Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 73, pp. 1–9.

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

DECAY ESTIMATES FOR SOLUTIONS OF EVOLUTIONARY

DAMPED p-LAPLACE EQUATIONS

FARID BOZORGNIA, PETER LEWINTAN

Abstract. In this note, we study the asymptotic behavior, as t tends to
infinity, of the solution u to the evolutionary damped p-Laplace equation

utt + aut = ∆pu

with Dirichlet boundary conditions. Let u∗ denote the stationary solution

with same boundary values, then we prove the W 1,p-norm of u(t)−u∗ decays

for large t like t
− 1

(p−1)p , in the degenerate case p > 2.

1. Introduction and problem setting

Let Ω ⊂ Rn be a bounded Lipschitz domain, p ≥ 2 and g ∈ W 1,p(Ω). Consider
the minimization of the functional

E (u) :=
1

p

∫
Ω

|∇u|p dx, (1.1)

over the class C := {u : u−g ∈W 1,p
0 (Ω)}. The minimizer denoted by u∗(x) satisfies

the following Euler-Lagrange-equation in the weak sense:

−∆pu
∗ = 0 in Ω,

u∗ = g on ∂Ω.
(1.2)

The first order flow of E (v), i.e. vt+∂vE (v) = 0, can be considered as a classical
steepest descent flow for solving the minimization problem (1.1). In the degenerate
case p > 2 the authors of [9] obtained the sharp decay rate

sup
x∈Ω
|v(t, x)− u∗(x)| = O

(
t−

1
p−2
)

as t→∞.

Their proof is based on the Moser iteration applied to the difference v(t, x)−u∗(x),
which itself is not a solution, thus bounding the L∞-norm in terms of the Lp-norm.

It is well known, that an improvement in the convergence rate may be gained
by considering the corresponding second order damped problem, cf. [6, 10, 3]
and references therein. Moreover, second order damped problems naturally appear
in modeling mechanical systems. For instance, the motion of a material point
with positive mass sliding on a profile defined by a function Φ under the action of

2010 Mathematics Subject Classification. 35B40, 35L70.

Key words and phrases. p-Laplace; telegraph equation; asymptotic behavior; convexity.
c©2021. This work is licensed under a CC BY 4.0 license.

Submitted July 8, 2021. Published September 10, 2021.

1



2 F. BOZORGNIA, P. LEWINTAN EJDE-2021/73

the gravity force, the reaction force, and the friction force can asymptotically be
approximated by the following second order dynamical system

ẍ(t) + λẋ(t) +∇Φ(x(t)) = 0 (1.3)

called heavy ball with friction system (HBF), cf. [2]. We refer to [7] and [4] to see
numerical algorithms based on the HBF system for solving some special problems,
e.g. large systems of linear equations, eigenvalue problems, nonlinear Schrödinger
problems, inverse source problems, and ill-posed problems. In [4] the authors have
shown advantages and superior convergence properties of such a dynamical func-
tional particle method compared to a first order dynamical system, and also to
several other iterative methods. So, it’s hardly surprising that second order dy-
namical equations play an important role in acceleration for convergence to steady
state solutions. In fact, the power of the use of the damped p-Laplace equation
in image denoising was investigated in [3]. However, an analysis as in [9] of the
asymptotic behavior, as t → ∞, of the solutions to a damped p-Laplace equation
was not done so far.

Our purpose here is to obtain the decay rate for large time of u − u∗ where u
denotes a solution to the evolutionary damped p-Laplace equation (The question
of existence of solutions will be the subject of a forthcoming note.), namely:

utt + a ut = ∆pu in (0,∞)× Ω,

u(0, x) = u0(x) in {0} × Ω,

ut(0, x) = 0 ∀x ∈ Ω,

u(t, x) = g(x) on [0,∞)× ∂Ω,

(1.4)

where a > 0 is constant and u0 ∈W 1,p(Ω), such that u0 − g ∈W 1,p
0 (Ω).

It is clear, that the solution of the damped equation (1.4) behaves for large time
like the stationary solution of (1.2). Moreover, we show the following rate of decay
for the W 1,p-norm of their difference.

Theorem 1.1. Let p ≥ 2, u∗ denote a solution to (1.2) and u a solution to (1.4).
For large time we have

‖u− u∗‖W 1,p(Ω) ≤ C · t−
1

(p−1)p ,

with a constant C = C(p,Ω, u0, a) > 0.

