Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 27, pp. 1–18.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2024.27

EXISTENCE OF SOLUTIONS TO STOCHASTIC p(t, x)-LAPLACE

EQUATIONS AND APPLICATIONS

CHEN LIANG, LIXU YAN, YONGQIANG FU

Abstract. In this article, we consider a stochastic p(t, x)-Laplace equation.
First we use the Galerkin method toobtain a unique weak solution. Then

we obtain optimal controls for the corresponding stochastic optimal control

problem

1. Introduction

In this article, we consider the stochastic p(t, x)-Laplace equation

du− div(|∇u|p(t,x)−2∇u)dt = f(u)dt+ g(t)dt+ σdW, (t, x) ∈ [0, T ]× Σ

u(t, x) = 0, (t, x) ∈ [0, T ]× ∂Σ

u(0, x) = u0, x ∈ Σ

(1.1)

where Σ ⊂ Rd is a bounded smooth domain, T ∈ (0,+∞), p(t, x) > 1, u0, g
are known functions, f is a continuous accretive operator, u : Rd → RN is a
vector-valued stochastic process, σ is a operator-valued function, {W (t)}t∈[0,T ] is a
E-valued Q-Brownian motion.

Then we consider the corresponding stochastic control system

du− div(|∇u|p(t,x)−2∇u)dt = f(u)dt+ g(t)dt+Avdt+ σdW, t ∈ (0, T ]

u(0, x) = u0

where Av is a control item. The cost function is

J(v) = E
{∫ T

0

∥Hu(v)− µd∥2L2dt+ (Kv, v)V
}

where u(v) is the solution of the stochastic control system, H,K are linear operators,
µd is a fixed stochastic process.

The theory of partial differential equations with variable growth has a wide
range of applications in solving non-standard exponential growth nonlinear prob-
lems. Stochastic partial differential equations have a wide range of applications in
financial mathematics, physics, engineering technology. In recent years, with the

2020 Mathematics Subject Classification. 35K92, 60H15.
Key words and phrases. Stochastic p(t, x)-Laplace equation; weak solution; optimal control;

Galerkin method.
©2024. This work is licensed under a CC BY 4.0 license.

Submitted October 7, 2023. Published March 27, 2024.

1



2 C. LIANG, L. YAN, Y. FU EJDE-2024/27

development of stochastic analysis, stochastic partial differential equations have
developed rapidly.

Ahemd [1] studied the case p(t, x) = 2,

du = D∆udt+ f(x, u)dW, (t, x) ∈ (0, T ]× Σ,

∂u/∂v = 0, (t, x) ∈ (0, T ]× ∂Σ,

u(0, x) = u0(x), x ∈ Σ.

Ahemd proved that there exists a unique weak solution for a stochastic Laplace
equation under suitable assumptions. Then the existence of optimal controls for
the corresponding stochastic optimal control problem was obtained. Different from
Laplace operator, even though p(t, x) ≡ p ̸= 2, p(t, x)-Laplace is a nonlinear op-
erator. Majee [8] studied p-Laplace equations and obtained the existence of weak
solutions under multiplicative noise. Based on the variational calculus and the
convexity of the costing function, the existence of optimal controls for the corre-
sponding stochastic optimal control problems was obtained. Sapountzoglou and
Zimmermann [10, 11] also discussed a stochastic p-Laplace equation and they ob-
tained solutions for the stochastic p-Laplace equation under additive noise and
multiplicative noise.

Zimmermann et al [3] discussed the stochastic p(t, x)-Laplace equation

du− div
(
|∇u|p(t,x)−2∇u

)
dt = h(t, x, u)dW, (t, x) ∈ (0, T )× Σ,

u(t, x) = 0, (t, x) ∈ (0, T ]× ∂Σ,

u(0, x) = u0(x), x ∈ Σ.

By using singular perturbation theory and a fixed point theorem, thy obtained
the existence and uniqueness of solutions for the stochastic p(t, x)-Laplace equa-
tions under additive noise and multiplicative noise. Zimmermann and Vallet [12]
used similar methods to consider stochastic p(ω, t, x)-Laplace equations and got the
corresponding results which are similar to [3].

2. Preliminaries

In this section, we recall some concepts of variable exponent Lebesgue spaces
and Sobolev spaces and some Banach spaces which involve stochastic variables; see
[9, 6] for details.

Let Σ ⊂ Rd be a bounded and smooth domain. p : Σ → [1,+∞) is a continuous
function. Let p+ = supx∈Σ p(x), p

− = infx∈Σ p(x). For each function u, the
modular is

ρp(x)(u) =

∫
Σ

|u(x)|p(x)dx.

The variable exponent Lebesgue space is defined by

Lp(x)(Σ) =
{
u is a measurable function : ρp(x)(u) <∞

}
with the norm

∥u∥Lp(x)(Σ) = inf
{
λ > 0 : ρp(x)

(u
λ

)
⩽ 1

}
.

Then the space Lp(x)(Σ) is a Banach space. Note

min
{
∥u∥p

−

Lp(x)(Σ)
, ∥u∥p

+

Lp(x)(Σ)

}
⩽ ρp(x)(u) ⩽ max

{
∥u∥p

−

Lp(x)(Σ)
, ∥u∥p

+

Lp(x)(Σ)

}
,



EJDE-2024/27 STOCHASTIC p(t, x)-LAPLACE EQUATIONS 3

so norm convergence is equivalent to modular convergence. If the exponent p is

bounded, the conjugate exponent p∗(x) = p(x)
p(x)−1 ; when p(x) = 1 the conjugate

exponent is p∗(x) = ∞. If 1 < p− ⩽ p+ < +∞, Lp(x)(Σ) is a reflexive Banach
space and its dual space is Lp∗(x)(Σ).

Definition 2.1. We call an exponent p : Σ → R a globally log-Hölder continuous
function if p satisfies the following conditions:

(1) There exists a positive constant α1 such that

|p(x)− p(y)| ⩽ α1

log(e + 1/|x− y|)
for all points x, y ∈ Σ;

(2) There exists a positive constant α2 such that

|p(x)− p∞| ⩽ α2

log(e + 1/|x|)
for all points x ∈ Σ.

