Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 80, pp. 1–16.

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

DOI: 10.58997/ejde.2025.80

LOCAL AND GLOBAL SOLVABILITY OF FRACTIONAL POROUS MEDIUM

EQUATIONS IN CRITICAL BESOV-MORREY SPACES

AHMED EL IDRISSI, HALIMA SRHIRI, BRAHIM EL BOUKARI, JALILA EL GHORDAF

Abstract. In this article we study fractional porous medium equations in Besov-Morrey spaces.

Using the Littlewood-Paley theory and the smoothing effect of the heat semi-group, we obtain
local well-posedness of this model. Also, we obtain global well-posedness for small initial data

in the critical Besov-Morrey spaces Ṅ
−2m+n

p

p,h,∞ (Rn) with 1/2 < m < 1, max{1, n
2m

} < p < ∞
and 1 ≤ h ≤ p.

1. Introduction

We study the fractional porous medium equation given by the nonlinear diffusion model with
fractional Laplacian operators,

ut − µ∆u = κ∇ · (u∇Ku) in Rn × (0,∞),

Ku = (−∆)−mu in Rn × (0,∞),

u(x, 0) = u0(x) for x ∈ Rn,

(1.1)

where n ≥ 1, u = u(x, t) denotes the density or concentration, and therefore non-negative, u0 is the
initial data, ∇ is the gradient operator, µ > 0 is the dissipative coefficient which corresponds the
viscous case, while µ = 0 represents the inviscid case, κ = ±1, and here for simplify the notation,
we take µ = κ = 1, (−∆)−m is the inverse fractional Laplacian operator, and 0 < m < 1, that is,
the abnormal (normal) diffusion is modeled by a fractional power of the Laplacian. We mention
here that, The interest in using fractional Laplacians in modeling diffusive processes has a wide
literature, especially when one wants to model long-range diffusive interaction, and this interest
has been activated by the recent progress in the mathematical theory, see [28, 9, 7, 33, 13, 15] and
related references cited therein.

When µ = 0, κ = 1 and m = 1, the model (1.1) corresponds to the mean field equation

ut = ∇ · (u∇Ku) in Rn × (0,∞),

Ku = (−∆)−1u in Rn × (0,∞),

u(x, 0) = u0(x) for x ∈ Rn,

(1.2)

which was introduced for the first time by Lin and Zhang [24]. They demonstrated the existence
and uniqueness of positive L∞ solution in two dimensions. There are many studies on well-
posedness results of this equation. For instance, refer to [31, 37] and related references cited
therein.

2020 Mathematics Subject Classification. 35K55, 35K15, 30H25.
Key words and phrases. Nonlinear diffusion; fractional porous medium equation;

local and global well-posedness; Besov-Morrey spaces; fractional Laplacians.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted April 13, 2025. Published August 4, 2025.

1



2 A. EL IDRISSI, H. SRHIR, B. EL BOUKARI, J. EL GHORDAF EJDE-2025/80

When µ = 1, κ = −1 and m = 1, system (1.1) reduces to the classical Keller-Segel system

ut −∆u+∇ · (u∇Ku) = 0 in Rn × (0,∞),

−∆Ku = u in Rn × (0,∞),

u(x, 0) = u0(x) for x ∈ Rn,

(1.3)

which describes a model of chemotaxis. System (1.3) has been introduce by Keller and Segel
[19]. The well-posedness of Keller-Segel models has been studied by several researchers in various
spaces. Biler and Karch [4] have established, in the critical Lebesgue space L

n
2 (Rn), the existence

of both local and global solutions of this equation with small initial data. Additionally, they have
demonstrated the finite-time blowup of non-negative solutions with specific initial data that satisfy
high-concentration or large-mass conditions. Making use of the Chemin mono-norm methods, Zhao

