EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6266
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Global Well Posedness for Damped Wave Models with
Damping in the Memory

Tayeb Hadj Kaddour1, Ali Hakem2, Abdelkader Benali3, Ibrahim Alraddadi4,∗,
Hijaz Ahmad4,5,6,7, Taha Radwan8, Dragan Pamucar9,∗

1 Department of Mathematics, Faculty of Exact Science and Informatics, Hassiba Benbouali
University of Chlef, Chlef, Algeria
2 Department of Technology, Laboratory ACEDP, Djilali Liabès University of Sidi Belabbes,
Sidi Belabbes, Algeria
3 Department of Mathematics, Faculty of Exact Science and Informatics, Laboratory of
Mathematics and its Applications LMA, Hassiba Benbouali University of Chlef, Algeria
4 VSB–Technical University of Ostrava, CEET, ENET Centre, 17. listopadu 2172/15, 708
00 Ostrava–Poruba, Czech Republic
5 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah,
Saudi Arabia
6 Operational Research Center in Healthcare, Near East University, Nicosia/TRNC, 99138
Mersin 10, Turkey
7 Department of Mathematics, College of Science, Korea University, 145 Anam-ro,
Seongbuk-gu, Seoul 02841, South Korea
8 Department of Management Information Systems, College of Business and Economics,
Qassim University, Buraydah 51452, Saudi Arabia
9 Széchenyi István University, Győr, Hungary

Abstract. This paper aims to study the Cauchy problem for damped wave models with a dis-
sipative memory term. The main objective is to establish global (in time) well-posedness results
for both energy solutions and higher-regularity solutions, determine the critical exponent in the
Fujita sense, and investigate the influence of nonlinear memory on the Fujita exponent. Using
modern tools from harmonic analysis and the Banach fixed point method, we show several results
by taking into consideration different regularity properties of the initial data.
2020 Mathematics Subject Classifications: 35L15, 35L71, 35B44, 35L05, 33C15
Key Words and Phrases: Damped wave equation, nonlinear memory, Cauchy problem, energy
solutions, Matsumura type estimates, fixed point theorem

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6266

Email addresses: t.hadjkaddour@univ-chlef.dz (T. Hadj Kaddour),
hakemali@yahoo.com (A. Hakem), benali4848@gmail.com (A. Benali),
ialraddadi@iu.edu.sa (I. Alraddadi), hijaz.ahmad@iu.edu.sa (H. Ahmad),
t.radwan@qu.edu.sa (T. Radwan), pamucar.dragan@sze.hu (D. Pamucar)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 2 of 24

1. Introduction

In this paper, we are interesting in the study of a Cauchy problem for a damped wave
model with dissipative term as fractional semilinear power-type non-linearity on the right-
hand side. The most important gaol is to study carefully the influence of the dissipative
memory term on the range of global (in time) existence of energy solutions for different
regularities. In order to satisfy our aim we prove several results of global existence and
find the critical exponent in Fujita sense for our model by using matsumura type estimates
and Banach fixed point theorem. For physical application we refer the reader to check
Eg. [1–5] and the references therein. First, let us introduce some crucial notations used
throughout this paper
Notations

• (x)+ = max{x, 0}: The positive part of x.

• ut: The first partial differential of u with respect to t.

• ∗x: The convolution product with respect to the x−variable.

• p∗ or pFuj : The critical exponent in Fujita sense.

• |D|σ: Pseudo-differential operator of order σ.

• Ḣ: Homogeneous Sobolev space.

2. Background and preliminaries

2.1. Background and Motivation

Recently the damped wave equation

utt −∆u+ ut =

∫ t

0
(t− τ)−γ |u(τ, ·)|pdτ, (t, x) ∈ (0,∞)× Rn, (1)

is considered by A. Fino in [6] where he proved the global (in time) existence of energy
solution by using the weighted energy method and blow-up results by the test function
method. In particular he showed that the critical exponent in Fujita sense in the L1 ∩
L2−theory for the Cauchy problem of equation (1) is

p∗(n, γ) = max
{
pγ(n);

1

γ

}
, where pγ(n) = 1 +

2(2− γ)

(n− 2(1− γ))+
, (2)

where (x)+ stands for the positive part of x,this means that (n− 2(1− γ))+ = max{n−
2(1−γ), 0}. In [7], where the same model is considered, M. D’Abbicco has improved some
results of global existence by using Matsumura type estimates but he didn’t investigate
blow-up. In particular he proposed the same critical exponent proposed in [6]. In fact the
authors proved in [6] and [7] that the term utt does not influence the critical exponent since



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 3 of 24

Cazenave et al proved in [8] that the critical exponent is the same (2) for the corresponding
heat equation with non linear memory

ut −∆u =

∫ t

0
(t− τ)−γu(τ, ·)|u(τ, ·)|p−1dτ, (t, x) ∈ (0,∞)× Rn. (3)

In the other hand, the semilinear Cauchy problem corresponding to (1)

utt −∆u+ ut = |u|p, (4)

is considered by many authors, we refer the reader to eg. [9], [10] where they showed
that the critical exponent in the L1 ∩ L2 theory for Cauchy problem of (4) is the Fujita
exponent

pFuj = 1 +
n

2
. (5)

In the recent paper [11], we proved that some class of effective damping influence the
critical exponent in Fujita sense. In this paper we will show that the dissipative nonlinear
memory dominates the damping term ut by considering the following Cauchy problem:

utt −∆u+ ut =
∫ t
0 (t− τ)−γ |ut(τ, x)|p dτ, (t, x) ∈ (0,∞)× Rn,

u(0, x) = u0(x), ut(0, x) = u1(x), x ∈ Rn,
(6)

where γ ∈ (0, 1) and p > 1. Here, the dissipative memory has to be seen as Riemann-
Liouville fractional integral to a constant up of |ut|p. For this reason the non linear term
in (6) is interpreted as a fractional power type non linearity.

Finally, we finish this discussion by the connexion between the semilinar damped wave
equation and the damped wave equation with non linear memory by noting that∫ t

0
(t− s)−γ |u(s, ·)|p ds −→ Γ(1− γ)|u(t, ·)|p as γ −→ 1, (7)

in distribution sense, where Γ is the Euler gamma function. For this reason we have

lim
γ→1

p∗(n, γ) = pFuj , (8)

2.2. Representation of the solution

The presence of the nonlinear term on the right-hand side suggests to us the apply the
Duhamel’s principle where the study of the Cauchy problem (6) reduces to the study of
the following Cauchy problem:

utt −∆u+ ut = 0, (t, x) ∈ (0,∞)× Rn,
u(0, x) = u0(x), ut(0, x) = u1(x), x ∈ Rn,

(9)

and the family of parameter-dependent Cauchy problems

vtt −∆v + vt = 0, (t, x) ∈ (τ,∞)× Rn,
v(τ, x) = 0, x ∈ Rn, τ ≥ 0,

vt(τ, x) = h(τ, x) :=
∫ τ
0 (τ − s)−γ |ut(s, x)|p ds, x ∈ Rn, τ ≥ 0.

(10)



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 4 of 24

Denoting v = v(t, τ, x) the solution of Cauchy problem (10), and ulin = ulin(t, x) the
solution of Cauchy problem (9), then the solution u = u(t, x) of the Cauchy problem (6)
is formally given by

u(t, x) = ulin(t, x) + unl(t, x), (11)

where
unl(t, x) :=

∫ t

0
v(t, τ, x) dτ.