If the exponent p is globally log-Hölder continuous, C∞
0 (Σ) is dense in Lp(x)(Σ).

The variable exponent Sobolev space is defined by

W 1,p(x)(Σ) =
{
u ∈ Lp(x)(Σ) : ∇u ∈ (Lp(x)(Σ))d

}
with the norm

∥u∥W 1,p(x)(Σ) = ∥u∥Lp(x)(Σ) + ∥∇u∥(Lp(x)(Σ))d .

Note that W 1,p(x)(Σ) is a Banach space. If 1 < p− ⩽ p+ < +∞, W 1,p(x)(Σ) is

reflexive. W
1,p(x)
0 (Σ) is the closure of C∞

0 (Σ) under the norm ∥ · ∥W 1,p(x)(Σ). If the

exponent p is globally log-Hölder continuous, C∞
0 (Σ) is dense in W 1,p(x)(Σ).

Definition 2.2. Let ΣT = (0, T ) × Σ, and p,m : ΣT → (1,+∞) be globally
log-Hölder continuous. X(ΣT ) is defined by

X(ΣT ) =
{
u ∈ Lm(t,x)(ΣT ) : ∇u ∈ (Lp(t,x)(ΣT ))

d,

u(t, x) ∈W
1,p(t,x)
0 (Σ) for a.e. t ∈ [0, T ]

}
.

with the norm

∥u∥X(ΣT ) = ∥u∥Lm(t,x)(ΣT ) + ∥∇u∥(Lp(t,x)(ΣT ))d .

Note thatX(ΣT ) is a reflexive Banach space, and C∞
0 (ΣT ) and C

∞
0 ([0, T ], C∞

0 (Σ))
are dense in X(ΣT ).

For a vector-valued function u = (u1, u2, . . . , uN )T, we can define the space

(Lp(x)(Σ))N =
{
u :

N∑
i=1

∥ui∥Lp(x)(Σ) <∞
}

with the norm

∥u∥(Lp(x)(Σ))N =

N∑
i=1

∥ui∥Lp(x)(Σ).

Similarly, we define the space(
W 1,p(x)(Σ)

)N
=

{
u ∈ (Lp(x)(Σ))N : ∇u ∈ (Lp(x)(Σ))d×N

}



4 C. LIANG, L. YAN, Y. FU EJDE-2024/27

with the norm

∥u∥(W 1,p(x)(Σ))N = ∥u∥(Lp(x)(Σ))N + ∥∇u∥(Lp(x)(Σ))d×N .

Then we have the vector-valued function space

X(ΣT ) =
{
u ∈

(
Lm(t,x)(ΣT )

)N
: ∇u ∈

(
Lp(t,x)(ΣT )

)d×N
,

u(t, x) ∈
(
W

1,p(t,x)
0 (Σ)

)N
a.e. t ∈ [0, T ]

}
with the norm

∥u∥X(ΣT ) = ∥u∥(Lm(t,x)(ΣT ))N + ∥∇u∥(Lp(t,x)(ΣT ))d×N

Next we will recall some Banach spaces which involve stochastic variables. Let
(Ω,F ,P) be a complete probability space with a filtration Ft∈[0,T ]. Let

LF0
2 (Ω, X) =

{
u is F0 adapted X-valued stochastic variable : E∥u∥2X <∞

}
,

LFT
2 (Ω, X) =

{
u is FT adapted X-valued stochastic variable : E∥u∥2X <∞

}
,

L∞(Ω) =
{
ξ is measurable RN -valued stochastic variable :

inf{M : P(|ξ| > M) <∞}
}
,

C1([0, T ], (C∞
0 (Σ))N ) =

{
φ is a continuous function on ΣT : φ(t),

dφ(t)

dt
∈ (C∞

0 (Σ))N
}
.

For each positive constant p ∈ [1,+∞), let

LF
p ([0, T ], X)

=
{
u is a Ft∈[0,T ] adapted stochastic process : E

∫ T

0

∥u(t)∥pXdt <∞
}

with the norm

∥u∥LF
p ([0,T ],X) =

(
E
∫ T

0

∥u(t)∥pXdt
)1/p

.

When p = +∞, we let

LF
∞([0, T ], X)

=
{
u is a Ft∈[0,T ] adapted stochastic process : ess supt∈[0,T ] E∥u(t)∥2X <∞

}
with the norm

∥u∥LF
∞([0,T ],X) = ess supt∈[0,T ]

(
E∥u(t)∥2X

)1/2
.

For any p ∈ [1,+∞], LF
p ([0, T ], X) is a Banach space. When p ∈ (1,+∞),

LF
p ([0, T ], X) is a reflective Banach space. When p < +∞, LF

p ([0, T ], X) is a
separable Banach space.

Next we define the space Lp(x)(Ω× Σ)

Lp(x)(Ω× Σ) =
{
u : E

{∫
Σ

|u|p(x)dx
}
< +∞

}
with the norm

∥u∥Lp(x)(Ω×Σ) = inf
{
λ > 0 : E

{∫
Σ

|u
λ
|p(x)dx

}
< +∞

}
.



EJDE-2024/27 STOCHASTIC p(t, x)-LAPLACE EQUATIONS 5

Note that Lp(x)(Ω× Σ) is a reflective Banach space. Now we define the space

LF
p(t,x)(Ω× ΣT )

=
{
u is a Ft∈[0,T ] adapted stochastic process : E

{∫
ΣT

|u|p(t,x)dxdt
}
< +∞

}
with the norm

∥u∥LF
p(t,x)

(Ω×ΣT ) = inf
{
λ > 0 : E

{∫
ΣT

|u
λ
|p(t,x)dxdt

}
< +∞

}
.

for 1 < p− ⩽ p+ < +∞, LF
p(t,x)(Ω × ΣT ) is a reflective Banach space. The fol-

lowing theorem gives a relation between almost everywhere convergence and weak
convergence in LF

p(t,x)(Ω× ΣT ).

