Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 09, pp. 1–13.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE OF GLOBAL WEAK SOLUTIONS FOR A

p-LAPLACIAN INEQUALITY WITH STRONG DISSIPATION IN

NONCYLINDRICAL DOMAINS

JORGE FERREIRA, ERHAN PIŞKIN, MOHAMMAD SHAHROUZI,
SEBASTIÃO CORDEIRO, CARLOS ALBERTO RAPOSO

Abstract. In this work, we obtain global solutions for nonlinear inequalities
of p-Laplacian type in noncylindrical domains, for the unilateral problem with

strong dissipation

u′′ −∆pu−∆u′ − f ≥ 0 in Q0,

where ∆p is the nonlinear p-Laplacian operator with 2 ≤ p < ∞, and Q0 is

the noncylindrical domain. Our proof is based on a penalty argument by J. L.
Lions and Faedo-Galerkin approximations.

1. Introduction

Let Ω ⊂ Rn be an open and bounded set with smooth boundary Γ, T be a
positive real, fixed but arbitrary, and Q0 = Ω × (0, T ) be the cylinder with side
border Σ0 = Γ0 × (0, T ). J. L. Lions [11] considered the problem

u′′ −∆u− f ≥ 0 in Q0,

u′ ≥ 0 in Q0,

u = 0 on Σ0,

u(0) = u0, u′(0) = u1 in Ω.

(1.1)

If K = {v ∈ H1
0 (Ω); v(x) ≥ 0 a.e. on Ω}, then (1.1) can be reformulated as

〈u′′(t), v − u′(t)〉+ 〈−∆u(t), v − u′(t)〉 ≥ 〈f(t), v − u′(t)〉, ∀v ∈ K,
u′(t) ∈ K a.e.,

u(0) = u0, u′(0) = u1.

(1.2)

We consider the p-Laplacian operator ∆pu = div(|∇u|p−2∇u), which can be
extended to a monotone, bounded, hemicontinuos and coercive operator between
the spaces W 1,p

0 (Ω) and its dual by

−∆p : W 1,p
0 (Ω)→W−1,q(Ω), 〈−∆pu, v〉p =

∫
Ω

|∇u|p−2∇u · ∇v dx.

2020 Mathematics Subject Classification. 35Q55, 35B44, 26A33, 35B30.
Key words and phrases. Global solution; weak solutions; p-Laplacian inequality;

strong dissipation; noncylindrical domain.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted February 4, 2021. Published January 27, 2022.

1



2 J. FERREIRA, E. PIŞKIN, M. SHAHROUZI, S. CORDEIRO, C. A. RAPOSO EJDE-2022/09

The existence of a global solution for the wave equation of p-Laplacian type

utt −∆pu = 0 (1.3)

without an additional dissipation term, is an open problem. For n = 1, Derher
[5] gave the finite time existence of solution and showed, by a generic counter-
example, that the global solution not can be expected. Later, adding a strong
dissipation (−∆u′) in (1.3), the well-posedness and asymptotic behavior it was
studied by Greenberg [9]. Nevertheless, when the strong damping is replaced by
weaker damping (ut), existence and uniqueness of a global solution are only known
for n = 1, 2. See [4, 22]. Gao and Ma [8] proved global existence of solution and
asymptotic behavior under the intermediate damping (−∆)αut with 0 < α ≤ 1.
The memory damping was analyzed by Raposo et al. [18], p-Laplacian damping
was studied by Pereira et al. [16] and a thermoelastic effect was considered in
[20]. For wave coupled systems of the p-laplacian type see [15]. For other works
on this subject, we cite [14] and the references therein. For a brief review of the
literature on non-cylindrical domain, we cite [1, 7, 11, 17]. Unilateral problem is
very interesting because in general, dynamic contact problems are characterized
by nonlinear hyperbolic variational inequalities. For contact problems in elasticity
and finite element method see Kikuchi-Oden [10] and reference therein. For contact
problem viscoelastic materials, see Rivera and Oquendo [21]. For dynamic contact
problems with friction, for instance, problems involving unilateral contact with
dry friction of Coulomb, see Ballard and Basseville [3]. The study of variational
inequalities in bounded domains has been analyzed by several authors, for example,
see [2, 6, 12, 13].

In this work we consider the following p-Laplacian unilateral problem with strong
dissipation,

u′′ −∆pu−∆u′ − f ≥ 0 in Q0,

u′ ≥ 0 in Q0,

u = 0 on Σ0,

u(0) = u0 u′(0) = u1 in Ω.