By using Fourier transform argument, the solution ulin = ulin(t, x) of Cauchy problem
(9) is given by

ulin(t, x) = E0(t, 0, x) ∗x u0(x) + E1(t, 0, x) ∗x u1(x), (12)

where E0 = E0(t, 0, x) and E1 = E1(t, 0, x) are the fundamental solutions for the Cauchy
problem (9), that is, E0 corresponds to the initial data (u0, u1) = (δ0, 0) and E1 =
E1(t, 0, x) corresponds to the initial data (u0, u1) = (0, δ0), where δ0 denotes the Dirac
Delta-distribution supported in 0. Here and in the sequel, ∗x denotes for the convolution
with respect to the x−variables.
Similarly, the solution v = v(t, τ, x) of the Cauchy problem (10) is given by

v(t, τ, x) = E1(t, τ, x) ∗x h(τ, u),
where h(τ, u) =

∫ τ
0 (τ − s)−γ |ut(s, x)|p ds, t > τ ≥ 0,

(13)

since v(τ, x) = 0 for all x ∈ Rn and τ ≥ 0. By using (11), (12) and (13) the solution of
the Cauchy problem (6) is formally given as a solution of the fixed point equation

u(t, x) = E0(t, 0, x) ∗x u0(x) + E1(t, 0, x) ∗x u1(x) +
∫ t

0
E1(t, τ, x) ∗x h(τ, u) dτ.

Since the coefficients of the linear problem (9) associated to the cauchy problem (6) are
constants then the considered model (6) is invariant by translation. For this reason we
can make a shift of the solution in the non linear part. In this way, the solution of Cauchy
problem (6) is formally represented as

u(t, x) = E0(t, 0, x) ∗x u0(x) + E1(t, 0, x) ∗x u1(x)

+

∫ t

0
E1(t− τ, 0, x) ∗x h(τ, u) dτ. (14)

Now, let us introduce some notations that will be used in the sequel. For all T > 0 we
denote by X(T ) the space of solutions to the Cauchy problem (6). For all u ∈ X(T ) we
define the mapping N as follows:

N : u ∈ X(T ) → N(u)

= E0(t, 0, x) ∗ u0(x) + E1(t, 0, x) ∗ u1(x) +
∫ t

0
E1(t− τ, 0, x) ∗x h(τ, u) dτ.



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 5 of 24

Our main strategy is to prove well-posedness results for solutions to (6) as solutions of the
fixed point equation u = N(u) by proving the following inequalities for all u, v ∈ X(T ):

‖Nu‖X(T ) ≤ C‖(u0, u1)‖Aσ,m + C‖u‖pX(T ), (15)

‖Nu−Nv‖X(T ) ≤ C‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
, (16)

where the norm ‖·‖X(T ) in X(T ) will be defined later in a suitable way and makes X(T ) as
a Banach space. For the further considerations we introduce the scale {Am,σ} of Banach
spaces with σ ≥ 1 and m ∈ [1, 2), where

Am,σ = Am,σ(Rn) =
(
Hσ(Rn) ∩ Lm(Rn)

)
×
(
Hσ−1(Rn) ∩ Lm(Rn)

)
. (17)

2.3. Outline of this paper

This paper is structured to help the reader clearly understand its objectives. It consists
of six sections and an appendix. Section 1 provides the work and outlines its purpose.
Section 2 presents the background and preliminaries, including the strategy of proof. In
Section 3, we states the main results, which are proved in Section 4. For brevity, the
proof of Theorem 4 is omitted and breviary discussed since it follows the same arguments
as the proof of Theorem 3. Section 5 concludes the paper, highlighting in particular its
novelty and main contributions. Section 6 contains our acknowledgements, followed by
a bibliography arranged in alphabetical order. The paper ends with an appendix, which
gathers the main tools used, especially in Section 4, together with additional remarks.

Throughout the present paper we write f . g when there exists a constant C > 0 such
that f ≤ Cg, and f ≈ g when g . f . g. Nonnegative constants C or Cj , j ∈ N, are
always supposed to be independent of T > 0. For the sake of brevity we sometimes put
for all n ∈ N∗ and (k, j) ∈ N2

J (k,j)
n (t) =

∫ t

0
(1 + t− τ)−

n
4
− k

2
−j

∫ τ

0
(τ − s)−γ(1 + s)−βdsdτ. (18)

3. Main result

3.1. Global existence of energy solution in low dimension

In term of comparison, the dissipative term ut does not influence the critical exponent
since the two models (3) and (1) have the same critical exponent in Fujita sense. In the
other hand, one can note that the non linear memory influence the critical exponent by
comparing the critical exponent of (1) and (4) and there is continuity with respect to γ
(see (8)). In this section we study the influence of the damping term in the memory on
the critical exponent by comparing the critical exponent of Cauchy problem (6) with the
one obtained for (1). For this reason, we begin by taking m = σ = 1 in (17). In Section
3.3, we will see that the range of admissible p for global existence can not be improved
even by adding an additional regularity. So, let us fix in this section m = σ = 1 in (17),



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 6 of 24

that is the data are from the energy space without additional regularity (ie. m = 1 in
(17)), and set for all n ≥ 1 and γ ∈ (0, 1)

pγ = pγ(n) :=
1

γ
. (19)

Then we have
Theorem 1. Assume that γ ∈ (1/2, 1) and p > pγ for n = 1 or γ ∈ (0, 1) and p > pγ for
n = 2. Then there exists ε > 0 such that for any initial data

(u0, u1) ∈ A1,1 := H1 ∩ L1 × L2 ∩ L1 with ‖(u0, u1)‖A1,1 ≤ ε,

there exists a unique global (in time) solution to Cauchy problem (6)

u ∈ C
(
[0,∞),H1

)
∩ C1

(
[0,∞), L2

)
.

Moreover, the solution satisfies the following Matsumura type decay estimates

‖u(t, ·)‖L2 . (1 + t)−
n
4
−γ+1‖(u0, u1)‖A1,1 ,

‖∇u(t, ·)‖L2 .

{
(1 + t)

1
4
−γ‖(u0, u1)‖A1,1 if n = 1,

(1 + t)−γ log(2 + t)‖(u0, u1)‖A1,1 if n = 2,

‖∂j
t∇ku(t, ·)‖L2 . (1 + t)−γ‖(u0, u1)‖A1,1 for all j ≥ 1 and k ≥ 2.

3.2. Global existence in high dimension

In Section 3.1 we presented global existence for low dimension (n = 1 and n = 2). In
that case the admissible range for p is superiorly unbounded. In space dimensional n ≥ 3,
there appears an upper bound for the admissible values for p. This upper bound is caused
by the application of Gagliardo-Nirenberg inequality (see condition 108). So, let us, first,
state the global (in time) existence results
Theorem 2. Let n ≥ 3. Assume that γ ∈ (n−2

n , 1) and pγ < p ≤ pGN := n
n−2 . Then there

exists ε > 0 such that for any initial data

(u0, u1) ∈ A1,1 := H1 ∩ L1 × L2 ∩ L1 with ‖(u0, u1)‖A1,1 ≤ ε,

there exists a unique global (in time) solution to Cauchy problem (6)

u ∈ C
(
[0,∞),H1

)
∩ C1

(
[0,∞), L2

)
.

Moreover, the solution satisfies the following Matsumura type decay estimates

‖u(t, ·)‖L2 .

 (1 + t)
1
4
−γ‖(u0, u1)‖A1,1 if n = 3,

(1 + t)−γ log(2 + t)‖(u0, u1)‖A1,1 if n = 4,
(1 + t)−γ‖(u0, u1)‖A1,1 if n ≥ 5,

(20)

‖u(t, ·)‖Ḣ1/2 .