Theorem 2.3 ([5]). Let p be a bounded globally log-Hölder continuous function with
p(t, x) > 1. If {un} is bounded in LF

p(t,x)(Ω× ΣT ) and un → u a.e. (ω, t, x) ∈ Ω×
ΣT , then there exist a subsequence {un}, such that un → u weakly in LF

p(t,x)(Ω×ΣT ).

Similarly, the above spaces can be extended to the case of vector-valued function
spaces. Hence we introduce the space LF (Ω, X(ΣT )).

Definition 2.4.

LF (Ω, X(ΣT )) =
{
u ∈ (LF

m(t,x)(Ω× ΣT ))
N ,∇u ∈ (LF

p(t,x)(Ω× ΣT ))
d×N ,

u(ω, t, x) ∈ X(ΣT ), a.e. ω ∈ Ω
}

with the norm

∥u∥LF (Ω,X(ΣT )) = ∥u∥(LF
m(t,x)

(Ω×ΣT ))N + ∥∇u∥(LF
p(t,x)

(Ω×ΣT ))d×N

Note that LF (Ω, X(ΣT )) is a reflective Banach space. In this article we set
m(t, x) = 2. Let E be a separable Hilbert space.

Theorem 2.5 ([7]). Let Q ∈ L (E) be a symmetric nonnegative operator, TrQ <
∞. B is an E-valued Q-Wiener process. For each t ∈ [0, T ], y ∈ E we have:

(1) B is E-valued Gauss process and

E(B(t), y)E = 0, E(B(t), y)2E = t(Qy, y).

(2) B has the expression

B(t) =

∞∑
j=1

√
λjβj(t)ej , (2.1)

where {ei}∞i=1 is an orthonormal basis of E, {λi}∞i=1 is the sequence of
eigenvalues of Q. βi(t) is a sequence of Brownian motions which are in-
dependent from each other on probability space

(
Ω,F ,P,Ft∈[0,T ]

)
. The

series converges strongly to B in LF
2 (Ω, C([0, T ], E)).

(3) Let O be a separable Hilbert space. If σ(t) ∈ L (E,O) (t ∈ [0, T ]), then∫ T

0

σ(s)dB(s) =

∞∑
j=1

√
λj

∫ T

0

σ(s)(ej)dβj(s). (2.2)

The series converges strongly to
∫ T

0
σ(s)dB(s) in LF

2 (Ω, C([0, T ], O)).

Finally we recall the Crandal-Liggett theorem.



6 C. LIANG, L. YAN, Y. FU EJDE-2024/27

Theorem 2.6 ([2]). Let Y be a Banach space, L is a m-accretive operator, ∆n is

a partition of [0, T ], and |∆n| → 0 as n → ∞. If u0 ∈ D(L), then there exists a
u ∈ C ([0, T ],Y ) and a nonlinear operator semigroup {T (t)}t⩾0, such that

u(t) = T (t)u0.

If un is the implicit interpolation approximation of u, then

∥un(t)− u(t)∥Y → 0, as n→ ∞

uniformly on [0, T ].

We denote by ∥ · ∥L2 the norm (L2(Σ))N ; denote by ∥ · ∥L2q(x) the norm of
(L2q(x)(Σ))N ; denote by ∥ · ∥L2(Ω×ΣT ) the norm of (LF

2 (Ω × ΣT ))
N ; denote by

∥ · ∥Lp(t,x)(Ω×ΣT ) the norm of (LF
p(t,x) (Ω× ΣT ))

d×N
; denote by (·, ·)L2 the product

of (L2(Σ))N .

3. Existence and uniqueness of weak solutions

Let E be a separable Hilbert space, σ(t) be a bounded linear operator from E
to (L2(Σ))N and

∥σ(t)∥ ⩽M, ∀t ∈ [0, T ]. (3.1)

Let Q ∈ L (E) be a symmetric nonnegative operator, {W (t)}t∈[0,T ] be a E-valued

Q-Brown motion defined on
(
Ω,F ,P,Ft∈[0,T ]

)
.

Fix ω ∈ Ω, t ∈ [0, T ], f is a continuous accretive operator from
(
L2q(x)(Σ)

)N
to(

(L2q(x)(Σ))∗
)N

, where q(x) is continuous and q(x) ⩾ 1. Additionally, f satisfies
the following conditions:

(H1) There exist c1 ∈ [0,+∞) and c2 ∈ (0,+∞), such that

⟨f(u), u⟩(L2q(x))∗,L2q(x) ⩽ c1∥u∥2L2 − c2∥u∥2L2q(x) .

(H2) f(u) with respect to u is a completely continuous operator from the space

LF
2

(
[0, T ], (L2q(x)(Σ))N

)
to

(
LF
2

(
[0, T ], (L2q(x)(Σ))N

))∗
.

(H3) For each u, v ∈ (L2q(x)(Σ))N ,

⟨f(u)− f(v), u− v⟩(L2q(x))∗,L2q(x) ⩽ 0.

Next we give the concept of weak solutions for system (1.1).

Definition 3.1. An RN -valued stochastic process

u ∈ LF
∞

(
[0, T ], (L2(Σ))N

)
∩ LF (Ω, X(ΣT )) ∩ LF

2

(
[0, T ], (L2q(x)(Σ))N

)
is a weak solution of (1.1), if for each φ ∈ C1([0, T ], (C∞

0 (Σ))N ), u satisfies

(u(T ), φ(T ))L2 − (u0, φ(0))L2 −
∫ T

0

(
u(t),

dφ

dt

)
L2dt

+

∫ T

0

∫
Σ

|∇u|p(t,x)−2∇u∇φdxdt

=

∫ T

0

⟨f(u(t)), φ⟩(L2q(x))∗,L2q(x)dt+

∫ T

0

(g(t), φ(t))L2dt

+

∫ T

0

(φ(t), σ(t)dW (t))L2

(3.2)



EJDE-2024/27 STOCHASTIC p(t, x)-LAPLACE EQUATIONS 7

Under the above conditions, we use Galërkin’s method to prove that equation
(1.1) admits a unique weak solution. The main result of this section reads as follows.

