that n
= 1, m = 0. It is enough to show
that v
′
= D
x
v ∈ G
σ
H
s
, T for some T 0.
We need an important auxiliary assertion, whose proof is essentially contained in [27].
L
EMMA
7. Let g ∈ G
θ
R
p
: R, 1 ≤ θ ≤ σ , g0 = 0. Then there exists a
positive continuous nondecreasing function Gt, t ≥ 0 such that
50 kD
α
gvw k
s
≤ |D
α
gv |
∞
kwk
s
+ G|v|
∞ α
α
θ
kwk
s −1
+ k∇vk
s s
−1
for all v ∈ L
∞
R
n
: R
p
, v
′
∈ H
s
R
n
: R
p
, w ∈ H
s
R
p
, α ∈ Z
p +
, provided s n2
+ 1. Proof of Theorem 2. Write u
= v
′
. We observe that f v
′
= f
′
v u and the hy-
potheses imply that gv : = f
′
v ∈ L
∞
R and u
∈ H
s
R . Thus differentiating k
Perturbative methods 117
times we obtain that u satisfies P D
k
u = D
k
gvu + D
k
w
′
which leads to D
k
u = P
−1
D
k
gvu + P
−1
D
k
w
′
. Hence, since m
≥ 1 and P
−1
D is bounded in H
s
R we get the following estimates
kD
k
u k
s
≤ CkD
k −1
gvu k
s
+ kD
k −1
w
′
k
s
≤
k −1
X
j =0
k − 1
j kD
j
gvD
k −1− j
u k
s
+ kD
k −1
w
′
k
s
≤
k −1
X
j =0
k − 1
j
j
X
ℓ =0
C
ℓ −1
ℓ ×
X
p1+···+ pℓ= j
p
1
≥1,··· , p
ℓ
≥1 ℓ
Y
µ =1
kD
p
µ
−1
u k
s
p
µ
kD
ℓ
gvD
k −1− j
u k
s
+ kD
k −1
w
′
k
s
. 51
Now, by Lemma 7 and 51 we get, with another positive constant C, T
k
k
σ
kD
k
u k
s
≤ CT k
−σ
kvk
s k
−1
X
j =0
k − 1
j
−σ +1 j
X
ℓ =0
C
ℓ −1
ℓ
σ −θ
× X
p1+···+ pℓ= j
p
1
≥1,··· , p
ℓ
≥1 ℓ
Y
µ =1
T
p
µ
|D
p
µ
−1
u |
s
p
µ σ
G |v|
∞ ℓ
T
k −1− j
|D
k −1− j
u |
s
k − 1 − j
σ
+ kw
′
k
σ, T
;s
. 52
Next, we conclude as in [8]. R
EMARK
3. As a corollary from our abstract theorem we obtain apparently new results on the analytic G
1 un
R regularity of traveling waves of the Kuramoto–
Sivashinsky equation cf. [41], and the Fisher–Kolmogorov equation and its generaliza- tions cf. [37], [31].
5. Decay estimates in Gelfand–Shilov spaces