[36] obtained well-posedness results in the Besov spaces Ḃ
−2+n

p
p,r (Rn) with 1 ≤ p, r ≤ ∞. Making

use of the smoothing effect of the heat semigroup, Iwabuchi [17] proved the global well-posedness

of the system (1.3) in Ḃ
−2+n

p
p,∞ (Rn) where n ≥ 1 and max{1, n/2} < p < ∞, under the condition

of smallness of the initial data. He also demonstrated, with a sufficient condition, the local well-

posedness in Ḃ
−2+n

p
p,∞ (Rn) [18]. Later, by the same method, Nogayama and Sawano [25] extended

these well-posedness results, where they established global well-posedness in the Besov-Morrey

spaces Ṅ−2m+n
p

p,h,∞ (Rn) with max{1, n
2 } < p < ∞ and 1 ≤ h ≤ p, and local well-posedness in closed

subspaces of these spaces. And here we mention that certain aspects of these results were also
extended to the fractional power bipolartype drift-diffusion system. Further information on this
topic can be found in [23, 11] and the relevant references cited therein.

When κ = 1 and 0 < m < 1, system (1.1) has been derived starting from the same origin as the
fractional porous medium equation initially proposed by Caffarelli and Vázquez [6]. Indeed, the
model is conceived through the incorporation of the dissipative term µ(−∆)v into the continuity
equation

∂tu+∇ · (uV ) = 0, (1.4)

where V = −∇p is the velocity, and p represents the gas pressure which is related to v through
a linear integral operator p = Ku, with kernel K(x, y) = c|x− y|−(d−2m). For µ = 0, that is, the
fractional porous medium equation in the inviscid case, we have many studies on this equation. In
[6] the authors stated that the notable feature of this equation is the finite speed of propagation,
and they established, the existence of a weak solution with bounded initial data that exponentially
decays at infinity, the property of compact support, and also the relevant integral estimates. See
[31, 28, 37] for more information on this equation. For the viscous case (µ > 0), El Idrissi et

al.[10] considered the system (1.1) for initial data in the critical Besov spaces Ḃ
−2m+ d

p
p,∞ (Rn) with

1
2 < m < 1 and max{1, n

2m} < p < ∞. They established sufficient conditions for the existence
and uniqueness of local solutions, and also proved the existence of global solutions for small initial
data in the same spaces. Furthermore, the well-posedness of this model has been demonstrated in

various functional settings: in the Besov spaces Ḃ
−2m+n

p
p,r (Rn) with 1 ≤ p, r ≤ ∞, in the Fourier-

Besov spaces FḂ
−2m+ n

p′
p,r (Rn) with 1 ≤ p, r ≤ ∞, in the critical Fourier-Besov-Morrey spaces

FṄ
−2m+λ

p+3(1− 1
p )

p,λ,r (R3) with 1 ≤ p < ∞ and 1 ≤ r ≤ ∞, and in the critical variable exponent

Besov-Morrey spaces Ṅ
−2m+ n

q(·)
r(·),q(·),h (Rn) with 1−ε

2 < m < 1 + n
2q(·) , 0 < ε < 1, 1 ≤ r(·) ≤ q(·) < ∞

and 1 ≤ h ≤ ∞. These results were obtained by Xiao and Zhang [35], El Idrissi et al. [8], Toumlilin
[34], and El Idrissi et al. [12], respectively, by applying the Chemin mono-norm methods.

Using a different method from the latter, which is the smoothing effect of the heat semigroup,
we seek to establish the existence of local solutions for initial data in a closed subspace of Besov-
Morrey spaces, which will be define later. Also we show the existence of global solutions of system

(1.1) in the critical Besov-Morrey spaces Ṅ−2m+n
p

p,h,∞ (Rn) with 1
2 < m < 1, max{1, n

2m} < p < ∞,
1 ≤ h ≤ p, under the smallness condition of initial data. The idea of our work is motivated by the
papers [17, 18, 25], which dealt with the classical Keller-Segel systems.