Theorem 3.2. Let p(t, x) be a bounded globally log-Hölder continuous function and
p(t, x) > 1. If (H1)–(H3) and (3.1) hold, then equation (1.1) has a unique weak
solution.

u ∈ LF
∞

(
[0, T ], (L2(Σ))N

)
∩ LF (Ω, X(ΣT )) ∩ LF

2

(
[0, T ], (L2q(x)(Σ))N

)
for any u0 ∈ LF0

2

(
Ω, (L2(Σ))N

)
and g ∈ LF

2

(
[0, T ], (L2(Σ))N

)
.

Proof. This proof is divided into four steps.

Step 1: Uniqueness of a weak solution. Assume that two solutions satisfy

u, v ∈ LF
∞

(
[0, T ], (L2(Σ))N

)
∩ LF (Ω, X(ΣT )) ∩ LF

2

(
[0, T ], (L2q(x)(Σ))N

)
with initial states u0, v0 and g1, g2. Since u, v satisfy system (1.1) in the weak sense,
by integrating by parts, we deduce that

1

2
∥u− v∥2L2 +

∫ t

0

∫
Σ

(
|∇u|p(s,x)−2∇u− |∇v|p(s,x)−2∇v

)
(∇u−∇v)dxds

=
1

2
∥u0 − v0∥2L2 +

∫ t

0

⟨f(u)− f(v), u− v⟩(L2q(x))∗,L2q(x)ds

+

∫ t

0

(g1 − g2, u− v)L2 ds+

∫ t

0

(u− v, σdW )L2 .

As ∫ t

0

∫
Σ

(
|∇u|p(s,x)−2∇u−∇v|p(s,x)−2∇v

)
(∇u−∇v)dxds ⩾ 0,

by (H2), we have

1

2
∥u− v∥2L2 ⩽

1

2
∥u0 − v0∥2L2 +

∫ t

0

(g1 − g2, u− v)L2 ds+

∫ t

0

(u− v, σdW )L2

and further after taking the expectation we have

1

2
E∥u− v∥2L2 ⩽

1

2
E∥u0 − v0∥2L2 + E

∫ t

0

(g1 − g2, u− v)L2 ds.

When u0 = v0 and g1 = g2, we deduce u = v.

Step 2: Existence of solutions for finite dimensional truncated systems. We choose

an orthonormal basis {ei}∞i=1, such that {ei}∞i=1 ⊂ (C∞
0 (Σ))N ⊂

⋃∞
n=1 Vn

(C1(Σ))N

,
where Vn = span{e1, e2, . . . , en}. Let {ei} be an orthonormal basis of E, Wn is an
n-dimensional Brown motion. We consider the truncation of system (1.1):

dun − div
(
|∇un|p(t,x)−2∇un

)
dt = f(un)dt+ gn(t)dt+ σdWn, (t, x) ∈ (0, T ]× Σ,

un(t, x) = 0, (t, x) ∈ [0, T ]× ∂Σ,

un(0, x) =

n∑
j=1

(u0, ej)L2ej , x ∈ Σ,

(3.3)



8 C. LIANG, L. YAN, Y. FU EJDE-2024/27

where

un(t) =

n∑
j=1

θjn(t)ej , un(0) =

n∑
j=1

(u0, ej)L2ej =

n∑
j=1

θjn(0)ej ,

gn(t) =

n∑
j=1

(g(t), ej)L2ej , Wn(t) =

n∑
j=1

(W (t), ej)Eej .

{θjn(t)} are unknown functions. Let L,F and G be n-dimensional vectors, A is a
n× n matrix, whose entries are

Li(θ) ≜
∫
Σ

∣∣∣ n∑
j=1

θjn∇ej
∣∣∣p(t,x)−2( n∑

j=1

θjn∇ej
)
∇eidx,

Fi(θn) ≜
〈
f(

n∑
j=1

θjnej), ei
〉
(L2q(x))∗,L2q(x) ,

Gi(t) ≜
n∑

j=1

(g(t), ej)L2 ei, aij(t) ≜
√
λj(ei, σēj)L2 ,

where 1 ⩽ i, j ⩽ n. We consider the n dimensional stochastic system

dθn = Lθndt+ F (θn)dt+Gdt+AdWn, t ∈ [0, T ] (3.4)

We claim F is a m-accretive operator on RN . On one hand, because f is accretive,
we obtain F is a accretive operator. On the other hand, F is continuous, according
to [4, Appendix D Corollary D.10], we can obtain F is m-accretive. Let ∆k =

{
0 =

t0k < t1k <, . . . , < tkk = T
}
be the kth uniform partition of [0, T ], denote δk ≜ |Πk|,

the sequence of approximate solutions {θn,k(t)} is given by

θn,k
(
tik
)
= (I − δkF )

−1 [
θn,k

(
ti−1
k

)
+ δkLθn,k

(
ti−1
k

)
+ δkG

(
ti−1
k

)
+A

(
ti−1
k

) (
Bn

(
tik
)
−Bn

(
ti−1
k

)) ]
,

(3.5)

where i = 1, 2, . . . , k. By Theorem 2.6, there exists θn ∈ CF ([0, T ],Rn), such that

θn,k(t) → θn(t) strongly in Rn,

uniformly on [0, T ]. Thus

θn =
(
θ1n, θ

2
n, . . . , θ

n
n

)T
is a solution of (3.4), so un =

∑n
j=1 θ

j
nej is a solution of the system (3.3).