Our proof is based on a careful analysis of the error term

e(t) :=

∫
Ω

a2

2
w2 + awwt + w2

t + 2
(1

p
|∇u|p − 1

p
|∇u∗|p

)
dx (1.5)

where we have set w = u − u∗. Note that our error term is chosen in such a way
that it is compatible to our problem and we can estimate the error in terms of its
derivative. Moreover, the fact

d

dt

∫
Ω

1

p
|∇u|p dx = −

∫
Ω

ut∆pudx,

cf. page 4, justifies the appearance of the last term in the error. It is worth men-
tioning that with our argumentation scheme we can improve the decay rate in the
linear case p = 2 and obtain the classical result from [8], cf. the discussion in section
3.1.



EJDE-2021/73 DECAY ESTIMATES 3

2. Basic results

Let us briefly introduce the notation used throughout this work. The Euclidean
norm in Rn is denoted by | · |. A generic positive constant is represented by capital
or lower case letter c, possibly varying from line to line. We often write u(t)(x) for
u(t, x).

Given a real Banach space X, the (Banach) space Lp(0, T ;X) consists of all
measurable functions u : [0, T ]→ X such that

‖u‖Lp(0,T ;X) =
(∫ T

0

‖u(t)‖pX dt
)1/p

<∞ , 1 ≤ p <∞ ,

L∞(0, T ;X) is the space of all measurable u : [0, T ]→ X such that

‖u‖L∞(0,T ;X) = ess spuut∈[0,T ] ‖u(t)‖X <∞.

The Banach space W 1,p(0, T ;X), for 1 ≤ p ≤ ∞, consists of all u ∈ Lp(0, T ;X)
such that ∂tu exists in the weak sense and belongs to Lp(0, T ;X).

Recall that for u ∈W 1,p(0, T ;X) we have u ∈ C0([0, T ];X) and

max
0≤t≤T

‖u(t)‖X ≤ c(T )‖u‖W 1,p(0,T ;X).

For further reading and elaborated clarifications on spaces involving time, we refer
the reader to [5, Sec. 5.9.2].

Throughout this work, we use the following inequalities:

• let p ≥ 2. For all a, b ∈ Rn we have

22−p|a− b|p ≤ 〈|a|p−2a− |b|p−2b, a− b〉, (2.1)

• for p ≥ 2 and with an adequate constant c(p) ∈ (0, 1]:

|b|p ≥ |a|p + p〈|a|p−2a, b− a〉+ c(p)|b− a|p, (2.2)

• furthermore, for ‖f‖Lp(Ω) ≤M and ‖g‖Lp(Ω) ≤M the estimate∫
Ω

∣∣|f |p − |g|p∣∣dx ≤ c(p,Ω)Mp−1‖f − g‖Lp(Ω) (2.3)

holds, cf. [11, p. 75].

Firstly, let us define the concept of weak solutions to the evolutionary damped
p-Laplace equation.

Definition 2.1. We say that u ∈W 1,p
loc (0,∞;W 1,p(Ω)) is a solution to

utt + aut = ∆pu

if ∫ ∞
0

∫
Ω

−ut φt − auφt + |∇u|p−2〈∇u,∇φ〉dxdt = 0,

for each φ ∈ C∞0 ((0,∞)× Ω).

In the following, let us denote by u∗ a solution to (1.2), and by u a solution to
(1.4). Moreover, we set

E(t) :=

∫
Ω

1

2
u2
t (t, x) +

1

p
|∇u(t, x)|p dx.



4 F. BOZORGNIA, P. LEWINTAN EJDE-2021/73

Corollary 2.2. E(·) is non-increasing, or rather in the weak sense we have

E′(t) = −a
∫

Ω

u2
t dx. (2.4)

Proof. A multiplication of
utt + a ut = ∆pu

by ut followed by an integration over Ω gives∫
Ω

uttut dx−
∫

Ω

(∆pu)ut dx = −a
∫

Ω

u2
t dx. (2.5)

Moreover, an integration by parts yields

−
∫

Ω

ut ∆pudx =
d

dt

∫
Ω

1

p
|∇u|p dx,

note that there is no time dependence of ut on the boundary. In view of∫
Ω

utt ut dx =
1

2

d

dt

∫
Ω

ut
2 dx,

we combine the last two equalities with (2.5) to achieve the desired relation (2.4):

E′(t) =
d

dt

∫
Ω

1

2
u2
t dx+

d

dt

∫
Ω

1

p
|∇u|p dx = −a

∫
Ω

u2
t dx. �

The above considerations were formal and can all be made rigorous, cf. [12, p.
156ff]. In view of (2.4), we show that the gradient of u (with respect to space) is
bounded by the initial data and that ut tends to zero for large times.