EJDE-2025/80 FRACTIONAL POROUS MEDIUM EQUATIONS 3

The Besov-Morrey space Ṅ−2m+n
p

p,h,∞ is critical for the system (1.1). In fact, if u(x, t) is the

solution of system (1.1), then

uλ(x, t) := λu(λx, λ2t)

is also a solution of the same equation and

∥u(·, 0)∥
Ṅ

−2m+n
p

p,h,∞

∼ ∥uλ(·, 0)∥
Ṅ

−2m+n
p

p,h,∞

.

The Besov-Morrey space, introduced by Kozono and Yamazaki [22], extends the concept of
classical Besov spaces by incorporating Morrey spaces as a fundamental element, thus providing a
larger functional framework. In particular, Besov-Morrey spaces are strictly broader than classical
Besov spaces (also refer to Remark 2.11). It is important to note that replacing the Lp-norm by the
Mp

h-norm is not sufficient to ensure a direct transition from Besov spaces to Besov-Morrey spaces.
One of the main difficulties comes from the collapse of certain essential embedding features when
working in the Besov-Morrey framework (see, for example, [14, 25] for a more detailed discussion).
Within this extended functional framework, numerous recent works have been carried out on a
variety of fluid dynamics systems. The reader is referred to [14, 20, 22, 25, 29] and related references
cited therein for further information. However, Besov-Morrey spaces are generally not separable.
Therefore, the compatibility of initial data must be taken into account. With this in mind, in order
to consider the local existence result for the the system (1.1), we impose a vanishing condition

on the high frequency components of the initial data u0. i.e., we take u0 ∈ ˜̇N s
p,h,r = ˜̇N s

p,h,r(Rn),
where

˜̇N s
p,h,r :=

{
f ∈ Ṅ s

p,h,r : lim
N→∞

∥
∑
j>N

∆̇jf∥Ṅ s
p,h,r

= 0
}
,

which is a subspace of Besov-Morrey spaces (the idea is inspired by [18]). See Section 2 for
definitions of other notation.

To address the system (1.1), we consider the integral equation

u = et∆u0 +

∫ t

0

e(t−t′)∆∇ ·
(
u∇(−∆)−mu

)
dt′, (1.5)

where et∆ := F−1(e−t|ξ|2F) and (−∆)−m := F−1(|ξ|−2mF) are the heat semigroup and the
inverse fractional Laplacian operators, respectively. By using the contraction mapping approach
to the map below, we can solve (1.5),

Ψ(u) := et∆u0 +

∫ t

0

e(t−t′)∆∇ ·
(
u∇(−∆)−mu

)
dt′. (1.6)

Throughout this paper, C will represent constants that may differ at different places, E ≲ F
denotes the existence of a constant C > 0 such that E ≤ CF and E ∼ F denotes the existence of
constants C1, C2 > 0 such that C1F ≤ E ≤ C2F . BC (0, T ;X) is the set of all functions bounded
on [0, T ) and continuous on (0, T ) with values in the space X. We define for v ∈ S(Rn), the
Fourier transform as

Fv(ξ) = v̂(ξ) :=
1

(2π)n/2

∫
Rd

e−ix·ξv(x)dx,

and its inverse Fourier transform as

F−1v(x) :=
1

(2π)n/2

∫
Rd

eix·ξv(ξ)dξ.

Organization of the paper: In Section 2, some basic facts about Littlewood-Paley theory and
some product laws in Besov-Morrey spaces are presented, and then presenting the statements of
our two main results. To prove these results, we give their key estimates and prove them in Section
3. Lastly, Theorem 2.9 and Theorem 2.10 are established in Section 4.

2. Preliminaries and main results

We introduce some basic knowledge of Littlewood-Paley theory and Besov-Morreyspaces and
reviews some lemmas that are pertinent to our purposes.