Step 3: A priori estimate. Integrating by parts on (3.3) we obtain

E∥un(t)∥2L2 + E
∫ t

0

∫
Σ

|∇un|p(s,x)dxds+ 2C2E
∫ t

0

∥un(s)∥2L2q(x)ds

⩽ E∥un(0)∥2L2 + CE
∫ t

0

∥un(s)∥2L2ds+ 2CεE
∫ t

0

∥gn(s)∥2L2ds

. (3.6)

Since u0 ∈ LF0
2

(
Ω, (L2(Σ))N

)
, g ∈ LF

2

(
[0, T ], (L2(Σ))N

)
,

E∥un(0)∥2L2 = E
∥∥ n∑

j=1

(u0, ej)L2ej
∥∥2
L2

= E
( n∑

j=1

|(u0, ej)L2 |2
)



EJDE-2024/27 STOCHASTIC p(t, x)-LAPLACE EQUATIONS 9

⩽ E
( ∞∑

j=1

|(u0, ej)L2 |2
)

= E∥u0∥2L2 = ∥u0∥LF0
2 (Ω,(L2(Σ))N )

,

E
∫ T

0

∥gn(t)∥2L2dt = E
∫ T

0

∥∥ n∑
j=1

(g(t), ej)L2ej
∥∥2
L2dt

= E
∫ T

0

( n∑
j=1

|(g(t), ej)L2 |2
)
dt

⩽
∫ T

0

E
( ∞∑

j=1

|(g(t), ej)L2 |2
)
dt

= E
∫ T

0

∥g(t)∥2L2dt

= ∥g∥2LF
2 ([0,T ],(L2(Σ))N ),

It follows that the first term and third term on the right=hand side of (3.3) are
bounded. Then by Gronwall’s inequality, we obtain

E∥un(t)∥2L2 ⩽ C, (3.7)

where C = C
(
∥u0∥LF0

2 (Ω,L2)
, ∥g∥LF

2 ([0,T ],L2), T, ε, c1, c2
)
. As T is a fixed positive

number,

E
∫ T

0

∥un(t)∥2L2dt ⩽ C.

By (3.6), we arrive at

∥∇un∥Lp(t,x)(Ω×ΣT ) ⩽ C, E
∫ T

0

∥un(t)∥2L2q(x)dt ⩽ C.

Hence, {un} is bounded in LF
∞

(
[0, T ], (L2(Σ))N

)
∩LF (Ω, X(ΣT ))∩LF

2

(
[0, T ], (L2q(x)(Σ))N

)
.

By Eberlein-Smulian theorem and Alaoglu theorem, there exists a subsequence (still
denoted by {un}) and a stochastic process u such that

un → u weakly ∗ in LF
∞

(
[0, T ], (L2(Σ))N

)
, (3.8)

un → u weakly in LF (Ω, X(ΣT )) , (3.9)

un → u weakly in LF
2

(
[0, T ], (L2q(x)(Σ))N

)
, (3.10)

Step 4: Limit process. We prove u is a weak solution of (1.1) by showing u satisfies
(3.2). For any φ ∈ C1

(
[0, T ], (C∞

0 (Σ))N
)
and ξ ∈ L∞(Ω), from (3.3) we obtain

0 = E{ξ(un(0), φ(0))L2} − E{ξ(un(T ), φ(T ))L2}+ E
{
ξ

∫ T

0

(
un(t),

dφ

dt

)
L2dt

}
− E

{
ξ

∫ T

0

∫
Σ

|∇un|p(t,x)−2∇un∇φdxdt
}

+ E
{
ξ

∫ T

0

⟨f(un(t)), φ⟩(L2q(x))∗,L2q(x)dt
}
+ E

{
ξ

∫ T

0

(gn(t), φ)L2dt
}



10 C. LIANG, L. YAN, Y. FU EJDE-2024/27

+ E
{
ξ

∫ T

0

(φ, σ(t)dWn(t))L2

}
= I1 − I2 + I3 − I4 + I5 + I6 + I7.

Next we analyze the limits of I1, . . . , I7.
(1) Consider I1. Noting un(0) is the n-dimensional truncation of u(0), we obtain

∥u0 − un(0)∥2L2 =
∥∥ ∞∑

j=n+1

(u0, ej)L2ej
∥∥2
L2

=
( ∞∑

j=n+1

|(u0, ej)L2 |2
)

⩽
( ∞∑

j=1

|(u0, ej)L2 |2
)

= ∥u0∥2L2 .

Since

∥un(0)− u(0)∥2L2 → 0 as n→ ∞,

by using dominated convergence Theorem, we obtain

E∥un(0)− u(0)∥2L2 → 0 as n→ ∞,

that is

un(0) → u0 strongly in LF0
2

(
Ω, (L2(Σ))N

)
.

Since φ(0) ∈ (C1(Σ)N ) ⊂ LF0
2

(
Ω, (L2(Σ))N

)
, we derive that

E
{
ξ(un(0), φ(0))L2

}
→ E

{
ξ(u0, φ(0))L2

}
as n→ ∞. (3.11)

(2) Similarly, for I6, we obtain

gn → g strongly in LF
2

(
[0, T ], (L2(Σ))N

)
.

In view of φ ∈ C1
(
[0, T ], (C∞

0 (Σ))N
)
⊂ LF

2 ([0, T ],
(
L2(Σ))N

)
, we obtain

E
{
ξ

∫ T

0

(gn(t), φ(t))L2dt
}
→ E

{
ξ

∫ T

0

(g(t), φ(t))L2dt
}

as n→ ∞. (3.12)

(3) Consider I3. By

un → u weakly∗ in LF
∞

(
[0, T ], (L2(Σ))N

)
and dφ

dt ∈ C
(
[0, T ], (C∞

0 (Σ))N
)
⊂ LF

1

(
[0, T ], (L2(Σ))N

)
, we obtain

E
{
ξ

∫ T

0

(
un(t),

dφ

dt

)
L2dt

}
→ E

{
ξ

∫ T

0

(
u(t),

dφ

dt

)
L2dt

}
(3.13)

as n→ ∞.
(4) Consider I5. Because

un → u weakly in LF (Ω, X(ΣT )) ,

By H(2), we know that

f(un) → f(u) strongly in
(
LF
2

(
[0, T ], (L2q(x)(Σ))N

))∗
.



EJDE-2024/27 STOCHASTIC p(t, x)-LAPLACE EQUATIONS 11

Since φ ∈ C1
(
[0, T ], (C∞

0 (Σ))N
)
⊂ LF

2

(
[0, T ], (L2q(x)(Σ))N

)
, thus we obtain

E
{
ξ

∫ T

0

⟨f(un(t)), φ⟩(L2q(x))∗,L2q(x)dt
}
→ E

{
ξ

∫ T

0

⟨f(u(t)), φ⟩(L2q(x))∗,L2q(x)dt
}

(3.14)
as n→ ∞.

(5) Consider I7. According to Theorem 2.5, we know that

Wn →W strongly in LF
2 (Ω, C([0, T ], E)) .