{
(1 + t)−γ log(2 + t)‖(u0, u1)‖A1,1 if n = 3,
(1 + t)−γ‖(u0, u1)‖A1,1 if n ≥ 4,

(21)

‖∇u(t, ·)‖L2 . (1 + t)−γ‖(u0, u1)‖A1,1 , (22)
‖∂tu(t, ·)‖L2 . (1 + t)−γ‖(u0, u1)‖A1,1 . (23)



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 7 of 24

3.3. Global existence of high regular solution with additional regularity
m ∈ [1, 2)

In this section we require an additional regularity on the data. However, we choose
m ∈ (1, 2) and σ > 1 in the data space (17). The purpose of this section is to show that
the regularity of the data does not influence the critical exponent, although the additional
regularity improve somehow the range of admissible p as it is well known. First, let us set
for all n ≥ 3, m ∈ (1, 2), γ ∈ (0, 1) and σ > 1

pγ,m(n) =
1

γ
. (24)

Then we have

Theorem 3. Assume that n ≥ 3, σ ∈ (1, n2 ), γ ∈ (0, 1) and m ∈ (1, 2) such that

n

2

( 1

m
− 1

2

)
< 1,

n

2

( 1

m
− 1

2

)
+

σ

2
> 1, (25)

and p satisfies the condition

p > max
{
dσe; pγ,m(n)

}
, (26)

Then, there exists a positive constant ε0 such that for any initial data

(u0, u1) ∈ Am,σ(Rn) satisfying ‖(u0, u1)‖Am,σ ≤ ε for all ε ≤ ε0,

there is a uniquely determined global (in time) energy solution

u ∈ C
(
[0,∞),Hσ

)
∩ C1

(
[0,∞),Hσ−1

)
to the Cauchy problem (6). Moreover, the solution satisfies the following estimates:

‖u(t, ·)‖L2 . (1 + t)1−
n
2
( 1
m
− 1

2
)−γ‖(u0, u1)‖Am,σ ,

‖|D|σu(t, ·)‖L2 . (1 + t)−γ‖(u0, u1)‖Am,σ ,

‖ut(t, ·)‖L2 . (1 + t)−γ‖(u0, u1)‖Am,σ ,

‖|D|σ−1ut(t, ·)‖L2 . (1 + t)−γ‖(u0, u1)‖Am,σ .

The second result we have is

Theorem 4. Let us assume n ≥ 3, σ ∈ (1, n2 ), γ ∈ (0, 1) and m ∈ [1, 2) such that

n

2

( 1

m
− 1

2

)
≥ 1, (27)

and p satisfies the condition

max
{
dσe; pγ,m(n)

}
< p ≤ pGN :=

n

n− 2
, (28)



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 8 of 24

Then, there exists a positive constant ε0 such that for any initial data

(u0, u1) ∈ Am,σ(Rn)satisfying ‖(u0, u1)‖Am,σ ≤ ε for all ε ≤ ε0,

there is a uniquely determined global (in time) energy solution

u ∈ C
(
[0,∞),Hσ

)
∩ C1

(
[0,∞),Hσ−1

)
to the Cauchy problem (6). Moreover, the solution satisfies the following estimates:

‖u(t, ·)‖L2 .

{
(1 + t)−γ log(2 + t)‖(u0, u1)‖Am,σ if n ≥ 4 and m = 2n

n+4 ,

(1 + t)−γ‖(u0, u1)‖Am,σ else ,

‖|D|σu(t, ·)‖L2 . (1 + t)−γ‖(u0, u1)‖Am,σ ,

‖ut(t, ·)‖L2 . (1 + t)−γ‖(u0, u1)‖Am,σ ,

‖|D|σ−1ut(t, ·)‖L2 . (1 + t)−γ‖(u0, u1)‖Am,σ .

4. Proof of the results

4.1. Proof of Theorems 1 and 2

4.1.1. Proof of Theorem 1

Let us introduce, for T > 0, the space of energy solutions

X(T ) = C
(
[0, T ],H1(R)

)
∩ C1

(
[0, T ], L2(R)

)
(29)

with the norm

‖u‖X(T ) = sup
0≤t≤T

{
(1 + t)

n
4
+γ−1‖u(t, ·)‖L2 + `n(t)

−1(1 + t)γ‖∇u(t, ·)‖L2

+(1 + t)γ‖ut(t, ·)‖L2 + (1 + t)γ‖∇ut(t, ·)‖L2

}
, (30)

where
`n(t) =

{
(1 + t)

1
4 if n = 1,

log(2 + t) if n = 2.
(31)

First, let us prove the inequality (15).
The inequality

‖ulin‖X(T ) . ‖(u0, u1)‖A1,1

is an immediate consequence of Proposition 1 and Corollary 1 for τ = 0 and h(0, u(x)) =
u1(x). It remains to show the inequality

‖unl‖X(T ) . ‖u‖pX(T ). (32)

Taking into account the results of Corollary 1 we have

‖unl(t, ·)‖L2 .
∫ t

0
(1 + t− τ)−

n
4

∫ τ

0
(τ − s)−γ‖|ut(s, ·)|p‖L1∩L2 dsdτ. (33)



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 9 of 24

Since

‖|ut(s, ·)|p‖L1∩L2 = ‖ut(s, ·)‖pLp∩L2p = ‖ut(s, ·)‖pLp + ‖ut(s, ·)‖pL2p ,

then, it is obvious that one has to estimate the norms:

‖|ut(s, ·)‖pLp and ‖|ut(s, ·)‖pL2p .

For this purpose, we apply the classical Gagliardo-Nirenberg inequality (107). In this way
we obtain for j = 1, 2 the chain of inequalities

‖ut(s, ·)‖pLjp . ‖ut(s, ·)‖
p(1−θj(p))

L2 ‖∇ut(s, ·)‖
pθj(p)

L2

. (1 + s)−γp‖u‖pX(T )

. (1 + s)−β‖u‖pX(T ), (34)

where
θj(p) = n

(1
2
− 1

jp

)
, (35)

θj(p) ∈ [0, 1] if and only if p ≥ 2 since n ≤ 2, and

β = γp. (36)

Taking j = 1 and j = 2 in (34), respectively, we find

‖u(s, ·)‖p
Lp∩L2p . (1 + s)−β‖u‖pX(T ). (37)

Noting that β > 1 if and only if p >
1

γ
. Including (37) in (33) we find

‖unl(t, ·)‖L2 . J (0,0)
n (t)‖u‖pX(T ), (38)

where J
(0,0)
n (t) is defined by (18) (case: k = j = 0). Since β > 1 we estimate J

(0,0)
n (t) by

using Lemma 1 as follows:
J (0,0)
n (t) . (1 + t)−

n
4
+1−γ . (39)

Then, introducing the estimate (39) into (38) we find

‖unl(t, ·)‖L2 . (1 + t)−
n
4
+1−γ‖u‖pX(T ). (40)

Now we deal with ‖∂j
t∇kunl(t, ·)‖L2 for all k, j ∈ N such that 1 ≤ k+ j ≤ 2. Again, taking

into account the results of Corollary 1 we have the estimate

‖∂j
t∇kunl(t, ·)‖L2 .

∫ t

0
(1 + t− τ)−

n
4
− k

2
−j

∫ τ

0
(τ − s)−γ‖|ut(s, ·)|p‖L1∩L2 dsdτ.

Using the estimate (37) we get

‖∇k∂j
t u

nl(t, ·)‖L2 . J (k,j)
n (t)‖u‖pX(T ), (41)



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 10 of 24

where J
(k,j)
n (t) is defined by (18). Since β > 1 we may estimate J

(k,j)
n (t), after using

Lemma 1 as follows:

J (k,j)
n (t) .

 (1 + t)−
3
4 if n = k = 1 and j = 0,

(1 + t)−γ log(2 + t) if n = 2, k = 1 and j = 0,
(1 + t)−γ if n = 1, 2 and k + j ≥ 1.

(42)

Putting (42) into (41) we derive the estimates

‖∇unl(t, ·)‖L2 .