4 A. EL IDRISSI, H. SRHIR, B. EL BOUKARI, J. EL GHORDAF EJDE-2025/80

2.1. Littlewood-Paley theory and Besov-Morrey spaces. We start by recalling the Littlewood-
Paley decomposition (see [3] for more details). Let φ ∈ S(Rn) be a smooth radial function such
that

0 ≤ φ ≤ 1,

suppφ ⊂
{
ξ ∈ Rn :

3

4
≤ |ξ| ≤ 8

3

}
,∑

j∈Z

φ
(
2−jξ

)
= 1, for all ξ ̸= 0,

and we denote φj(ξ) = φ(2−jξ). So, for all u ∈ S ′(Rn), let’s set the frequency localization
operators for all j ∈ Z, to be as below

∆̇ju = F−1φj ∗ u and Ṡju =
∑

k≤j−1

∆̇ku. (2.1)

Then, we have the homogeneous Littlewood-Paley decomposition

u =
∑
j∈Z

∆̇ju for all u ∈ S ′(Rn)/P(Rn),

with P(Rn) denoting the collection of all polynomials over Rn. One observes here that ∆̇j has

frequency [|ξ| ∼ 2j} and that Ṡj has frequency [|ξ| ≲ 2j}, and one also notes that the quasi-
orthogonality property holds for the Littlewood-Paley decomposition, that is, for every u, v ∈
S ′(Rn)/P(Rn),

∆̇i∆̇ju = 0 if |i− j| ≥ 2,

∆̇i

(
Ṡj−1u∆̇jv

)
= 0 if |i− j| ≥ 5.

(2.2)

Next, before we give the definition of Besov-Morrey spaces introduced by Kozono and Yamazaki
[22], we first present that of Morrey spaces, which serve as foundation for these spaces. Refer to
[22, 29, 30, 16, 2, 32], for more information on these spaces.

Definition 2.1. Let 1 ≤ h ≤ p < ∞. The Morrey space Mp
h = Mp

h(Rn) is defined to be the set

of all u ∈ Lh
loc(Rn) such that

∥u∥Mp
h
:= sup

x0∈Rd, R>0

R
d
p−

d
h ∥u∥Lh(B(x0,R)) < ∞,

where B(x0, R) represents an open ball in Rd with center x0 and radius R.

The space Mp
h equipped with the norm ∥ · ∥Mp

h
is a Banach space. Furthermore, we have

Mp
h1

↪→ Mp
h2

for 1 ≤ h2 ≤ h1 ≤ p < ∞, Mp
p = Lp(Rn) for 1 < p ≤ ∞, and that M1

1 = M(Rn)

where M(Rn) stands for the space of finite Radon measures on Rd.

Definition 2.2. For s ∈ R, 1 ≤ h ≤ p ≤ ∞, 1 ≤ r ≤ ∞ and u ∈ S ′(Rn)/P, we set

∥u∥Ṅ s
p,h,r

:=

{(∑
j∈Z 2jsr∥∆̇ju∥rMp

h

)1/r
if r < ∞,

supj∈Z 2js∥∆̇ju∥Mp
h

if r = ∞.

Then the homogeneous Besov-Morrey space Ṅ s
p,h,r = Ṅ s

p,h,r(Rn) is defined by

Ṅ s
p,h,r =

{
u ∈ S ′(Rn)/P : ∥u∥Ṅ s

p,h,r
< ∞

}
.

Note that the homogeneous Besov space Ḃs
p,r(Rn) is the particular case of p = h;

Definition 2.3. For s ∈ R, 1 ≤ p, r ≤ ∞ and u ∈ S ′(Rn)/P, we set

∥u∥Ḃs
p,r

:=

{(∑
j∈Z 2jsr∥∆̇ju∥rLp

)1/r
if r < ∞,

supj∈Z 2js∥∆̇ju∥Lp if r = ∞.

Then the homogeneous Besov space Ḃs
p,r(Rn) is defined by

Ḃs
p,r(Rn) =

{
u ∈ S ′(Rn)/P : ∥u∥Ḃs

p,r
< ∞

}
.