As

E
∫ T

0

(φ(t), σdWn(t))L2 = E
∫ T

0

(
φ(t),

n∑
i=1

√
λiσ(ēi)dβi(t)

)
L2

and φ ∈ C1
(
[0, T ], (C∞

0 (Σ))N
)
is a deterministic function, by Theorem 2.5(3), we

obtain

E
{
ξ

∫ T

0

(φ(t), σdWn(t))
}
→ E

{
ξ

∫ T

0

(φ(t), σdW (t))
}

as n→ ∞. (3.15)

(6) Consider I4 and I2. Since

E
{∫

ΣT

||∇un|p(t,x)−2∇un|p
∗(t,x)dxdt

}
= E

{∫
ΣT

|∇un|p(t,x)dxdt
}
⩽ C,

there exists a subsequence (still denoted by {un}) and a stochastic process η, such
that

|∇un|p(t,x)−2∇un → η weakly in
(
LF
Lp∗(t,x)(Ω× ΣT )

)d×N
,

and further

E
{
ξ

∫ T

0

∫
Σ

|∇un|p(t,x)−2∇un∇φdxdt
}
→ E

{
ξ

∫ T

0

∫
Σ

η∇φdxdt
}

(3.16)

as n→ ∞. In view of (3.7), we obtain

E
{
∥un(T )∥2L2

}
⩽ C,

therefore there exists a function û ∈ LFT
2

(
Ω, (L2(Σ))N

)
such that

un(T ) → û weakly in LFT
2

(
Ω, (L2(Σ))N

)
.

Now we prove u(T ) = û. For any ψ ∈ (C∞
0 (Σ))

N
and any ϕ ∈

(
C1[0, T ]

)N
, we

have

0 = −(un(T ), ψϕ(T ))L2 + (un(0), ψϕ(0))L2 +

∫ T

0

(
un(t), ψ

dϕ

dt

)
L2

dt

−
∫ T

0

∫
Σ

|∇un|p(t,x)−2∇un∇ϕψdxdt+
∫ T

0

⟨f(un(t)), ψϕ⟩(L2q(x))∗,L2q(x))dt

+

∫ T

0

(gn(t), ψϕ)L2dt+

∫ T

0

(ϕψ, σdWn(t))L2 .



12 C. LIANG, L. YAN, Y. FU EJDE-2024/27

Letting n→ ∞, we obtain

0 = −(û, ψϕ(T ))L2 + (u0, ψϕ(0))L2 +

∫ T

0

(
u(t), ψ

dϕ

dt

)
L2dt

−
∫ T

0

∫
Σ

η∇ϕψdxdt+
∫ T

0

⟨f(u(t)), ψϕ⟩(L2q(x))∗,L2q(x)dt

+

∫ T

0

(g(t), ψϕ)L2dt+

∫ T

0

(ϕψ, σdW (t))L2

(3.17)

for any ϕ ∈
(
C1

0 ([0, T ])
)N ⊂

(
C1[0, T ]

)N
and further

0 =

∫ T

0

(
u(t), ψ

dϕ

dt

)
L2dt−

∫ T

0

∫
Σ

η∇ϕψdxdt

+

∫ T

0

⟨f(u(t)), ψϕ⟩(L2q(x))∗,L2q(x)dt+

∫ T

0

(g(t), ψϕ)L2dt+

∫ T

0

(ϕψ, σdW (t))L2 .

A density argument and the definition of derivatives with respect to time in the
distributional sense imply

0 =

∫ T

0

(
u(t),

dφ

dt

)
L2
dt−

∫ T

0

∫
Σ

η∇φdxdt

+

∫ T

0

⟨f(u(t)), φ⟩(L2q(x))∗,L2q(x)dt+

∫ T

0

(g(t), φ)L2dt+

∫ T

0

(φ, σdW (t))L2 .

For each φ ∈ (C∞
0 (ΣT ))

N
, du

dt satisfies

0 = −
∫ T

0

(du
dt
, φ

)
L2dt−

∫ T

0

∫
Σ

η∇φdxdt

+

∫ T

0

⟨f(u(t)), φ⟩(L2q(x))∗,L2q(x)dt+

∫ T

0

(g(t), φ)L2dt

+

∫ T

0

(φ, σdW (t))L2 .

Then we obtain∫ T

0

(
du

dt
, φ

)
L2

dt =

∫ T

0

∫
Σ

div ηφdxdt+

∫ T

0

⟨f(u(t)), φ⟩(L2q(x))∗,L2q(x)dt

+

∫ T

0

(g(t), φ)L2dt+

∫ T

0

(φ, σdW (t))L2

≜ ⟨S, φ⟩.

Furthermore, for any ψ ∈ (C∞
0 (Σ))

N
and any ϕ ∈

(
C1[0, T ]

)N
, we obtain

−
∫ T

0

(
u(t), ψ

dϕ

dt

)
L2

dt+

∫ T

0

∫
Σ

η∇ϕψdxdt−
∫ T

0

⟨f(u(t)), ψϕ⟩(L2q(x))∗,L2q(x)dt

−
∫ T

0

(g(t), ψϕ)L2dt−
∫ T

0

(ϕψ, σdW (t))L2

= −
∫ T

0

(
u(t), ψ

dϕ

dt

)
L2dt− ⟨S, ϕψ⟩



EJDE-2024/27 STOCHASTIC p(t, x)-LAPLACE EQUATIONS 13

= −
∫ T

0

(
u(t), ψ

dϕ

dt

)
L2dt−

∫ T

0

(du
dt
, ψϕ

)
L2dt

= (u(0), ϕψ(0))L2 − (u(T ), ϕψ(T ))L2 .

In view of (3.17), we obtain u(T ) = û and

un(T ) → u(T ) weakly in LFT
2

(
Ω, (L2(Σ))N

)
.