(1.4)

We prove the existence of solutions for (1.4) by using the penalty method.

2. Penalty method

When using the penalization technique as in [11], a difficulty may appear since
the term 〈u′′(t), v − u′(t)〉 makes sense only when u′′(t) ∈ H−1(Ω), which is not
always possible to obtain. For this reason, the result obtained is the weak formula-
tion of (1.4), namely: if K ⊂ W 1,p

0 (Rn) is a closed and convex subset with 0 ∈ K,
and

V = {v ∈ L2(0, T ;W 1,p
0 (Ωt)); v

′ ∈ L2(0, T,W−1,p′

0 (Ωt)), v(t) ∈ K a.e.},

K = {v ∈W 1,p
0 (Ω); v(x) ≥ 0 a.e. in Ω},

equation (1.4) can be reformulated as

〈u′′(t), v − u′(t)〉+ 〈∆pu(t), v − u′(t)〉+ 〈−∆u′(t), v − u′(t)〉 ≥ 〈f(t), v − u′(t)〉,
u(0) = u0, u′(0) = u1,

(2.1)
for u′(t) ∈ K a.e. and for all v ∈ K.



EJDE-2022/09 p-LAPLACIAN INEQUALITIES IN NONCYLINDRICAL DOMAINS 3

Then, the existence of u : Q→ R, with

u(0) = u0 ∈W 1,p
0 (Ω0), u′(0) = u1 ∈ L2(Ω0) ∩K, u′(t) ∈ K a.e. in (0, T )

and ∫ s

0

〈v′(t) + ∆pu(t)−∆u′(t)− f(t), v(t)− u′(t)〉

≥ 1

2
|v(s)− u′(s)|2 − 1

2
|v(0)− u′(0)|2,∀s ∈ (0, T ), ∀v ∈ V .

(2.2)

It is easy to check that if u′ ∈ V , then (2.1) and (2.2) are equivalent. However,
we shall find a solution for (1.4) in the sense of (2.2). Thus, the objective of this
work is to obtain the existence of global weak solution to (1.4) considering Q as a
non-cylindrical domain, as in fact, Lions [11] provides existence and uniqueness of
weak solutions and/or regular for operators of the parabolic-hyperbolic type in the
noncylindrical domain.

By D(Ω) we denote the space of infinitely differentiable functions with compact
support contained in Ω. The inner product and norm in L2(Ω) and H1

0 (Ω) will be

represented by (· , ·), | · |, ‖·‖, respectively, and by 〈·, ·〉 the duality between W 1,p
0 (Ω)

and W−1,p′(Ω).
If T > 0 and X is a Banach space with the norm ‖·‖X , we denote by Lp(0, T ;X),

1 ≤ p < +∞, the Banach space of vector functions u : (0, T ) → X that are
measurable and ‖u(t)‖X ∈ Lp(0, T ) with the norm

‖u‖Lp(0,T ;X) =
[ ∫ T

0

‖u(t)‖pXdt
]1/p

,

and by L∞(0, T ;X) the Banach space of vector functions u : (0, T ) → X that are
measurable and ‖u(t)‖X ∈ L∞(0, T ) with the norm

‖u‖L∞(0,T ;X) = esssup0<t<T ‖u(t)‖X .

Let Ω be an open, connected and bounded subset of Rn with regular boundary
Γ, Q ⊂ Q0 a noncylindrical domain. We will use the following notation

Ωs = Q ∩ {t = s} for 0 < s < T, Ω0 = intRn(Q ∩ {t = 0}),
ΩT = intRn(Q ∩ {t = T}), Γs = ∂Ωs,

Σ = ∪0<s<TΓs, ∂Q = Ω0 ∪ Σ ∪ ΩT the boundary of Q.

It is clear that Ω0 6= ∅. Our hypotheses on Q are:

(H1) Ωt is monotonically increasing, that is, Ω∗t ⊂ Ω∗s if t < s, where Ω∗t =
Proj{t=0}Ωt.

(H2) For each t ∈ [0, T ], Ωt has the following regularity property: if u ∈W 1,p
0 (Ω)

e u = 0 a.e. in Ω \ Ω∗t , then u|Ω∗t ∈ H
1
0 (Ω∗t ).

To simplify the notation, we identity Ω∗t with Ωt.
Let us define

Lq(0, T ;Lp(Ωt)) = {w ∈ Lq(0, T ;Lp(Ω)) : w = 0 a.e. in Q0 \Q}.