EJDE-2025/80 FRACTIONAL POROUS MEDIUM EQUATIONS 5

For s ∈ R, 1 ≤ h0 ≤ h ≤ p < ∞ and 1 ≤ r0 ≤ r ≤ ∞, we have the following embeddings (refer
to [22, 29]):

Ṅ 0
p,h,1 ↪→ Mp

h ↪→ Ṅ 0
p,h,∞, (2.3)

Ṅ s
p,h,r0 ↪→ Ṅ s

p,h,r, (2.4)

Ṅ s
p,h,r ↪→ Ṅ s

p,h0,r. (2.5)

Throughout this paper, the following Bony paraproduct decomposition will be used,

uv = Ṫuv + Ṫvu+ Ṙ(u, v), (2.6)

with

Ṫuv =
∑
j

Ṡj−1u∆̇jv, Ṙ(u, v) =
∑
j

∑
|j−l|≤1

∆̇ju∆̇lv.

For more details, see [3, 5].

2.2. Essential lemmas. We invoke the following lemmas.

Lemma 2.4 ([29]). (1) (Hölder’s inequality) Let 1 ≤ p, p1, p2 < ∞ and 1 ≤ h, h1, h2 ≤ ∞
satisfying h ≤ p, hi ≤ pi(i = 1, 2), 1

p = 1
p1

+ 1
p2

and 1
h = 1

h1
+ 1

h2
. Then for all u ∈ Mp1

h1

and v ∈ Mp2

h2
, there exists a constant C such that

∥uv∥Mp
h
≤ C∥u∥Mp1

h1

∥v∥Mp2
h2

. (2.7)

(2) (Young’s inequality) Let 1 ≤ h ≤ p < ∞. Then for all u ∈ L1 and v ∈ Mp
h, one has

∥u ∗ v∥Mp
h
≤ ∥u∥L1∥v∥Mp

h
. (2.8)

(3) (Sobolev embedding) Let 1 ≤ h ≤ p < ∞, 1 ≤ h0 ≤ p0 < ∞, 1 ≤ r ≤ ∞, and let s, s0 ∈ R
with s < s0. If

s0 −
n

p0
= s(·)− n

p
and

h0

p0
=

h

p
,

then

Ṅ s0
p0,h0,r

↪→ Ṅ s
p,h,r. (2.9)

Lemma 2.5. ∂α
ξ is a bounded operator from Ṅ s+|α|

p,h,r to Ṅ s
p,h,r.

Proof. Using that [|ξ| ∼ 2j} for all j ∈ Z, one has

∥∂α
ξ u∥Ṅ s

p,h,r
=

(∑
j∈Z

2srj∥∆̇j∂
α
ξ u∥rMp

h

)1/r

=
(∑

j∈Z

2srj∥|ξ|α∆̇ju∥rMp
h

)1/r

≲
(∑

j∈Z

2srj2|α|jr∥∆̇ju∥rMp
h

)1/r

≲ ∥u∥Ṅ s+|α|
p,h,r

.

□

Remark 2.6. Lemma 2.5 gives the following inequalities:

∥∇mu∥Ṅ s
p,h,r

≲ ∥u∥Ṅ s+m
p,h,r

,

∥∇m · u∥Ṅ s
p,h,r

≲ ∥u∥Ṅ s+m
p,h,r

,

∥∆mu∥Ṅ s
p,h,r

≲ ∥u∥Ṅ s+2m
p,h,r

.



