for each u ∈ H
1
O , where c
S
0 is a constant that depends on the dimension and 2
∗
=
2d d
−2
if d
2, while 2
∗
may be any number in ]2, ∞[ if d = 2 and 2
∗
= ∞ if d = 1. Therefore one has kuk
2
∗
,2;t
≤ c
S
k∇uk
2,2;t
, for each t
≥ 0 and each u ∈ L
2 l oc
R
+
; H
1
O
. And if u ∈ L
∞ l oc
R
+
; L
2
O T
L
2 l oc
R
+
; H
1
O
, one has
kuk
2, ∞;t
∨ kuk
2
∗
,2;t
≤ c
1
kuk
2 2,
∞;t
+ k∇uk
2 2,2;t
1 2
, with c
1
= c
S
∨ 1. One particular case of interest for us in relation with this inequality is when p
1
= 2, q
1
= ∞ and p
2
= 2
∗
, q
2
= 2. If I = I 2, ∞, 2
∗
, 2 , then the corresponding set of associated conjugate numbers is I
′
= I
′
2, ∞, 2
∗
, 2 = I 2, 1,
2
∗
2
∗
−1
, 2 , where for d = 1 we make the convention that
2
∗
2
∗
−1
= 1. In this particular case we shall use the notation L
;t
:= L
I;t
and L
∗ ;t
:= L
I
′
;t
and the respective norms will be denoted by
kuk
;t
:= kuk
I;t
= kuk
2, ∞;t
∨ kuk
2
∗
,2;t
, kuk
∗ ;t
:= kuk
I
′
;t
. Thus we may write
kuk
;t
≤ c
1
kuk
2 2,
∞;t
+ k∇uk
2 2,2;t
1 2
, 3
for any u ∈ L
∞ l oc
R
+
; L
2
O T
L
2 l oc
R
+
; H
1
O
and t ≥ 0 and the duality inequality becomes
Z
t
Z
O
u s, x v s, x d x ds ≤ kuk
;t
kvk
∗ ;t
, for any u
∈ L
;t
and v ∈ L
∗ ;t
.
2.2 Hypotheses
Let {B
t
:= B
j t
j ∈{1,··· ,d
1
}
}
t ≥0
be a d
1
-dimentional Brownian motion defined on a standard filtered probability space Ω,
F , F
t t
≥0
, P .
Let A be a symmetric second order differential operator given by A := −L = −
P
d i, j=1
∂
i
a
i, j
∂
j
. We assume that a is a measurable and symmetric matrix defined on
O which satisfies the uniform ellipticity condition
λ|ξ|
2
≤ X
i, j
a
i, j
xξ
i
ξ
j
≤ Λ|ξ|
2
, ∀x ∈ O , ξ ∈ R
d
, 4
where λ and Λ are positive constants. The energy associated with the matrix a will be denoted by
E w, v =
d
X
i, j=1
Z
O
a
i, j
x∂
i
wx ∂
j
vx d x. 5
It’s defined for functions w, v ∈ H
1
O , or for w ∈ H
1 l oc
O and v ∈ H
1
O with compact support. 506
We consider the semilinear stochastic partial differential equation 1 for the real-valued random field u
t
x with initial condition u0, . = ξ., where ξ is a F -measurable random variable with
values in L
2 l oc
O . We assume that we have predictable random functions
f :
R
+
× Ω × O × R × R
d
→ R , h
: R
+
× Ω × O × R × R
d
→ R
d
1
g =
g
1
, ..., g
d
: R
+
× Ω × O × R × R
d
→ R
d
We define f
·, ·, ·, 0, 0 := f ,
h ·, ·, ·, 0, 0 := h
and g
·, ·, ·, 0, 0 := g = g
1
, ..., g
d
. We considere the following sets of assumptions :
Assumption H: There exist non negative constants C, α, β such that
i | f t, ω, x, y, z − f t, ω, x, y
′
, z
′
| ≤ C | y − y
′
| + |z − z
′
|
ii
P
d
1
j=1
|h
j
t, ω, x, y, z − h
j
t, ω, x, y
′
, z
′
|
2
1 2
≤ C | y − y
′
| + β |z − z
′
|,
iii
P
d i=1
|g
i
t, ω, x, y, z − g
i
t, ω, x, y
′
, z
′
|
2
1 2
≤ C | y − y
′
| + α |z − z
′
|.
iv the contraction property as in [5] :
α + β
2
2 λ .
Moreover we introduce some integrability conditions on f , g
, h and the initial data
ξ :
Assumption HD local integrability conditions on f , g
and h :
E Z
t
Z
K
| f
t
x| + |g
t
x|
2
+ |h
t
|
2
d x d t ∞
for any compact set K ⊂ O , and for any t ≥ 0.
Assumption HI local integrability condition on the initial condition :
E Z
K
|ξx|
2
d x ∞
for any compact set K ⊂ O .
Assumption HD
E f
∗ ;t
2
+ g
2 2,2;t
+ h
2 2,2;t
∞, for each t
≥ 0. Sometimes we shall consider the following stronger forms of these conditions:
507
Assumption HD2
E f
2 2,2;t
+ g
2 2,2;t
+ h
2 2,2;t
∞, for each t
≥ 0.
Assumption HI2 integrability condition on the initial condition :
E kξk
2 2
∞.
Remark 1. Note that 2, 1 is the pair of conjugates of the pair 2, ∞ and so 2, 1 belongs to the set I
′
which defines the space L
∗ ;t
. Since kvk
2,1;t
≤ p
t kvk
2,2;t
for each v ∈ L
2,2
[0, t] × O , it follows that L
2,2
[0, t] × O ⊂ L
2,1;t
⊂ L
∗ ;t
, and
kvk
∗ ;t
≤ p
t kvk
2,2;t
, for each v ∈ L
2,2
[0, t] × O . This shows that the condition HD is weaker than HD2.
The Lipschitz condition H is assumed to hold throughtout this paper, except the last section de-
voted to Burgers type equations. The weaker integrability conditions HD and HI are also as-
sumed to hold everywhere in this paper. The other stronger integrability conditions will be men- tioned whenever we will assume them.
2.3 Weak solutions