When 1 ≤ q <∞ we consider the norm

‖w‖Lp(0,T ;Lp(Ωt)) =
[ ∫ T

0

‖w(t)‖qLp(Ωt)
dt
]1/q

,



4 J. FERREIRA, E. PIŞKIN, M. SHAHROUZI, S. CORDEIRO, C. A. RAPOSO EJDE-2022/09

which coincides with ‖w‖Lq(0,T ;Lp(Ω)). And when q =∞ we consider

‖w‖L∞(0,T ;Lp(Ωt)) = ess sup0<t<T ‖w(t)‖Lp(Ωt).

Note that Lq(0, T ;Lp(Ωt)) is a closed subspace of Lq(0, T ;Lp(Ω)) for 1 ≤ q ≤
∞. Analogously we define Lq(0, T ;W 1,p

0 (Ωt)), 1 ≤ q ≤ ∞. It is also true that

Lq(0, T ;W 1,p
0 (Ωt)) is a closed subspace of Lq(0, T ;W 1,p

0 (Ω)).

3. Existence of a global weak solution

Theorem 3.1. Let f ∈ L2(0, T, L2(Ωt)), u0 ∈ W 1,p
0 (Ω0), u1 ∈ L2(Ω0) ∩K, with

K being a convex and closed subset of W 1,p
0 (Ω), and 0 ∈ K. Lets us suppose that

(H1) and (H2) are satisfied. Then there exists a function u : Q→ R satisfying

u ∈ L∞(0, T ;W 1,p
0 (Ωt)), (3.1)

u′ ∈ L∞(0, T ;L2(Ωt)) ∩ L2(0, T ;H1
0 (Ωt)), (3.2)

u′(t) ∈ K a.e. in (0, T ), (3.3)

u(t)→ ũ0 in H1
0 (Ω) if t→ 0, (3.4)

u′(t)→ ũ1 in L2(Ω) if t→ 0, (3.5)∫ s

0

〈v′(t) + ∆pu(t)−∆u(t)− f(t), v(t)− u′(t)〉dt

≥ 1

2
|v(s)− u′(s)|2L2(Ωs) −

1

2
|v(0)− v(1)|L2(Ω0), ∀s ∈ (0, T ), ∀v ∈ V,

(3.6)

u(0) = u0, u′(0) = u1 in Ω0. (3.7)

with ũ0 and ũ1 being extensions of u0 and u1 to Ω that vanish outside Ω0.

Theorem 3.1 will be proved by using the Faedo-Galerking method, penalty op-
erator associated to the convex set and penalty method from Lions [11]. At First
we find a solution of penalized problem in the cylinder Q0 and then we show that
the restriction to the noncylinder domain Q is indeed weak solution for the original
problem.

To this end, let ũ0 ∈ W 1,p
0 (Ω), ũ1 ∈ L2(Ω), and f̃ ∈ L2(Q0) be the extensions

to zero outside Ω0 of u0, u1, and f , respectively. Let us also consider a penalty
function for noncylindrical domains:

M(x, t) =

{
0 in Q ∪ Ω0 × {0},
1 in Q0 \ (Q ∪ Ω0 × {0}).

Let PK : Hr
0 (Ω)→ K be the projection operator: for u ∈ Hr

0 (Ω), Pku is the unique
element in K such that

‖u− PKu‖ ≤ ‖u− k‖, ∀k ∈ K,

where r is a fixed integer with r > 1 + n
2 −

n
p such that Hr

0 (Ω) ↪→ W 1,p
0 (Ω) con-

tinuously. Let J be the duality operator from Hr
0 (Ω) into H−1(Ω) relatively to the

identity from R+ to R+. That is,

〈Ju, u〉 = ‖Ju‖H−r(Ω)‖u‖,
‖J(u)‖H−r(Ω) = ‖u‖.



EJDE-2022/09 p-LAPLACIAN INEQUALITIES IN NONCYLINDRICAL DOMAINS 5

We consider now β : H1
0 (Ω) → H−1(Ω) defined by β(u) = J(u − PKu). The

operator β is a penalty operator associated to K, thus satisfies

β is monotone, bounded, Hemicontinuous and

K = {v ∈ H1
0 (Ω);β(v) = 0}.

(3.8)

The proof of Theorem 3.1 is a consequence of the following theorem.