(1 + t)−

3
4 ‖u‖pX(T ) if n = k = 1 and j = 0,

(1 + t)−γ log(2 + t)‖u‖pX(T ) if n = 2, k = 1 and j = 0,

(1 + t)−γ‖u‖pX(T ) if n = 1, 2 and 1 ≤ k + j ≤ 2.

(43)

Then, the inequality (32) is concluded from (43), (40) and the definition (30) of the
norm in X(T ).

Now, let us turn to the inequality (16).

By definition of the operator N and the results of Corollary 1, we have

‖(Nu−Nv)(t, ·)‖L2 .
∫ t

0
(1 + t− τ)−

n
4

×
∫ τ

0
(τ − s)−γ

∥∥|ut(s, ·)|p − |vt(s, ·)|p
∥∥
L1∩L2 dsdτ. (44)

Then we have to estimate

‖|ut(s, ·)|p − |vt(s, ·)|p‖L1 and ‖|ut(s, ·)|p − |vt(s, ·)|p‖L2 .

First, we have by Hölder’s inequality for j = 1, 2,

‖|ut|p − |vt|p‖Lj . ‖ut − vt‖Ljp

(
‖ut‖p−1

Ljp + ‖vt‖p−1
Ljp

)
. (45)

Then, the aim is to estimate the following terms for j = 1 and j = 2:

‖ut − vt‖Ljp , ‖ut‖p−1
Ljp and ‖vt‖p−1

Ljp .

Let us begin with the estimate of the term ‖ut(s, ·) − vt(s, ·)‖Ljp . By using Gagliardo-
Nirenberg inequality (107) with k = 1 and q = p we estimate

‖ut(s, ·)− vt(s, ·)‖Ljp . ‖ut(s, ·)− vt(s, ·)‖
1−θj(p)

L2

∥∥|∇k(ut − vt)(s, ·)
∥∥θj(p)
L2

. (1 + s)−γ(1−θj(p))(1 + s)−γθj(p)‖u− v‖X(T )

. (1 + s)−γ‖u− v‖X(T ), (46)

where θj(p) is defined (35). We use the same tools to estimate ‖ut(s, ·)‖p−1
Ljp . We get

‖ut(s, ·)‖p−1
Ljp . ‖ut(s, ·)‖

(p−1)(1−θj(p))

L2 ‖∇ut(s, ·)‖
(p−1)θj(p)

L2



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 11 of 24

. (1 + s)−γ(p−1)(1−θj(p))(1 + s)−γ(p−1)θj(p)‖u‖p−1
X(T )

. (1 + s)−γ(p−1)‖u‖p−1
X(T ). (47)

In the same way we derive

‖vt(s, ·)‖p−1
Ljp . (1 + s)−γ(p−1)‖v‖p−1

X(T ). (48)

Finally, by (48), (47) and (46) we get from (45), for j = 1, 2 the estimates

‖|ut(s, ·)|p − |vt(s, ·)|p‖Lj . (1 + s)−β‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
, (49)

where β is given by (36). As a consequence, it follows from (49) that

‖|ut(s, ·)|p − |vt(s, ·)|p‖L1∩L2 . (1 + s)−β‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
. (50)

Plugging the estimate (50) in (44) we find

‖(Nu−Nv)(t, ·)‖L2 . J (0,0)
n (t)‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
,

where J
(0,0)
n (t) is defined by (18) (case: k = j = 0). Then, using the estimate (39) we may

conclude that

‖(Nu−Nv)(t, ·)‖L2 . (1 + t)−
n
4
+1−γ‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
. (51)

Now, let us turn to estimate the term ‖∂j
t∇k(Nu−Nv)(t, ·)‖L2 for 1 ≤ k + j ≤ 2. As we

did above we have after using the results of Corollary 2 we have

‖∂j
t∇k(Nu−Nv)(t, ·)

∥∥
L2 .

∫ t

0
(1 + t− τ)−

n
4
− k

2
−j

×
∫ τ

0
(τ − s)−γ‖|ut(s, ·)|p − vt(s, ·)|p‖L1∩L2 dsdτ. (52)

Including the estimate (50) into (52) we find∥∥∇k∂j
t (Nu−Nv)(t, ·)

∥∥
L2 . J (k,j)

n (t)‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
, (53)

where J
(k,j)
n (t) is defined by (18). Using the estimate (42) for J

(k,j)
n (t) we find from (53)

the estimate∥∥∇k∂j
t (Nu−Nv)(t, ·)

∥∥
L2

.


(1 + t)−

3
4 ‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
if n = k = 1 and j = 0,

log(2+t)
(1+t)γ ‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
if n = 2, k = 1 and j = 0,

(1 + t)−γ‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
if n = 1, 2 and 1 ≤ k + j ≤ 2.

(54)

Finally, the desired inequality (16) is concluded from the estimates (54), (51) and the
definition (30) for the norm in X(T ). This ends the proof of Theorem 1.



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 12 of 24

4.1.2. Proof of Theorem 2

The proof of Theorem 2 is similar to that of Theorem 1, except, in Theorem 2 an upper
bound appears for the range of admissible values of p. This upper bound is caused by
the application of Gagliardo-Nirenberg inequality. More precisely, in formula (35), the
condition θj(p) ∈ [0, 1] implies

2

j
≤ p ≤ 2n

(n− 2)j
, (55)

according to (108). Since condition (55) must hold simultaneously for j = 1, 2 we conclude
that 2 ≤ p ≤ n

n−2 . The parameter n
n−2 is called Gagliardo-Nirenberg exponent and,

usually, denoted by pGN . Since the proofs are essentially the same we omit the details of
proof of Theorem 2.

4.2. Proof of Theorems 3 and 4

4.2.1. Proof of Theorem 3

Let us introduce, for T > 0, the space of energy solutions

X(T ) = C
(
[0, T ],Hs(R)

)
∩ C1

(
[0, T ],Hs−1(R)

)
(56)

with the norm

‖u‖X(T ) = sup
0≤t≤T

{
(1 + t)

n
2
( 1
m
− 1

2
)+γ−1

(
‖u(t, ·)‖L2 + (1 + t)γ‖|D|σu(t, ·)‖L2

)
+(1 + t)γ‖ut(t, ·)‖L2 + (1 + t)γ‖|D|σ−1ut(t, ·)‖L2

}
. (57)

As usual, the aim is to prove the inequalities (15) and (16). Let us start with the first
one. The inequality

‖ulin‖X(T ) . ‖(u0, u1)‖A1
1,0

is concluded directly by Proposition 2 from [11] and Corollary 2 for τ = 0 and h(0, u(x)) =
u1(x). It remains to show the inequality

‖unl‖X(T ) . ‖u‖pX(T ). (58)

Thanks to the results of Proposition 2 we have for κ = 0, σ (with the convention |D|κ = Id
if κ = 0),

‖|D|κunl(t, ·)‖L2 .
∫ t

0
(1 + t− τ)−

n
2
( 1
m
− 1

2
)−κ

2

×
∫ τ

0
(τ − s)−γ

∥∥|u(s, ·)|p∥∥
Lm∩L2∩Ḣσ−1 dsdτ. (59)

For this reason we shall estimate the norms

‖|u(s, ·)|p‖Lm , ‖|u(s, ·)|p‖L2 , and ‖|u(s, ·)|p‖Ḣσ−1 .



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 13 of 24

First, we apply fractional Gagliardo-Nirenberg inequality we get for j = m and j = 2
the estimates

‖|ut(s, ·)|p‖Lj = ‖ut(s, ·)‖pLjp . ‖ut(s, ·)‖
p(1−θσ,j(p))

L2

∥∥∣∣D∣∣σut(s, ·)∥∥pθσ,j(p)L2

. (1 + s)−β‖u‖pX(T ), (60)

where
β = γp, (61)

and θσ,j(p) =
n
σ (

1
2 − 1

jp) ∈ [0, 1] if and only if{
2
j ≤ p if n ≤ 2σ,
2
j ≤ p ≤ 2n

(n−2σ)j if n > 2σ.