By the weak lower semi-continuity of the norm, we obtain

lim inf
n→∞

(
E∥un(T )∥2L2

)
⩽ E∥u(T )∥2L2 . (3.18)

Since φ(T ) ∈
(
C1(Σ)N

)
⊂ LFT

2

(
Ω, (L2(Σ))N

)
,

E
{
ξ(un(T ), φ(T ))L2

}
→ E

{
ξ(u(T ), φ(T ))L2

}
as n→ ∞. (3.19)

Combining (3.11), (3.13), (3.14), (3.12), (3.15), (3.16), and (3.19), we have

0 = (u0, φ(0))L2 − (u(T ), φ(T ))L2 +

∫ T

0

(
u(t),

dφ

dt

)
L2dt

−
∫ T

0

∫
Σ

η∇φdxdt+
∫ T

0

⟨f(u(t)), φ⟩(L2q(x))∗,L2q(x)dt

+

∫ T

0

(gn(t), φ)L2dt+

∫ T

0

(φ, σ(t)dWn(t))L2 .

(3.20)

Next we prove that η = |∇u|p(t,x)−2∇u. By (3.20), we know that u is a weak
solution of the problem

du− div ηdt = f(u)dt+ g(t)dt+ σdW, (t, x) ∈ [0, T ]× Σ,

u(t, x) = 0, (t, x) ∈ [0, T ]× ∂Σ,

u(0, x) = u0, x ∈ Σ.

(3.21)

Integrating by parts, on (3.21) we obtain

0 =
1

2
∥u0∥2L2 −

1

2
∥u(T )∥2L2 −

∫ T

0

∫
Σ

η∇udxdt

+

∫ T

0

⟨f(u(t)), u⟩(L2q(x))∗,L2q(x)dt+

∫ T

0

(g(t), u)L2dt

+

∫ T

0

(u, σ(t)dW (t))L2 .

(3.22)

From

0 ⩽ E
{∫

ΣT

(
|∇un|p(t,x)−2∇un − |∇u|p(t,x)−2∇u

)
(∇un −∇u)dxdt

}
=

1

2
E
{
∥un(0)∥2L2

}
− 1

2
E
{
∥un(T )∥2L2

}
+ E

{∫ T

0

⟨f(un(t)), un(t)⟩(L2q(x))∗,L2q(x)dt
}

+ E
{∫ T

0

(gn(t), un(t))L2 dt
}
+ E

{∫ T

0

(un(t), σ(t)dWn(t))L2

}
+ E

{∫
ΣT

|∇un|p(t,x)−2∇un∇u− |∇u|p(t,x)−2∇u(∇un −∇u)dxdt
}
,



14 C. LIANG, L. YAN, Y. FU EJDE-2024/27

and (3.16) and (3.20), we have

0 ⩽
1

2
E
{
∥u0∥2L2

}
− 1

2
E
{
∥u(T )∥2L2

}
+ E

{∫ T

0

⟨f(u(t)), u(t)⟩(L2q(x))∗,L2q(x)dt
}

+ E
{∫ T

0

(g(t), u(t))L2 dt
}
+ E

{∫ T

0

(u(t), σ(t)dW (t))L2

}
+ E

{∫
ΣT

η∇udxdt
}
= 0.

Furthermore,

E
{∫

ΣT

(
|∇un|p(t,x)−2∇un − |∇u|p(t,x)−2∇u

)
(∇un −∇u) dxdt

}
→ 0

as n→ +∞,

E
{∫

ΣT

|∇un −∇u|p(t,x)dxdt
}

⩽ CE
{∫

ΣT

(
|∇un|p(t,x)−2∇un − |∇u|p(t,x)−2∇u

)
(∇un −∇u) dxdt

}
→ 0 as n→ +∞.

Therefore,

∇un → ∇u strongly in
(
LF
Lp(t,x)(Ω× ΣT )

)d×N
.

Thus there exists a subsequence (still denoted by {un}) such that

∇un → ∇u, a.e.(ω, t, x) ∈ Ω× ΣT .

Furthermore,

|∇un|p(t,x)−2∇un → |∇u|p(t,x)−2∇u, a.e. (ω, t, x) ∈ Ω× ΣT

By Theorem 2.5, we obtain η = |∇u|p(t,x)−2∇u.
In summary, for each φ ∈ C1

(
[0, T ], (C∞

0 (Σ))N
)
, we obtain

(u(T )), φ(T ))L2 − (u0, φ(0))L2 −
∫ T

0

(
u,

dφ

dt

)
L2dt

+

∫ T

0

∫
Σ

|∇u|p(t,x)−2∇u∇φdxdt

=

∫ T

0

⟨f(u), φ⟩(L2q(x))∗,L2q(x)dt+

∫ T

0

(g(t), φ)L2dt+

∫ T

0

(φ, σ(t)dW (t))L2 ,

i.e., u is a solution of (1.1). □

4. Existence of optimal controls

For a real Hilbert space V , the set V = LF
∞ ([0, T ], V ) is the control function

space, and the linear operator A ∈ L
(
V , LF

2

(
[0, T ], (L2(Σ))N

))
is the control

item. We consider the stochastic control problem

du− div
(
|∇u|p(t,x)−2∇u

)
dt = f(u)dt+ g(t)dt+Avdt+ σdW, t ∈ (0, T ],

u(0, x) = u0.

(4.1)



EJDE-2024/27 STOCHASTIC p(t, x)-LAPLACE EQUATIONS 15

Define the solution map as follows:

Φ : v → u(v)

LF
∞([0, T ], V ) → LF

∞
(
[0, T ], (L2(Σ))N

)
∩ LF (Ω, X(ΣT )) ∩ LF

2

(
[0, T ], (L2q(x)(Σ))N

)
,

where u(v) is the solution of (4.1) called the state of the control problem (4.1). The
observed state is denoted by H(u(v)) where

H ∈ L
(
LF (Ω, X(ΣT )), L

F
2

(
[0, T ], (L2(Σ))N

))
is a linear operator. A fixed stochastic process µd ∈ LF

2

(
[0, T ], (L2(Σ))N

)
is called

the desired state. The cost function is defined as

J(v) = E
{∫ T

0

∥Hu(v)− µd∥2L2dt+ (Kv, v)V
}
, (4.2)

where the operator K ∈ L (V, V ) satisfies

(Kv(t), v(t))V = (v(t),Kv(t))V ⩾ k∥v(t)∥2V
for k ∈ [0,+∞). Let Vad ⊂ V be an admissible set. We call v0 ∈ Vad the optimal
control if

J(v0) = min
v∈Vad

J(v).