Theorem 3.2. Suppose the hypotheses of Theorem 3.1 are satisfied. Then for each
µ > 0 there exists a function uµ : Q0 → R satisfying

uµ ∈ L∞(0, T ;W 1,p
0 (Ω)), (3.9)

u′µ ∈ L∞(0, T ;L2(Ω)) ∩ L2(0, T ;H1
0 (Ω)), (3.10)∫ s

0

[〈v′(t) + ∆puµ(t)−∆u′µ(t)− f̃ , v(t)− u′µ(t) +
1

µ
〈M(t)u′µ(t), v(t)〉〉] dt

≥ 1

2
|v(s)− u′µ(s)|2 − 1

2
|v(0)− u1|2, ∀t ∈ (0, T ),

∀µ, ∀v ∈ L2(0, T ;W 1,P
0 (Ω)) such that v′ ∈ L2(0, T,W−1,p′(Ω).

(3.11)

Before prove the main theorem, we present the existence of a special basis.

4. Galerkin basis

According [19], we will show that there exists a Hilbert space Hs
0(Ω) with 0 < s

such that Hs
0(Ω) ↪→ W p

0 (Ω) is continuous and Hs
0(Ω) ↪→ L2(Ω) is continuous and

compact.
For v ∈ H1(Rn) we consider Fourier transform of v,

v̂(ξ) =
1

(2π)n/2

∫
Rn

e−(ξ.x)iv(x) dx

and

Hs(Rn) = {v ∈ L2(Rn) : (1 + ‖ξ‖s/2v̂(ξ)) ∈ L2(Rn)}.
Since Ω is a bounded open set with sufficiently smooth boundary, we have Hs(Ω)
is the set of restrictions on Ω of the functions v ∈ Hs(Rn), then

‖v‖Hs(Ω) = inf{‖V ‖Hs(Rn) : V = v a.e. in Ω}

and

Hs
0(Ω) = C∞0 (Ω)

Hs(Ω)
.

We need

Wm,q
0 (Ω) ↪→Wm−k,qk

0 (Ω),
1

qk
=

1

q
− k

n
.

Choosing qk = p, m − k = 1 and q = 2 we obtain m = 1 + n
2 −

n
p . For s > m we

have

Hs
0(Ω) ↪→W 1,p

0 (Ω) ↪→ H1
0 (Ω) ↪→ L2(Ω)

from where our goal follows. Now, from spectral theory the problem

((vj , v))Hs
0 (Ω) = λj(vj , v), for all v ∈ Hs

0(Ω)

has solution and moreover {vj}j∈N precisely, is a Schauder basis forHs
0(Ω)∩Lr+1(Ω)

with elements that are orthogonal in L2(Ω).



6 J. FERREIRA, E. PIŞKIN, M. SHAHROUZI, S. CORDEIRO, C. A. RAPOSO EJDE-2022/09

5. Proof of the main theorem

The proof of Theorem 3.2 will be made in 4 steps.

5.1. Penalty approximated problem. Let {w1, w2, . . . } be a Schauder basis of
Hs

0(Ω) as demonstrate before, and for each m ∈ N let Vm = [w1, . . . , wm] be the
subspace generated by the m first vectors from this basis. Let 0 < ε < 1 fixed. We
wish to find

uεµm(x, t) := uεµm(t) =

m∑
1

gjεµm(t)wj(x),

where gjεµm(t) of the system of ODEs

(u′′εµm(t), wj) + 〈∆puεµm(t), wj〉+ (∇uεµm(t),∇wj)

+
1

ε
(β(uεµm(t)), wj) +

1

µ
(M(t)uεµm(t), wj)

= ( ˜f(t), wj), ∀wj ∈ Vm,

(5.1)

uεµm(0) = u0m → ũ0 strongly in W 1,p
0 (Ω), (5.2)

u′′εµm(0) = u1m → ũ1 strongly in L2(Ω). (5.3)

By Caratheodory the system (5.1)–(5.3) has a local solution uεµm(t) defined in
some interval [0, tm), 0 < tm < T .

5.2. A priori estimates I. Composing (5.1) with u′εµm(t) ∈ Vm and then inte-
grating from 0 to t < tm, we obtain

1

2

{
|u′εµm(t)|2 +

1

p
‖uεµm(t)‖pW01,p(Ω)

}
+

∫ t

0

‖u′εµm(s)‖2H1
0 (Ω) ds

+
1

ε

∫ t

0

(β(u′εµm(s)), u′εµm(s)) ds+
1

µ

∫ t

0

(M(t)u′εµm(s), u′εµm(s)) ds

=

∫ t

0

(f̃(s), u′εµm(s)) ds+
1

2
|u0m|2 +

1

p
‖u1m‖pW 1,p

0 (Ω)
.