Summarizing, we have

‖|u(s, ·)|p‖Lm∩L2 . (1 + s)−β‖u‖pX(T ), (62)

where β is given by (61) and {
2 ≤ p if n ≤ 2σ,
2 ≤ p ≤ n

n−2σ if n > 2σ.
(63)

To estimate the norm ‖|ut(s, ·)|p‖Ḣσ−1 we apply the fractional chain rule (110) as follows:

‖|ut(s, ·)|p‖Ḣσ−1 = ‖|D|σ−1|ut(s, ·)|p‖L2

. ‖ut(s, ·)‖p−1
Lq1 ‖|D|σ−1ut(s, ·)‖Lq2 for p > dσ − 1e, (64)

where
p− 1

q1
+

1

q2
=

1

2
. (65)

A possible choice is, for example, q1 = n(p − 1) and q2 = 2n
n−2 for n ≥ 3. The norm

‖ut(s, ·)‖p−1
Lq1 can be estimated by using classical Gagliardo-Nirenberg inequality (107). In

this way we may conclude

‖ut(s, ·)‖p−1
Lq1 . ‖ut(s, ·)‖

(p−1)(1−θσ,3(q1))

L2 ‖|D|σut(s, ·)‖
(p−1)θσ,3(q1)

L2

. (1 + s)−γ(p−1)‖u‖p−1
X(T ), (66)

where θσ,3(q1) =
n
σ (

1
2 − 1

q1
) ∈ [0, 1] if and only if

2 ≤ q1 ≤
2n

n− 2σ
if n > 2σ.

On the other hand, applying the fractional Gagliardo-Nirenberg inequality (106) gives

‖|D|σ−1ut(s, ·)‖Lq2 . ‖ut(s, ·)‖
1−θσ,σ−1(q2)

L2 ‖|D|σut(s, ·)‖
θσ,σ−1(q2)

L2



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 14 of 24

. (1 + s)−γ‖u‖X(T ), (67)

where θσ,σ−1(q2) =
n
σ (

1
2 − 1

q2
) + σ−1

σ ∈ [σ−1
σ , 1] if and only if

2 ≤ q2 ≤
2n

n− 2
if n ≥ 3.

Substituting the estimates (67) and (66) into (64) we obtain the estimate

‖|u(s, ·)|p‖Ḣσ−1 . (1 + s)−γp‖u‖pX(T ) = (1 + s)−β‖u‖pX(T ), (68)

where β is defined by (61). Consequently, from the estimates (68) and (62) we conclude
the estimate

‖|u(s, ·)|p‖Lm∩L2∩Ḣσ−1 . (1 + s)−β‖u‖pX(T ), (69)

where β is defined by (61). Notice that β > 1 if and only if p > pγ,m(n). Then, plugging
(69) into (59) we get

‖|D|κunl(t, ·)‖L2 . ‖u‖pX(T )

∫ t

0
(1 + t− τ))−

n
2
( 1
m
− 1

2
)−κ

2

×
∫ τ

0
(τ − s)−γ(1 + s)−β dsdτ . Jκ,0

n (t)‖u‖pX(T ), (70)

where Jκ,0
n (t) is defined by (18) (case (k, j) = (κ, 0)). So, due to Lemma 1 and the

assumptions of Theorem 3 we may estimate Jκ,0
n (t) for κ = 0 or κ = σ as follows:

Jκ,0
n (t) .

{
(1 + t)1−

n
2
( 1
m
− 1

2
)−γ if κ = 0,

(1 + t)−γ if κ = σ.
(71)

The estimates (71) lead to the following estimates:

‖unl(t, ·)‖L2 . (1 + t)1−
n
2
( 1
m
− 1

2
)−γ‖u‖pX(T ), (72)

and ∥∥|D|σunl(t, ·)
∥∥
L2 . (1 + t)−γ‖u‖pX(T ). (73)

Now let us turn to estimate the norms ‖unlt (t, ·)‖L2 and ‖|D|σ−1unlt (t, ·)‖L2 . Again, by
Proposition 2, we have for κ = 1, σ the estimates

‖|D|κ−1unlt (t, ·)‖L2 .
∫ t

0
(1 + t− τ)−

n
2
( 1
m
− 1

2
)−κ−1

2
−1

×
∫ τ

0
(τ − s)−γ

∥∥|u(s, ·)|p∥∥
Lm∩L2∩Ḣσ−1 dsdτ. (74)



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 15 of 24

Taking into account the estimate (69) we get from (74) the estimates

‖|D|κ−1unlt (t, ·)‖L2 . J (κ−1,1)
n (t)‖u‖pX(T ), (75)

where J
(κ−1,1)
n (t) is defined by (18) (case (k, j) = (κ − 1, 1)). Thanks to Lemma 1, the

integral J (κ−1,1)
n (t) is estimated In both cases κ = 1 and κ = σ, as follows:

J (κ−1,1)
n (t) . (1 + t)−γ . (76)

Including the estimate (76) into (75) we find for κ = 1

‖unlt (t, ·)‖L2 . (1 + t)−γ‖u‖pX(T ), (77)

and for κ = σ
‖|D|σ−1unlt (t, ·)‖L2 . (1 + t)−γ‖u‖pX(T ). (78)

Finally, the desired inequality (58) is concluded after using the estimates (72), (73) (77),
(78) and the definition (57) of the norm in X(T ). Summarizing we obtained (15).
Now, let us turn to the inequality (16). Taking into account the definition of the aaplication
N , we have by Proposition 2 for κ = 0 and σ the estimates

‖|D|κ(Nu−Nv)(t, ·)‖L2 .
∫ t

0
(1 + t− τ)−

n
2
( 1
m
− 1

2
)−κ

2

×
∫ τ

0
(τ − s)−γ

∥∥|ut(s, ·)|p − |vt(s, ·)|p
∥∥
Lm∩L2∩Ḣσ−1 dsdτ. (79)

Then, we have to estimate the norms∥∥|ut(s, ·)|p − |vt(s, ·)|p
∥∥
Lm ,

∥∥|ut(s, ·)|p − |vt(s, ·)|p
∥∥
L2 , and

∥∥|ut(s, ·)|p − |vt(s, ·)|p
∥∥
Ḣσ−1 .

Following the steps of proof of Theorem 1, we show immediately that

‖|ut(s, ·)|p − |vt(s, ·)|p‖Lm∩L2 . (1 + s)−β‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
, (80)

where β is as in (61). Then, it remains to estimate the norm ‖|ut(s, ·)|p − |vt(s, ·)|p‖Ḣσ−1 .
First, applying Leibniz formula from Proposition 3 allows us to conclude for p > dσe the
estimate

‖|ut(s, ·)|p − |vt(s, ·)|p‖Ḣσ−1 = ‖|D|σ−1
(
|ut|p − |vt|p

)
(s, ·)‖L2

.
∫ 1

0

∥∥|D|σ−1
[
(ut − vt)(ut − w(ut − vt))|ut − w(ut − vt)|p−2

]
(s, ·)

∥∥
L2 dw

.
∫ 1

0
‖|D|σ−1(ut − vt)(s, ·)‖Lr1

∥∥(ut − w(ut − vt))|ut − w(ut − vt)|p−2(s, ·)
∥∥
Lr2

dw

+

∫ 1

0
‖(ut − vt)(s, ·)‖Lr3

∥∥|D|σ−1
[
(ut − w(ut − vt))|ut − w(ut − vt)|p−2

]
(s, ·)

∥∥
Lr4

dw



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 16 of 24

(81)

with
1

r1
+

1

r2
=

1

r3
+

1

r4
=

1

2
. (82)

The term ‖|D|σ−1(ut − vt)(s, ·)‖Lr1 can be estimated by using the fractional Gagliardo-
Nirenberg inequality (106) in the form

‖|D|σ−1(ut − vt)(s, ·)‖Lr1 . ‖ut(s, ·)− vt(s, ·)‖1−θ31(r1)
L2 ‖|D|σ(ut − vt)(s, ·)‖θ31(r1)L2

. (1 + s)−γ‖u− v‖X(T ) (83)

with θ31(r1) =
n
σ

(
1
2 − 1

r1

)
+ σ−1

σ ∈ [σ−1
σ , 1] ∈ [0, 1] if and only if

2 ≤ r1 ≤
2n

n− 2
for n ≥ 3.