6 A. EL IDRISSI, H. SRHIR, B. EL BOUKARI, J. EL GHORDAF EJDE-2025/80

Lemma 2.7 ([25, 22, 21]). Let 1 ≤ h ≤ p < ∞, 1 ≤ h0 ≤ p0 < ∞, 1 ≤ r ≤ ∞ and s ∈ R. If

p ≤ p0 and h0

p0
= h

p . Then for all u ∈ Ṅ s
p,h,∞, one has

∥et∆u∥Ṅ s
p0,h0,r

≲ t
−n

2

(
1
p−

1
p0

)
∥u∥Ṅ s

p,h,r
. (2.10)

Lemma 2.8 ([25, 22, 21]). Let 1 ≤ h ≤ p < ∞, s ∈ R and ε > 0. Then for all u ∈ Ṅ s
p,h,∞, one

has
∥et∆u∥Ṅ s+ε

p,h,1
≲ t−

ε
2 ∥u∥Ṅ s

p,h,∞
. (2.11)

2.3. Main results. In this subsection, we state the two main theorems of this work. Our local

well-posedness theorem for the system (1.1) with u0 ∈
˜

Ṅ−2m+n
p

p,h,∞ is the following.

Theorem 2.9. Let n ≥ 1, 1/2 < m < 1 and 1 ≤ h ≤ p < ∞. Assume that u0 ∈
˜

Ṅ−2m+n
p

p,h,∞ . Then
we have the following results:

(1) Let max{1, n
2m} < p ≤ n and let p1, p2, h1, h2 be arbitrary real numbers satisfying

p ≤ p1, p2 < ∞, h1 ≤ p1, p2 ∈ (d,∞) ∩ [h2,∞),

1

p
<

1

p1
+

1

p2
,

1

h
=

1

h1
+

1

h2
,

h

p
=

h1

p1
=

h2

p2
.

(2.12)

Then, there exist δ > 0 and T > 0 such that the system (1.1) admits a unique time-local

solution u ∈ BC
(
0, T ; Ṅ−2m+n

p

p,h,∞
)
, and satisfy

sup
t∈(0,T )

∥u(t)∥
Ṅ

−2m+n
p

p,h,∞

< ∞,

sup
t∈(0,T )

tm− n
2p1 ∥u(t)∥Mp1

h1

+ sup
t∈(0,T )

t
1
2−

n
2p2 ∥|∇|1−2mu(t)∥Mp2

h2

≤ δ.

(2) Let n < p < ∞ and let ρ, ℏ be arbitrary real numbers satisfying

max{p, ℏ} ≤ ρ ≤ 2p,
ℏ
ρ
=

h

p
. (2.13)

Then, there exist δ > 0 and T > 0 such that the system (1.1) admits a unique time-local

solution u ∈ BC
(
0, T ; Ṅ−2m+n

p

p,h,∞
)
, and satisfy

sup
t∈(0,T )

∥u(t)∥
Ṅ

−2m+n
p

p,h,∞

< ∞, sup
t∈(0,T )

t
1
2−

n
2ρ ∥u(t)∥Ṅ 1−2m

ρ,ℏ,1
≤ δ.

Next, we present the global well-posedness theorem.

Theorem 2.10. Let n ≥ 1, 1/2 < m < 1 and 1 ≤ h ≤ p < ∞.

(1) Let max{1, n
2m} < p ≤ n and let p1, p2, h1, h2 be arbitrary real numbers satisfying (2.12).

Then there is a constant δ > 0 such that for any u0 ∈ Ṅ−2m+n
p

p,h,∞ satisfying ∥u0∥
Ṅ

−2m+n
p

p,h,∞

≤

δ, the system (1.1) has a unique global solution u ∈ BC
(
0,∞; Ṅ−2m+n

p

p,h,∞
)
, and satisfy

sup
t∈(0,∞)

∥u(t)∥
Ṅ

−2m+n
p

p,h,∞

+ sup
t∈(0,∞)

tm− n
2p1 ∥u(t)∥Mp1

h1

+ sup
t∈(0,∞)

t
1
2−

n
2p2 ∥|∇|1−2mu(t)∥Mp2

h2

≤ C0,

where C0 is a constant depending on δ.
(2) Let n < p < ∞ and let ρ, ℏ satisfy (2.13). Then there exists a constant δ > 0 such that

for any u0 ∈ Ṅ−2m+n
p

p,h,∞ satisfying ∥u0∥
Ṅ

−2m+n
p

p,h,∞

≤ δ, the system (1.1) has a unique global

solution u ∈ BC
(
0,∞; Ṅ−2m+n

p

p,h,∞
)
, and satisfies

sup
t∈(0,∞)