Thus, we have the following result.

Theorem 4.1. Let the assumptions in Theorem 3.2 be satisfied and let Vad is a
compact subset of V . Then stochastic control problem (4.1) with cost function (4.2)
has at least one optimal control v0 ∈ Vad.

Proof. Since Vad is compact, we need only to prove that Φ is continuous and J is
lower semi-continuous. Let {vk} ∈ Vad and

vk → v̄ in Vad.

Step 1: Φ is continuous. Suppose that {uk} and ū are weak solutions of (4.1).
Then uk − ū satisfies

duk − dū+ div
(
|∇ū|p(t,x)−2∇ū

)
dt− div

(
|∇uk|p(t,x)−2∇uk

)
dt

= f(uk)dt− f(ū)dt+Avkdt−Av̄dt

in the weak sense. After integrating by parts, we obtain

1

2
E∥uk − ū∥2L2 + E

∫ t

0

∫
Σ

(
|∇u|p(t,x)−2∇u− |∇ū|p(t,x)−2∇ū

)
(∇u−∇ū) dxds

= E
∫ t

0

⟨f(uk)− f(ū), uk − ū⟩(L2q(x))∗,L2q(x)ds+ E
∫ t

0

(Avk −Av̄, uk − ū)L2ds

By (H3) and Hölder’s inequality, we have

1

2
E∥uk − ū∥2L2 + E

∫ t

0

∫
Σ

(
|∇u|p(s,x)−2∇u− |∇ū|p(s,x)−2∇ū

)
(∇u−∇ū) dxds

⩽
(
E
∫ t

0

∥Avk −Av̄∥2L2ds
)1/2(

E
∫ t

0

∥uk − ū∥2L2ds
)1/2

.



16 C. LIANG, L. YAN, Y. FU EJDE-2024/27

Since {uk} is bounded in LF
∞

(
[0, T ], (L2(Σ))N

)
, there exists a constant M > 0,

such that

max
{
∥ū∥LF

∞([0,T ],(L2(Σ))N ), ∥u1∥LF
∞([0,T ],(L2(Σ))N ),

. . . , ∥uk∥LF
∞([0,T ],(L2(Σ))N ), . . .

}
⩽M.

Furthermore,

E
(∫ t

0

∥uk − ū∥2L2ds
)
⩽ C.

Hence we have

1

2
E∥uk − ū∥2L2 + E

∫ t

0

∫
Σ

(
|∇u|p(t,x)−2∇u− |∇ū|p(t,x)−2∇ū

)
(∇u−∇ū) dxds

⩽ C
(
E
∫ t

0

∥Avk −Av̄∥2L2ds
)1/2

.

Taking the limit, we obtain

lim
k→∞

sup
t∈[0,T ]

{1
2
E∥uk − ū∥2L2

+ E
∫ t

0

∫
Σ

(
|∇u|p(t,x)−2∇u− |∇ū|p(t,x)−2∇ū

)
(∇u−∇ū) dxds

}
= 0

which implies

uk → ū strongly in LF
∞

(
[0, T ], (L2(Σ))N

)
∩ LF (Ω, X(ΣT )).

Step 2: J is lower semi-continuous. We deenote

J(v) = E
{∫ T

0

∥Hu(v)− µd(t)∥2L2dt
}
+ E

{
(Kv(t), v(t))V

}
≜ J1(v) + J2(v).

As

J1(v̄) = E
{∫ T

0

∥Hū(v)− µd(t)∥2L2 − ∥Huk(v)− µd(t)∥2L2dt
}

+ E
{∫ T

0

∥Huk(v)− µd(t)∥2L2dt
}

and H ∈ L
(
LF (Ω, X(ΣT )), (L

2(Σ))N
)
, for any ε > 0, there exists Nε ∈ N such

that ∣∣∥Hū(v)− µd(t)∥2L2 − ∥Huk(v)− µd(t)∥2L2

∣∣ < ε

whenever k > Nε. Furthermore,

J1(v̄) ⩽ Tε+ E
{∫ T

0

∥Huk(v)− µd(t)∥2L2dt
}
= Tε+ J1(vk).

So we arrive at

J1(v̄) ⩽ lim inf
k→∞

J1(vk)

by the arbitrariness of ε.
From the convergence vk → v̄ in V , we derive that

vk(t) → v̄(t) a.e. t ∈ [0, T ]



EJDE-2024/27 STOCHASTIC p(t, x)-LAPLACE EQUATIONS 17

so {vk(t)} is bounded in V , which implies

(Kvk(t), vk(t))V ⩽ ∥K∥ ∥vk(t)∥2V < +∞.

By Fatou’s Lemma, we obtain

lim inf
k→∞

{
E (Kvk(t), vk(t))V

}
⩽

{
E lim inf

k→∞
(Kvk(t), vk(t))V

}
=

{
E lim inf

k→∞
(Kv̄(t), v̄(t))V

}
.

Furthermore,

J2(v̄) ⩽ lim inf
k→∞

J2(vk).

At last J(v̄) ⩽ lim infk→∞ J(vk). □

Acknowledgements. Lixu Yan was supported by the Fundamental Research Funds
for the Central Universities (No. 2572022BC06). The authors would like to thank
the referees for their helpful comments and suggestions.

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Chen Liang
School of Mathematics, Harbin Institute of Technology, Harbin, 150001, Heilongjiang,

China
Email address: liangchen3515@163.com

Lixu Yan
Department of Mathematics, Northeast Forestry University, Harbin, 150040, Heilongjiang,

China
Email address: yanlxmath@163.com



18 C. LIANG, L. YAN, Y. FU EJDE-2024/27

Yongqiang Fu (corresponding author)

School of Mathematics, Harbin Institute of Technology, Harbin, 150001, Heilongjiang,

China
Email address: fuyongqiang@hit.edu.cn


	1. Introduction
	2. Preliminaries
	3. Existence and uniqueness of weak solutions
	4. Existence of optimal controls
	Acknowledgements

	References