In the same way, by using the classical Gagliardo-Norenberg inequality, we estimate the
norm

∥∥(u− w(u− v))|ut − w(ut − vt)|p−2(s, ·)
∥∥
Lr2

as follows:∥∥(ut − w(ut − vt))|ut − w(ut − vt)|p−2(s, ·)
∥∥
Lr2

. ‖
(
ut − w(ut − vt)

)
(s, ·)‖p−1

L(p−1)r2

. ‖ut − w(ut − vt)‖(p−1)(1−θ32(r2))
L2 ‖|D|σ

(
ut − w(ut − vt)

)
‖(p−1)θ32(r2)
L2

. (1 + s)−γ(p−1)‖u− w(u− v)‖p−1
X(T ), (84)

where θ32(r2) =
n
σ

(
1
2 − 1

(p−1)r2

)
∈ [0, 1] if and only if

2

p− 1
≤ r2 ≤

2n

(p− 1)(n− 2σ)
if n > 2σ.

Due to the estimates (84) and (83), we may conclude the estimate

‖|D|σ−1(ut − vt)(s, ·)‖Lr1

∥∥(ut − w(ut − vt))|ut − w(ut − vt)|p−2(s, ·)
∥∥
Lr2

. (1 + s)−β− 1
σ
(1−am)(1−r)(2am+σ−1)‖u− v‖X(T )‖u− w(u− v)‖p−1

X(T )

. (1 + s)−β‖u− v‖X(T )‖u− w(u− v)‖p−1
X(T ), (85)

where β is as in (61). In order to estimate the norm ‖(ut − vt)(s, ·)‖Lr3 we apply the
classical Gagliardo-Nirenberg inequality (107) to obtain

‖(ut − vt)(s, ·)‖Lr3 . ‖(ut − vt)(s, ·)‖1−θ33(r3)
L2 ‖|D|σ(ut − vt)(s, ·)‖θ33(r3)L2

. (1 + s)−γ‖u− v‖X(T ), (86)

where θ33(r3) =
n
σ

(
1
2 − 1

r3

)
∈ [0, 1] if and only if

2 ≤ r3 ≤
2n

n− 2σ
if n > 2σ.



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 17 of 24

Now, we turn to estimate the norm
∥∥|D|σ−1

(
(ut − w(ut − vt))|ut − w(ut − vt)|p−2

)∥∥
Lr4

.
We can do this by applying the fractional chain rule (110). In this way we get∥∥|D|σ−1

(
(ut − w(ut − vt))|ut − w(ut − vt)|p−2

)∥∥
Lr4

.
∥∥ut − w(ut − vt)‖p−2

Lr5 ‖|D|σ−1(ut − w(ut − vt))
∥∥
Lr6

(87)

with
p− 2

r5
+

1

r6
=

1

r4
and p > dσe. (88)

In one hand the classical Gagliardo-Nirenberg inequality (107) allows us to get

‖(ut − w(ut − vt))(s, ·)‖p−2
Lr5

. ‖(ut − w(ut − vt))(s, ·)‖(p−2)(1−θ5(r5))
L2 ‖|D|σ

(
ut − w(ut − vt)

)
(s, ·)‖(p−2)θ5(r5)

L2

. (1 + s)−γ(p−2)‖u− w(u− v)‖p−2
X(T ) (89)

with θ5(r5) =
n
σ

(
1
2 − 1

r5

)
∈ [0, 1] if and only if

2 ≤ r5 ≤
2n

n− 2σ
if n > 2σ.

In the other hand, by applying the fractional chain rule (110) we may estimate the norm∥∥|D|σ−1(ut − r(ut − vt))
∥∥
Lr6

as follows:∥∥|D|σ−1(ut − w(ut − vt))
∥∥
Lr6

. ‖ut − w(ut − vt)‖1−θ6(r6)
L2

∥∥|D|σ(ut − w(ut − vt))
∥∥θ6(r6)
L2

. (1 + s)−γ‖u− w(u− v)‖X(T ) (90)

with θ6 =
n
σ

(
1
2 − 1

r6

)
+ σ−1

σ ∈ [σ−1
σ , 1] if and only if

2 ≤ r6 ≤
2n

n− 2
for n ≥ 3.

For r1 and r2 we may choose r1 = 2n
n−2 and r2 = n. For r3, · · · , r6, we may choose

r3 = r5 = n(p− 1), r6 = 2n
n−2 and, consequently, we find r4 =

2n(p−1)
n(p−1)−2 .

Therefore, by using the estimates (89) and (90) we get from (87) the estimate∥∥|D|σ−1
[
(u− w(u− v))|u− w(u− v)|p−2

]∥∥
Lr4

. (1 + s)−γ(p−1)‖u− w(u− v)‖p−1
X(T ). (91)

Hence, both estimates (91) and (86) imply the estimate

‖u− v‖Lr3

∥∥|D|σ−1[(u− w(u− v))|u− w(u− v)|p−2]
∥∥
Lr4

. (1 + s)−β− 1
σ
(2am+σ−1)(1−a)(1−r)‖u− v‖X(T )‖u− w(u− v)‖p−1

X(T )

. (1 + s)−β‖u− v‖X(T )‖u− w(u− v)‖p−1
X(T ), (92)



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 18 of 24

where β is defined by (61). Then, plugging (85) and (92) into (81) we get∥∥|u(s, ·)|p − |v(s, ·)|p)(s, ·)
∥∥
Ḣσ−1

. (1 + s)−β‖u− v‖X(T )

∫ 1

0
‖u− w(u− v)‖p−1

X(T ) dw

. (1 + s)−β‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
. (93)

Finally, thanks to the estimates (93) and (80) we conclude that∥∥|u(s, ·)|p − |v(s, ·)|p
∥∥
Lm∩L2∩Ḣσ−1

. (1 + s)−β‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
, (94)

where β is as (61). Next, including the estimate (94) into (79) we get

‖|D|κ(Nu−Nv)(t, ·)‖L2 . Jκ,0
n (t)‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
,

where Jκ,0
n (t) is given by (18). Hence, due to the estimates (71) we find

‖(Nu−Nv)(t, ·)‖L2 . (1 + t)1−
n
2
( 1
m
− 1

2
)−γ‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
, (95)

and

‖|D|σ(Nu−Nv)(t, ·)‖L2 . (1 + t)−γ‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
, (96)

respectively. Now, we turn to estimate the norms

‖∂t(Nu−Nv)(t, ·)‖L2 and ‖|D|σ−1∂t(Nu−Nv)(t, ·)‖L2 .