(5.4)

Using (5.2) and (5.3), the monotonicity of β, the definition of M , f̃ ∈ L2(Q0), and
Gronwall’s lemma in (5.3), we obtain

1

2
|u′εµm(t)|2 +

1

p
‖uεµm(t)‖pW01,p(Ω) +

∫ t

0

‖u′εµm(s)‖2H1
0 (Ω) ds

+
1

ε

∫ t

0

(β(u′εµm(s)), u′εµm(s)) ds+
1

µ

∫ t

0

(M(t)u′εµm(s), u′εµm(s))ds ≤ C

where C is a positive constant independent of ε, µ,m and t ∈ [0, tm). Hence we can
extend the solution uεµm(t) to the whole interval [0, T ], obtaining in addition

(uεµm) is bounded in L∞(0, T ;W 1,p
0 (Ω)), (5.5)

(u′εµm) is bounded in L∞(0, T ;L2(Ω)), (5.6)

(u′εµm) is bounded in L2(0, T ;H1
0 (Ω)), (5.7)

(uεµm(T )) is bounded in W 1,p
0 (Ω), (5.8)

(uεµm(T )′) is bounded in L2(Ω), (5.9)



EJDE-2022/09 p-LAPLACIAN INEQUALITIES IN NONCYLINDRICAL DOMAINS 7

( 1
√
µ
Mu′εµm

)
is bounded in L∞(0, T ;L2(Ω)). (5.10)

From the definition of β one can prove that β is Lipschitz and thus from (5.7), it
follows that

(β(u′εµm)) is bounded in L2(0, T ;H−1(Ω)). (5.11)

In addition, the operator ∆pu = div(|∇u|p−2∇u) is a bounded operator from

W 1,p
0 (Ω) to W−1,p′(Ω). Thus it follows from (5.5) that

(∆puεµm) is bounded in L∞(0, T ;W−1,p′(Ω)). (5.12)

We can thus extract subsequences from above sequences, denoted up to subindexes,
such that

uεµm
∗
⇀ uεµ in L∞(0, T ;W 1,p

0 (Ω)), (5.13)

u′εµm
∗
⇀ u′εµ in L∞(0, T ;L2(Ω)), (5.14)

u′εµm ⇀ u′εµ in L2(0, T ;H1
0 (Ω)), (5.15)

uεµm(T ) ⇀ uεµ(T ) in W 1,p
0 (Ω), (5.16)

u′εµm(T ) ⇀ u′εµ(T ) in L2(Ω). (5.17)

Since M ∈ L∞(Q0), it follows from (5.14) that

1
√
µ
Mu′εµm

∗
⇀

1
√
µ
Mu′εµ in L∞(0, T ;L2(Ω)), (5.18)

β(u′εµ) ⇀ χεµ in L2(0, T ;H−1(Ω)), (5.19)

∆puεµm
∗
⇀ ϕεµ in L∞(0, T ;W−1,p′(Ω)). (5.20)

5.3. A priori estimate II. Now we obtain an estimate for u′′εµm. It is done
through a standard argument on projections. Consider the projection operator
Pm : Hm

0 (Ω)→ Vm defined by

Pm[h] =

m∑
j=1

((h,wj))wj , h ∈ Hr
0 (Ω)

where ((·, ·)) stands for the inner product in Hr
0 (Ω). Let P ∗m ∈ L(H−r(Ω), H−r(Ω))

the self-adjoint extension of Pm. Since P ∗m[h] = Pm[h] = h, ∀h ∈ Vm, we conclude
from (5.1) that

(u′′εµm(t), w) = (P ∗m[f̃(t)], w)− 〈P ∗m[∆puεµm(t)], w〉+ 〈P ∗m[∆u′εµm(t)], w〉

− 1

ε
(P ∗m[β(u′εµm(t))], w)− 1

µ
(P ∗m[M(t)u′εµm(t)], w) ∀w ∈ Vm.

Thereby, using argument of denseness it follows from (5.7), (5.10), (5.11) and (5.12)
that

(u′′εµm) is bounded in L2(0, T ;H−r(Ω)) for each ε, µ. (5.21)

Taking into account the convergence obtained above, we can pass to the limit when
m→∞ in the approximated equation and obtain

u′′εµ + ϕεµ −∆u′εµ +
1

ε
χεµ +

1

µ
Mu′εµ = f̃ in L2(0, T ;W−1,p′(Ω)),

uεµ(0) = ũ0,

u′εµ(0) = ũ1.