Corollary 2.3. Let u be a solution to (1.4). Then:

(a) ‖ut(T )‖L2(Ω)
T→∞−−−−→ 0.

(b) For all T ≥ 0 it holds ‖∇u(T )‖Lp(Ω) ≤ ‖∇u0‖Lp(Ω).

Proof. Integrating (2.4) over (0, T ) we obtain∫
Ω

1

2
u2
t (T, x) dx+

∫
Ω

1

p
|∇u(T, x)|p dx+ a

∫ T

0

∫
Ω

u2
t (τ, x) dx dτ

≤
∫

Ω

1

p
|∇u0(x)|p dx.

(2.6)

Note that the right-hand side of inequality (2.6) is independent of T , hence, the
statement follows as T →∞. �

Remark 2.4. Taking the essential supremum with respect to time on both sides of
(2.6) shows

ut ∈ L∞(0,∞;L2(Ω)), and u ∈ L∞(0,∞;W 1,p(Ω)).

Recall that u∗ minimizes E (·). Hence, Corollary 2.3 ensures the boundedness of
the gradients of both u and u∗, more precisely

‖∇u∗‖Lp(Ω) ≤ ‖∇u‖Lp(Ω) ≤M (2.7)

where we have set M := ‖∇u0‖Lp(Ω).
Next, let us focus on the behavior of the energies. Since for large time the

dependence of u on time shrinks, cf. Corollary 2.3, the convergence of energies
should follow from the uniqueness of p-harmonic functions, and indeed, we have
the following result.



EJDE-2021/73 DECAY ESTIMATES 5

Lemma 2.5. Let u∗ and u be solutions of (1.2) and (1.4), respectively. Then

E (u)
t→∞−−−→ E (u∗).

Proof. Since u∗ is the unique minimizer of E (·), it suffices to show that

lim sup
t→∞

1

p

∫
Ω

|∇u|p dx ≤ 1

p

∫
Ω

|∇v|p dx (2.8)

for all v such that v − g ∈ W 1,p
0 (Ω). For that purpose we will basically follow the

proof of [1, Theorem 2.1 ]:

Let v ∈W 1,p(Ω) with v−g ∈W 1,p
0 (Ω) be given. Consider the auxiliary function

ϕ(t) :=
1

2

∫
Ω

(u(t, x)− v(x))
2

dx.

Then ϕ ∈W 2,1(0,∞), cf. Remark 2.4, and, as u fulfills (1.4), we have

ϕ′′(t) + aϕ′(t) =

∫
Ω

(u− v)∆pu+ u2
t dx = −

∫
Ω

|∇u|p−2〈∇u,∇u−∇v〉+ u2
t dx

(2.2)

≤
∫

Ω

1

p
|∇v|p − 1

p
|∇u|p + u2

t dx ≤
∫

Ω

1

p
|∇v|p +

3

2
u2
t dx− E(T )

for all t ∈ [0, T ], where we have used that E(·) is non-increasing. A multiplication
of both sides with eat, followed by an integration yields

ϕ′(t) ≤ e−atϕ′(0) +
1

a
(1− e−at)

(∫
Ω

1

p
|∇v|p dx− E(T )

)
+

3

2

∫ t

0

∫
Ω

e−a(t−τ)u2
t (τ, x) dxdτ.

Integrating once more and using the fact that E(T ) ≥ 1
p

∫
Ω
|∇u|p dx, implies

ϕ(T ) +
1

a2

(
aT − 1 + e−aT

) 1

p

∫
Ω

|∇u|p dx

≤ 1

a2

(
aT − 1 + e−aT

) 1

p

∫
Ω

|∇v|p dx + ϕ(0) +
1

a
(1− e−aT )ϕ′(0) + h(T )

(2.9)

where we have set

h(T ) :=
3

2

∫ T

0

∫ t

0

∫
Ω

e−a(t−τ)u2
t (τ, x) dx dτ dt

=
3

2a

∫ T

0

∫
Ω

u2
t (τ, x)(1− e−a(T−τ)) dx dτ.