Again by using Proposition 2 we have for κ = 1 and κ = σ the estimates

‖|D|κ−1∂t(Nu−Nv)(t, ·)‖L2 .
∫ t

0
(1 + t− τ)−

n
2
( 1
m
− 1

2
)−κ−1

2
−1

×
∫ τ

0
(τ − s)−γ

∥∥|ut(s, ·)|p − |vt(s, ·)|p
∥∥
Lm∩L2∩Ḣσ−1 dsdτ. (97)

Including the estimate (94) into (97) we get

‖|D|κ−1∂t(Nu−Nv)(t, ·)‖L2 . J (κ−1,1)
n (t)‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
, (98)

where J
(κ−1,1)
n (t) is defined by (18). Due to (76) we obtain for κ = 0 the estimate

‖∂t(Nu−Nv)(t, ·)‖L2 . (1 + t)−γ‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
, (99)

and for κ = σ the estimate∥∥|D|σ−1∂t(Nu−Nv)(t, ·)
∥∥
L2 . (1 + t)−γ‖u− v‖X(T )

(
‖u‖p−1

X(T ) + ‖v‖p−1
X(T )

)
. (100)

Finally, the estimates (100), (99), (96), (95) and the definition of the norm in X(T ) yield
the desired inequality (16). This ends the proof of Theorem 3.



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 19 of 24

Remark 1. Let us explain how to estimate the norm of u in Ḣ1/2 in the case n = 3 and
why the logarithmic term does appear. One can estimate ‖u‖Ḣ1/2 by using interpolation
argument. So, taking in (109)) σ = 1

2 , k1 = 0, k2 = 1 and θ = 1
2 we get

‖u(t, ·)‖Ḣ1/2 . ‖u(t, ·)‖
1
2

L2‖∇u(t, ·)‖
1
2

L2

. (1 + t)
1
2
( 1
4
−γ)− γ

2 ‖(u0, u1)‖A1
1,0

. (1 + t)
1
8
−γ‖(u0, u1)‖A1

1,0
. (101)

In the other hand, if we estimate the norm of u in Ḣ1/2 by using Corollary 1 and Lemma
1 we find

‖unl(t, ·)‖Ḣ1/2 .
(∫ t

0
(1 + t− τ)−

3
4
− 1

4

∫ τ

0
(τ − s)−γ(1 + s)−βdsdτ

)
‖u‖pX(T ),

for some β > 1. Since β > 1 and 3
4 + 1

4 = 1 we get the estimate∫ t

0
(1 + t− τ)−

3
4
− 1

4

∫ τ

0
(τ − s)−γ(1 + s)−βdsdτ . (1 + t)−γ log(2 + t).

As a consequence, we get

‖u(t, ·)‖Ḣ1/2 . (1 + t)−γ log(2 + t)‖(u0, u1)‖A1,1 . (102)

In term of comparison, the estimate (102) is more precise than the estimate (101). For
this reason, we consider the estimate (102) in Theorem 3.

4.2.2. Proof of Theorem 4

The proof of Theorem 4 is similar to the proof of Theorem 3. Expect in the case of
Theorem 4 we have

n

2

( 1

m
− 1

2

)
= 1 if and only if n ≥ 4 and m =

2n

n+ 4
.

In this case, the integral J (k,j)
n (t) is estimated for k = j = 0 as follows:

J (0,0)
n (t) =

{
(1 + t)−γ log(2 + t) if n ≥ 4 and m = 2n

n+4 ,

(1 + t)−γ else .

Remark 2. We refer the curious reader asking for the existence of the parameters qj,
j = 1, 2 and ri, i = 1, · · · , 6 which appears in the proof of Theorem 3 and Theorem 4 to
check [11] and some of the references therein.



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 20 of 24

5. Conclusion

We showed in Theorems 1, 3 and 4 the influence of the dissipative memory by its
influence on the critical exponent and the estimates of the norms of the solution and its
derivatives together since γ > 0. Indeed, it appears a loss of decay in the rate of the
estimate of these norms. In the other hand, let us consider the following Cauchy problem
of semi-linear dissipative wave equation:

utt −∆u+ ut = |ut|p t > 0, x ∈ Rn,
u(0, x) = u0(x), x ∈ Rn,
ut(0, x) = u1(x), x ∈ Rn,

(103)

Due to (7), it appears that Cauchy problem (103) has no critical exponent in Fujita sense
since

lim
γ→1

1

γ
= 1.

One can interpret this result by concluding that the solutions of Cauchy problem (103),
if they exist, then they are global in time. Consequently, any solution of Cauchy problem
(103) blows-up in finite time.

References

[1] E. A. Az-Zo’bi. New kink solutions for the van der waals p-system. Math. Meth. App.
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[2] E. A. Az-Zo’bi. Numeric-analytic solutions of mixed-type systems of balance laws.
Appl. Math. Comp., 255:133–143, 2015.

[3] E. A. Az-Zo’bi. On the reduced differential transform method and its application to
the generalized burgers-huxley equation. App. Math. sci., 8(177):8823–8831, 2014.

[4] E. A. Az-Zo’bi, A. Yildirim, and L. Akinyemi. Semi-analytic treatment of mixed
hyperbolic–elliptic cauchy problem modeling three-phase flow in porous media. Int.
Journ. Modern Phys., 35(29), 2021.

[5] E. A. Az-Zo’bi. Construction of solutions for mixed hyperbolic elliptic riemann initial
value system of conservation laws. 37(8):6018–6024, 2013.

[6] A. Fino. Critical exponent for damped wave equations with nonlinear memory. Hal
Arch. Ouv., 2010. Id: 00473941v2.

[7] M. D’Abbicco. The influence of a nonlinear memory on the damped wave equation.
Nonlinear Anal., 95:130–145, 2014.

[8] T. Cazenave, F. Dickstein, and F. D. Weissler. An equation whose fujita critical
exponent is not given by scaling. Nonlinear anal., 68:862–874, 2008.

[9] A. Matsumura. On the asymptotic behavior of solutions of semi-linear wave equations.
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[10] R. Ikehata and K. Tanizawa. Global existence of solutions for semilinear damped
wave equations in Rn with noncompactly supported initial data. Nonlinear Analysis:
Theory, Methods & Applications, 61(7):1189–1208, 2005.



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[11] T. Hadj Kaddour and M. Reissig. Global well posedness of effectively damped wave
models with nonlinear memory. Com. Pur & App. Anal., 20(5):2039–2064, 2021.

[12] S. Cui. Local and global existence of solutions to semilinear parabolic initial value
problems. Nonlinear Anal., 43(3):293–323, 2001.

[13] M. D’Abbicco, S. Lucente, and M. Reissig. Semilinear wave equations with effective
damping. Chin. Ann. Math., Serie B, 34:345–380, 2013.

[14] M. R. Ebert and M. Reissig. Methods for Partial Differential Equations. Qualitative
Properties of Solutions, Phase Space Analysis, Semilinear Models. Birkhäuser, Cham,
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[15] T. Runst and W. Sickel. Sobolev Spaces of Fractional Order, Nemytskij Operators,
and Nonlinear Partial Differential Equations. De Gruyter series in nonlinear analysis
and applications. Walter de Gruyter & Co., Berlin, 1996.

Appendix

5.1. Some auxiliary estimates for integrals

The following lemma is the key to estimate certain integrals that appear in the proof
of the global existence of small data solutions (Section 4), in particular those arising from
the nonlinear terms.

Lemma 1. Assume that 0 < γ < 1, a ≥ 0 and b > 1. Then we have∫ t

0
(1 + t− s)−a

∫ s

0
(s− τ)−γ(1 + τ)−b dτds

≤ C


(1 + t)−γ if a > 1,
(1 + t)−γ log(2 + t) if a = 1,
(1 + t)1−a−γ if a < 1.

Proof. Thanks to Lemma 4.1 from [12] and the fact that γ < 1, the integral with
respect to τ on [0, s] is estimated as follows:∫ s

0
(s− τ)−γ(1 + τ)−b dτ . (1 + s)−γ .

Finally, the statements of Lemma 1 are concluded after applying Lemma 4.1 from [12]
again.