8 J. FERREIRA, E. PIŞKIN, M. SHAHROUZI, S. CORDEIRO, C. A. RAPOSO EJDE-2022/09

It can be shown through the same arguments as in Ferreira-Ma [7] that ϕεµ = ∆puεµ
and reasoning likewise as in Rabello [17] that χεµ = β(u′εµ). Therefore, we obtain

u′′εµ + ∆puεµ −∆u′εµ +
1

ε
β(u′εµ) + V

1

µ
Mu′εµ

= V f̃ in L2(0, T ;W−1,p′(Ω)),

uεµ(0) = ũ0,

u′εµ(0) = ũ1.

(5.22)

We observe that the bounds obtained are independently on ε, µ and t, thus there
exist subsequences from previous sequences such that

uεµ
∗−−−⇀

ε→0
uεµ in L∞(0, T ;W 1,p

0 (Ω)), (5.23)

u′εµ
∗−−−⇀

ε→0
u′εµ in L∞(0, T ;L2(Ω)), (5.24)

u′εµ −−−⇀
ε→0

u′εµ in L2(0, T ;H1
0 (Ω)), (5.25)

1
√
µ
Mu′εµ

∗−−−⇀
ε→0

1
√
µ
Mu′εµ in L∞(0, T ;L2(Ω)), (5.26)∫ T

0

(β(u′εµ), u′εµ)dt −−−→
ε→0

0. (5.27)

From (5.22) we obtain

β(u′εµ) = ε
[
f −∆puεµ − u′′εµ + ∆u′εµ −

1

µ
Mu′εµ

]
in D′(0, T ;W−1,p′).

Thus, from the convergences (5.20), (5.23)-(5.26), it follows that

β(u′εµ) −−−→
ε→0

0 in D′(0, T ;H−r(Ω)).

In addition, from (5.26), since β is Lipschitz,

β(u′εµ) −−−⇀
ε→0

χ in L2(0, T ;H−1(Ω)).

Thereby we have χ = 0. On the other hand, thanks to the monotonicity and
hemicontinuity of β and (5.27), we prove that χ = β(uµ) and therefore we conclude
that

β(u′µ(t)) = 0 a.e. or u′µ ∈ K a.e. (5.28)



EJDE-2022/09 p-LAPLACIAN INEQUALITIES IN NONCYLINDRICAL DOMAINS 9

Let v ∈ L2(0, T ;W 1,p
0 (Ω)) such that v′ ∈ L2(0, T ;W−1,p′(Ω)). Therefore,

1

2
|v(s)− u′εµ(s)|2 − 1

2
|v(0)− u′εµ(0)|2

=
1

2

∫ s

0

d

dt
|v(t)− u′εµ(t)|2dt

=

∫ s

0

〈v′(t)− u′′εµ(t), v(t)− u′εµ(t)〉dt

=

∫ s

0

〈v′(t)−
[
f̃(t)−∆puεµ(t) + ∆u′εµ(t)− 1

ε
β(u′εµ)

− 1

µ
Mu′εµ(t)

]
, v(t)− u′εµ(t)〉dt

=

∫ s

0

〈v′(t)− f̃(t) + ∆puεµ(t)−∆u′εµ(t), v(t)〉dt

+

∫ s

0

〈v′(t)− f̃(t),−u′εµ(t)〉dt

+

∫ s

0

〈+∆puεµ(t), u′εµ(t)〉dt−
∫ s

0

〈−∆u′εµ(t), u′εµ(t)〉dt

+
1

ε

∫ s

0

〈β(u′εµ(t))− β(v), v(t)− u′εµ(t)〉dt︸ ︷︷ ︸
≤0

+

∫ s

0

1

µ
〈M(t)u′εµ(t), v(t)〉dt

+

∫ s

0

1

µ
〈M(t)u′εµ(t),−u′εµ(t)〉dt︸ ︷︷ ︸

≤0

.

(5.29)

Let Ψ = {ϕ ∈ C0[0, T ], ϕ(t) ≥ 0 ∀t ∈ [0, T ]}. Multiplying (5.29) by ϕ ∈ Ψ and
integrating from 0 to T , we obtain

∫ T

0

[
1

2
|v(s)− u′εµ(s)|2 +

1

p
‖uεµ(s)‖2

W 1,p
0

+

∫ s

0

‖u′εµ(t)‖2dt
]
ϕ(s)ds

≤
∫ T

0

ϕ(s)

∫ s

0

〈v′(t)− f̃(t) + ∆puεµ(t)−∆u′εµ(t), v〉 dt ds

+

∫ T

0

ϕ(s)

∫ s

0

〈v′(t)− f̃(t),−u′εµ(t)〉 dt ds

+
1

p
‖uεµ(0)‖2

W 1,p
0 (Ω)

∫ T

0

ϕ(s)ds+

∫ T

0

ϕ(s)

∫ s

0

1

µ
〈M(t)u′εµ(t), v(t)〉 dt ds

+
1

2

∫ T

0

|v(0)− u′εµ(0)|2ϕ(s)ds.