From Remark 2.4, the term h(T ) is bounded. Hence, dividing (2.9) by 1
a2

(
aT −1 +

e−aT
)

and letting T →∞ gives the desired estimate (2.8). �

On account of the convergence of the energies, we obtain the W 1,p convergence
of u to u∗.

Corollary 2.6. Let u and u∗ be as before, then we have

‖u− u∗‖W 1,p(Ω)
t→∞−−−→ 0.



6 F. BOZORGNIA, P. LEWINTAN EJDE-2021/73

Proof. By Poincaré’s inequality we have∫
Ω

|u− u∗|p dx ≤ c̃(p,Ω)

∫
Ω

|∇u−∇u∗|p dx. (2.10)

For the p-harmonic function u∗ it holds∫
Ω

|∇u∗|p−2〈∇u∗,∇u−∇u∗〉 = 0,

so that, by (2.2) we obtain

c(p)

∫
Ω

|∇u−∇u∗|p dx ≤
∫

Ω

|∇u|p − |∇u∗|p dx = p · (E (u)− E (u∗)). (2.11)

Thus, combining the above estimates we arrive at

‖u− u∗‖W 1,p(Ω) ≤ c(p,Ω) · |E (u)− E (u∗)| t→∞−−−→ 0

by Lemma 2.5. �

3. Proof of decay rates

We are now prepared to prove our main result.

Proof of Theorem 1.1. A multiplication of

utt + a ut = ∆pu−∆pu
∗

by w = w(t, x) := u(t, x) − u∗(x), and integrating by parts (note that w
∣∣
∂Ω

= 0)
yields∫

Ω

wtt w + awt w dx = −
∫

Ω

〈|∇u|p−2∇u− |∇u∗|p−2∇u∗,∇u−∇u∗〉dx

(2.1)

≤ −22−p
∫

Ω

|∇w|p dx.

Hence, multiplying both sides of the last inequality by a > 0 and adding
∫

Ω
aw2

t we
end up with

d

dt

∫
Ω

a2

2
w2 + awwt dx ≤

∫
Ω

aw2
t − 22−pa|∇w|p dx. (3.1)

Recall the definition of our error term

e(t) :=

∫
Ω

a2

2
w2 + awwt + w2

t + 2
(1

p
|∇u|p − 1

p
|∇u∗|p

)
dx. (1.5)

So, e ∈ W 1,1(0,∞) and due to the minimizing properties of u∗ = u∗(x), we have
that e(t) ≥ 0 for all t > 0. Since wt = ut we obtain

e′(t)
(3.1)

≤
∫

Ω

aw2
t − 22−pa|∇w|p dx+

d

dt

∫
Ω

u2
t + 2

1

p
|∇u|p dx

(2.4)
= −a

∫
Ω

w2
t + 22−p|∇w|p dx ≤ 0.

(3.2)



EJDE-2021/73 DECAY ESTIMATES 7

Moreover, again with wt = ut we have

e′(t)

a
=

∫
Ω

awwt + wwtt + w2
t dx+

2

a

d

dt

∫
Ω

1

2
w2
t +

1

p
|∇u|p dx

(2.4)
=

∫
Ω

awwt + wwtt − w2
t dx =

∫
Ω

w(∆pu−∆pu
∗)− w2

t dx

=

∫
Ω

w∆pu− w2
t dx,

(3.3)

since ∆pu
∗ = 0 in Ω. Using integration by parts (note that w

∣∣
∂Ω

= 0) we obtain∣∣ ∫
Ω

w∆pudx
∣∣ =

∣∣ ∫
Ω

|∇u|p−2〈∇u,∇w〉dx
∣∣ ≤ ∫

Ω

|∇u|p−1|∇w|dx

≤ ‖∇u‖p−1
Lp(Ω)‖∇w‖Lp(Ω)

(3.4)

by Hölder’s inequality. Using the boundedness of the gradient (2.7) we conclude
that

|e
′(t)

a
| (3.3)

=
∣∣∣ ∫

Ω

w∆pu− w2
t dx

∣∣∣
(3.4)

≤ ‖∇u‖p−1
Lp(Ω)‖∇w‖Lp(Ω) + ‖wt‖2L2(Ω)

(2.7)

≤ Mp−1‖∇w‖Lp(Ω) + ‖wt‖2L2(Ω)
t→∞−−−→ 0,

(3.5)

by Corollaries 2.6 and 2.3, respectively.
Our next goal is to estimate the error in terms of its derivative. From (2.3) we

arrive at

e(t) ≤
∫

Ω

(a2

2
+ a
)
w2 +

(a
4

+ 1
)
w2
t dx+ c(p,Ω, u0)‖∇w‖Lp(Ω) .