5.2. What about the condition β > 1 in Lemma 1?

The reader may ask for a corresponding result in the case b ∈ (0, 1] and how about the
condition β > 1. We confirm that the condition β > 1 allows to get optimal estimate for
the integral. The curious peoples are advised to check the discussion of this point in [11].



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 22 of 24

5.3. Decay estimates for solutions to auxiliary Cauchy problems

Let us announce some results on the decay estimates for Cauchy problems (9) and
(10). First, for problem (9), A. Matsumura proved in [9] the following proposition

Proposition 1. [9] If u is solution to Cauchy problem (9), then u and its derivatives
∂j
t∇κu satisfy for all j ∈ N and κ > 0 the following estimates:

‖u(t, ·)‖ ≤ C(1 + t)−
n
4
−κ

2
−j(‖(u0, u1)‖L1 + ‖u0‖Hκ+j + ‖u1‖H[κ+j−1]+

), (104)

for some constant C > 0.

The following result is an immediate consequence of Proposition 1 and the fact that
equation utt −∆u+ ut = 0 is invariant by translation.

Corollary 1. If v is solution to Cauchy problem (10), then v and its derivatives ∂j
t∇κv

satisfy for all j ∈ N and κ > 0 the following estimates:

‖v(t, ·)‖ ≤ C(1 + t− τ)−
n
4
−κ

2
−j(‖h(τ, ·)‖L1 + ‖h(τ, ·)‖H[κ+j−1]+

), (105)

for some constant C > 0.

The following result is a particular case of Proposition 5.2 from [11]. See also Theorem
3 in [13].

Corollary 2. [Proposition 5.1, [11]] Let us consider the Cauchy problem (10) with
h = h(τ, ·) ∈ Lm ∩Hσ−1 for some m ∈ [1, 2) and σ > 1. Then the energy solution belongs
to

C([τ,+∞),Hσ) ∩ C1([τ,+∞),Hσ−1),

and satisfies the following Matsumura type decay estimates:

‖v(t, ·)‖L2 ≤ C(1 + t− τ)−
n
2
( 1
m
− 1

2
)‖h(τ, ·)‖Lm∩Hσ−1 ,

‖|D|σv(t, ·)‖L2 ≤ C(1 + t− τ)−
n
2
( 1
m
− 1

2
)−σ

2 ‖h(τ, ·)‖Lm∩Hσ−1 ,

‖vt(t, ·)‖L2 ≤ C(1 + t− τ)−
n
2
( 1
m
− 1

2
)−1‖h(τ, ·)‖Lm∩Hσ−1 ,

‖|D|σ−1vt(t, ·)‖L2 ≤ C(1 + t− τ)−
n
2
( 1
m
− 1

2
)−σ−1

2
−1‖h(τ, ·)‖Lm∩Hσ−1 .

5.4. Main inequalities-tools from Harmonic Analysis

The following results can be found among other things in [14] or [15].

5.4.1. Fractional Gagliardo-Nirenberg inequality

Proposition 2. [11, 14, 15] Let 1 < p, p0, p1 < ∞ and κ ∈ [0, σ). Then the following
fractional Gagliardo-Nirenberg inequality holds for all u ∈ Lp0 ∩ Ḣσ

p1:

‖u‖Ḣκ
p
. ‖u‖1−θ

Lp0 ‖u‖θḢσ
p1

for κ

σ
≤ θ ≤ 1, (106)



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 23 of 24

where

θ =

1
p0

− 1
p + κ

p
1
p0

− 1
p1

+ σ
n

.

The following corollary is a particular case of Proposition 2.

Corollary 3. [7] Let u ∈ L2 ∩ Ḣk
2 . Then the following inequality holds:

‖u‖Lq . ‖u‖1−θk(q)
L2 ‖u‖θk(q)

Ḣk
2

, θk(q) =
n

k

(1
2
− 1

q

)
(107)

for any k ∈ (0, n2 ) and any q such that

2 ≤ q ≤ 2n

n− 2k
. (108)

The case q = 2n
n−2k reduces the inequality (107) to a well-known statement in the frame

of Sobolev embeddings.

Corollary 4. [11, 14, 15] For u ∈ Ḣk
2 , where q ∈ [2,∞) and k = n(12 − 1

q ), the following
inequality holds:

‖u‖Lq . ‖u‖Ḣk
2
.

Interpolation formulas are sometimes used to obtain suitable estimates. Here we recall
the relation

‖u‖Ḣσ
2
≤ ‖u‖1−θ

Ḣ
k1
2

‖u‖θ
Ḣ

k2
2

for some θ ∈ [0, 1] with k1(1− θ) + k2θ = σ. (109)

5.4.2. Fractional Leibniz rule

Proposition 3. [11, 14, 15] Let σ > 0, 1 ≤ r ≤ ∞ and 1 < p1, p2, q1, q2 ≤ ∞ satisfying

1

r
=

1

p1
+

1

p2
=

1

q1
+

1

q2
.

Then it holds the following fractional Leibniz rule:

‖|D|σ(fg)‖Lr . ‖|D|σf‖Lp1‖g‖Lp2 + ‖f‖Lq1‖|D|σg‖Lq2

for any f ∈ Ḣσ
p1 ∩ Lq1 and g ∈ Ḣσ

q2 ∩ Lp2.

5.4.3. Fractional chain rule

Proposition 4. [11, 14, 15] Let σ ∈ (0, 1), 1 < r, r1, r2 < ∞ and F a C1 function
satisfying for any τ ∈ [0, 1] and u, v ∈ R the inequality

|F ′(τu+ (1− τ)v)| ≤ µ(τ)(G(u) +G(v))),



T. Hadj Kaddour et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6266 24 of 24

for some continuous nonnegative function G and µ ∈ L1[0, 1]. Then,

‖F (u)‖Ḣσ
r
. ‖G(u)‖Lr1‖u‖Ḣσ

r2
,

for any u ∈ Ḣσ
r2 such that G(u) ∈ Lr1, provided that

1

r
=

1

r1
+

1

r2
.

In particular, to estimate norms like ‖|u|p‖Ḣs−1
r

or ‖±u|u|p−1‖Ḣs−1
r

we use the fractional
chain rule and the Gagliardo-Nirenberg inequality. In this way we may may conclude at
first for s ∈ (1, 2) and then by a straight-forward step to s ≥ 2 the estimate

‖ ± u|u|p−1‖Ḣs−1
r

+ ‖|u|p‖Ḣs−1
r

. ‖u‖p−1
Lq1 ‖|D|s−1u‖Lq2 , (110)

where
p− 1

q1
+

1

q2
=

1

r
, s > 1.

5.4.4. Fractional powers

The following tool is useful to estimate the p-power of a given function and the product
of two functions in Hs(Rn). This tool is meaningful in the case in which we have the
embedding

L∞(Rn) ↪→ Hs
r (Rn),

that is, when s > n
r .

Proposition 5. [11, 14, 15] Let r ∈ (1,∞), p > 1 and s ∈ (0, p). Let F (u) denote one of
the functions |u|p or ±u|u|p−1. Then, it holds the following inequality:

‖F (u)‖Hs
r
. ‖u‖Hs

r
‖u‖p−1

L∞

for any u ∈ Hs
r (Rn) ∩ L∞(Rn).

The next result is a direct consequence of the previous one for the case of homogeneous
Sobolev spaces.

Corollary 5. Let r ∈ (1,∞), p > 1 and s ∈ (0, p). Let F (u) denote one of the functions
|u|p or ±u|u|p−1. Then, it holds the following inequality:

‖F (u)‖Ḣs
r
. ‖u‖Ḣs

r
‖u‖p−1

L∞

for any u ∈ Ḣs
r (Rn) ∩ L∞(Rn).