(5.30)



10 J. FERREIRA, E. PIŞKIN, M. SHAHROUZI, S. CORDEIRO, C. A. RAPOSO EJDE-2022/09

Taking the limit inferior, it follows from (5.23)–(5.26) and from Banach-Steinhauss’
Theorem that∫ T

0

ϕ(s)

[
1

2
|v(s)− u′µ(s)|2 +

1

p
‖uµ(s)‖2 +

∫ s

0

‖u′µ(t)‖2dt
]
ds

≤
∫ T

0

ϕ(s)

∫ s

0

〈v′(t)− f̃(t) + ∆pu
′
µ(t)−∆u′µ(t), v(t)〉 dt ds

+

∫ T

0

ϕ(s)

∫ s

0

〈v′(t)− f̃(t),−u′µ(t)〉 dt ds

+
1

p
‖u(0)‖2

W 1,p
0

∫ T

0

ϕ(s)ds+

∫ T

0

ϕ(s)

∫ s

0

1

µ
〈M(t)u′µ(t), v(t)〉 dt ds

+

∫ T

0

ϕ(s)
1

2
|v(0)− u1|2 ds, ∀ϕ ∈ Ψ.

(5.31)

Thus considering

ϕ =

{
1 if t = s,

linear in (s− δ, s) and (s, s+ δ),

0 ≤ s ≤ 1, ϕ ∈ C0[0, T ], splitting the inequality (5.31) by δ > 0, taking the limit
with δ → 0, we obtain from the Lebesgue points Theorem for integrable functions∫ s

0

[〈v′(t)− f̃(t) + ∆puµ(t)−∆u′µ(t), v(t)− u′µ(t)〉]

+
1

µ
〈M(t)u′µ(t), v(t)〉dt

≥ 1

2
|v(s)− u′µ(s)|2 − 1

2
|v(0)− u1|2, ∀µ, a.e.

(5.32)

We obtain, therefore, the penalized inequality in cylinder domain Q0, what proves
Theorem 3.2.

5.4. Passage to the limit. It remains now passing to the limit when µ → 0 to
obtain the inequality in the noncylindrical domain Q and thus to have Theorem
3.1 proved.

From (5.23)-(5.26), Banach-Stainhaus’ Theorem and boundedness provided by
(5.5), (5.6),(5.7) and (5.10) independently on ε and µ, there exist subsequences
such that

uµ
∗
⇀ u in L∞(0, T ;W 1,p

0 (Ω)), (5.33)

u′µ
∗
⇀ u′ in L∞(0, T ;L2(Ω)), (5.34)

u′µ ⇀ u′ in L2(0, T ;H1
0 (Ω)), (5.35)

1
√
µ
Mu′µ ⇀ χ1 in L2(0, T ;L2(Ω)). (5.36)

From (5.35) we obtain

Mu′µ ⇀ χ2 in L2(0, T ;H1
0 (Ω)). (5.37)

We also have the convergence

βu′µ
∗
⇀ χ3 in L∞(0, T ;L2(Ω)). (5.38)



EJDE-2022/09 p-LAPLACIAN INEQUALITIES IN NONCYLINDRICAL DOMAINS 11

Since (Mu′µ, w) = (u′µ,Mw), it follows that χ2 = Mu′, thus

Mu′µ ⇀Mu′ in L2(0, T ;H1
0 (Ω)).

Since 1
µ

∫ T
0
|M(t)u′µ(t)|2dt ≤ C ∀µ, it follows that Mu′µ ⇀ 0 in L2(Q0). Hence,

Mu′ = 0 a.e. in Q0.