Using Lebesgue embedding and Poincaré’s inequality for the first term we obtain

e(t) ≤ c1(p,Ω, a)‖∇w‖2Lp(Ω) +
(a

4
+ 1
) ∫

Ω

w2
t dx+ c(p,Ω, u0)‖∇w‖Lp(Ω) .

Furthermore, in (3.2) we already aimed∫
Ω

w2
t + 22−p|∇w|p dx ≤ −e′(t)

a
.

All in all, we obtain

e(t) ≤ c2(p,Ω, a)
(
− e′(t)

a

)2/p

+
(a

4
+ 1
)(
− e′(t)

a

)
+ c3(p,Ω, u0)

(
− e′(t)

a

)1/p

.

Since

−e′(t)

a

t→∞−−−→ 0,

cf. (3.5), the error term e(t) ≥ 0 satisfies for large time a differential inequality

e(t) ≤ c4(p,Ω, u0, a)(−e′(t))
1
p ,

and we may rewrite this as

e′(t) ≤ −c5(p,Ω, u0, a) · e(t)p,

respectively, so by [8, Lemma 1.6] we obtain

e(t) ≤ c6(p,Ω, u0, a)t−
1

p−1 . (3.6)



8 F. BOZORGNIA, P. LEWINTAN EJDE-2021/73

By (3.6), (2.11) and Poincaré inequality we finally arrive at

‖u− u∗‖pW 1,p(Ω) ≤ c7(p,Ω, u0, a)t−
1

p−1 . �

3.1. Enhancement of the decay rate for p = 2. A crucial ingredient in our
proof of the decay rate was inequality (2.3) which we applied to estimate the dif-
ference of the energies. In fact, for p = 2 this relation can be improved to the
equality ∫

Ω

|∇u|2 − |∇u∗|2 dx =

∫
Ω

|∇u−∇u∗|2 dx

where we used the harmonicity of u∗. Hence, we obtain

e(t) ≤
∫

Ω

(a2

2
+ aε

)
w2 +

( a
4ε

+ 1
)
w2
t + |∇w|2 dx

≤
∫

Ω

((a2

2
+ aε

)
c̃(Ω) + 1

)
|∇w|2 +

( a
4ε

+ 1
)
w2
t dx

= c(a,Ω)

∫
Ω

w2
t + |∇w|2 dx

≤ c(a,Ω)
(
− e′(t)

a

)
(3.7)

where in the intermediate steps we used the Poincaré inequality, and ε > 0 was
chosen in such a way that the prefactors coincided. Relation (3.7) may be rewritten
as

e′(t) ≤ − a

c(a,Ω)
e(t) for all t > 0,

so, by Gronwall’s inequality, the error term fulfills

e(t) ≤ c · exp
(
− a

c(a,Ω)
t
)

and for the decay rate we arrive at

‖u− u∗‖2W 1,2(Ω) ≤ C · exp
(
− a

c(a,Ω)
t
)

for all t > 0,

which is a well known result, cf. [8, Theorem 2.1 a) ].

Acknowledgments. The authors are grateful to the Hausdorff Research Institute
for Mathematics (Bonn) for the support and hospitality during the trimester pro-
gram Evolution of Interfaces, where the work on this article was undertaken. Also
the authors want to thank John Anderson for helpful suggestions and discussion.

F. Bozorgnia was supported by the Portuguese National Science Foundation
through FCT and by European Unions Horizon 2020 research and innovation pro-
gram under Marie Skodowska-Curie grant agreement No. 777826 (NoMADS).

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Farid Bozorgnia

CAMGSD, Instituto Superior Técnico, University of Lisbon, Av. Rovisco Pais, 1049-001

Lisbon, Portugal
Email address: farid.bozorgnia@tecnico.ulisboa.pt

Peter Lewintan

Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127
Essen, Germany

Email address: peter.lewintan@uni-due.de


	1. Introduction and problem setting
	2. Basic results
	3. Proof of decay rates
	3.1. Enhancement of the decay rate for p=2
	Acknowledgments

	References