From the definition of M we obtain

u′ = 0 a.e. in Q0 \Q or

u′ = 0 a.e. in Ω \ Ωt

in [0, T ], which combined with (5.35) yields u′ ∈ L2(0, T ;H1
0 (Ωt)). Since u′ = 0 in

Q0 \Q and u(x, 0) = ũ0 = 0 in Ω\Ω0, it follows that u = 0 in Q0 \Q, which jointly
with (5.33),

u ∈ L∞(0, T ;W 1,p
0 (Ωt)). (5.39)

Again, from de monotonicity and hemicontinuity of β, and owing to the fact that
β(uµ) = 0 in L2(0, T ;H−1(Ω)), we conclude that

β(u′) = 0 a.e. or u′(t) ∈ K a.e. (5.40)

We have v ∈ L2(0, T ;W 1,p
0 (Ωt)) ↪→ L2(0, T ;H1

0 (Ωt)). Let v′ ∈ L2(0, T ;W−1,p′(Ωt)).
Hence: for almost every t ∈ (0, T ), v = 0 in Ω \ Ωt. Thus∫ s

0

(M(t)u′µ(t), v)dt =

∫ s

0

∫
Ω

M(t)u′µ(t)v(t)dxdt

=

∫ s

0

∫
Ωt

M(t)u′µ(t)v(t)dxdt = 0, ∀µ

because M = 0 in Ωt.
Taking the limit inferior in (5.32) in first member of the equation and the limit

in the second member when µ→ 0 and using the convergence obtained up to here,
it follows that∫ T

0

ϕ(s)
[1

2
|v(s)− u′(s)|2 +

1

p
‖u(s)‖p

W 1,p
0 (Ω)

+

∫ s

0

‖u′(t)‖2dt
]

≤
∫ T

0

ϕ(s)

∫ s

0

〈v′ − f + ∆pu−∆u′, v〉 dt ds+

∫ T

0

ϕ(s)

∫ s

0

〈v′ − f,−u′〉 dt ds

+

∫ T

0

ϕ(s)
1

p
‖u(0)‖2 ds+

∫ T

0

ϕ(s)
1

2
|v(0)− u1|2ds.

Thus, for almost s we have

1

2
|v(s)− u′(s)|2L2(Ωs) −

1

2
|v(0)− u′(0)|2L2(Ω0)

≤
∫ s

0

〈v′ − f + ∆p −∆u′, v〉dt+

∫ s

0

〈v′ − f,−u′〉 ds

+
1

p
‖u(0)‖p

W 1,p
0 (Ω0)

− 1

p
‖u(s)‖p

W 1,p
0 (Ωs)

ds−
∫ s

0

‖u′(t)‖2dt

=

∫ s

0

〈v′ − f + ∆pu−∆u′, v〉dt+

∫ s

0

〈v′ − f,−u′〉 ds

−
∫ s

0

〈∆pu, u
′〉dt−

∫ s

0

〈−∆u′, u′〉dt



12 J. FERREIRA, E. PIŞKIN, M. SHAHROUZI, S. CORDEIRO, C. A. RAPOSO EJDE-2022/09

=

∫ s

0

〈v′ − f + ∆pu−∆u′, v〉dt+

∫ s

0

〈v′ − f + ∆pu−∆u′,−u′〉dt

=

∫ s

0

〈v′ − f + ∆p −∆u′, v − u′〉dt

for all s ∈ (0, T ), and all v ∈ V .
To show the continuity of u′ we can use the same arguments as in [11].

Acknowledgments. The authors would like to express their gratitude to the
anonymous referees for their constructive comments and suggestions that allowed
improve this manuscript.

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Jorge Ferreira

Department of Exact Sciences, Federal Fluminense University, Volta Redonda, 27213-

145 RJ, Brazil
Email address: jorge ferreira@id.uff.br

Erhan Pişkin
Department of Mathematics, Dicle University, Diyarbakir, Turkey

Email address: episkin@dicle.edu.tr

Mohammad Shahrouzi

Department of Mathematics, Jahrom University, Jahrom, P.O. Box: 74137-66171, Iran

Email address: mshahrouzi@jahromu.ac.ir

Sebastião Cordeiro

Faculty of Exact Sciences and Technology, Federal University of Pará, Abaetetuba,
68440-000 PA, Brazil

Email address: sebastiao@ufpa.br

Carlos Alberto Raposo
Department of Mathematics and Statistics, Federal University of São João del-Rei,

São João del-Rei, 36307-352 MG, Brazil
Email address: raposo@ufsj.edu.br


	1. Introduction
	2. Penalty method
	3. Existence of a global weak solution
	4. Galerkin basis
	5. Proof of the main theorem
	5.1. Penalty approximated problem
	5.2. A priori estimates I
	5.3. A priori estimate II
	5.4. Passage to the limit
	Acknowledgments

	References