∥u(t)∥
Ṅ

−2m+n
p

p,h,∞

+ sup
t∈(0,∞)

t
1
2−

n
2ρ ∥u(t)∥Ṅ 1−2m

ρ,ℏ,1
≤ C0,



EJDE-2025/80 FRACTIONAL POROUS MEDIUM EQUATIONS 7

where C0 is a constant depending on δ.

Remark 2.11. The method used in this paper, for proving local and global well-posedness results,
is different from the one in [35] and [36]. There, the authors used the Chemin mono-norm method,

to study systems (1.1) and (1.1) with m = 1, respectively. In Besov spaces Ḃs
p,r, their functional

framework is distinct from our own. Indeed, one has Ṅ s
p,h,r ̸⊂ Ḃs′

p′,r′ for any 1 ≤ p′ < ∞,

1 ≤ r′ < ∞ and s′ ∈ R. However, the results of this work remain valid if we take Besov spaces
Ḃs

p,r instead of Besov-Morrey spaces Ṅ s
p,h,r. In fact, if we have p = h, then Ṅ s

p,p,r = Ḃs
p,r.

3. Key estimates

We get our critical estimates out of the way in this section. First, the following linear estimate
is used.

Lemma 3.1. Let 0 < T ≤ ∞, 1
2 < m < 1 and 1 ≤ h ≤ p < ∞.

(1) Let n
2m < p ≤ n and p1, p2, h1, h2 satisfy (2.12), and let ∥ · ∥XT

be given as

∥u∥XT
:= sup

t∈(0,T )

∥u(t)∥
Ṅ

−2m+n
p

p,h,∞

+ sup
t∈(0,T )

tm− n
2p1 ∥u(t)∥Mp1

h1

+ sup
t∈(0,T )

t
1
2−

n
2p2 ∥|∇|1−2mu(t)∥Mp2

h2

.
(3.1)

Then

∥et∆u0∥XT
≲ ∥u0∥

Ṅ
−2m+n

p
p,h,∞

. (3.2)

(2) Let p > n and ρ, ℏ satisfy (2.13), and let ∥ · ∥YT
be defined by

∥u∥YT
:= sup

t∈(0,T )

∥u(t)∥
Ṅ

−2m+n
p

p,h,∞

+ sup
t∈(0,T )

t
1
2−

n
2ρ ∥u(t)∥Ṅ 1−2m

ρ,ℏ,1
. (3.3)

Then

∥et∆u0∥YT
≲ ∥u0∥

Ṅ
−2m+n

p
p,h,∞

. (3.4)

Proof. (1) On the one hand, we have

∥et∆u0∥
Ṅ

−2m+n
p

p,h,∞

≲ ∥u0∥
Ṅ

−2m+n
p

p,h,∞

. (3.5)

On the other hand, since p1 ≥ p > n
2m and h1

p1
= h

p , using the embedding (2.3) and the embedding

(2.9), and Lemma 2.8 with ε = 2m− n
p1

> 0, we obtain

tm− n
2p1 ∥et∆u0∥Mp1

h1

≲ tm− n
2p1 ∥et∆u0∥

Ṅ
n
p

− n
p1

p,h,1

≲ tm− n
2p1 t−

2m− n
p1

2 ∥u0∥
Ṅ

−2m+n
p

p,h,∞

≲ ∥u0∥
Ṅ

−2m+n
p

p,h,∞

,

(3.6)

and since p2 > n, p ≤ p2, and
h2

p2
= h

p , using the embeddings (2.3) and (2.9), Remark 2.6 and

Lemma 2.8 with ε = 1− n
p2

> 0, one has

t
1
2−

n
2p2 ∥|∇|1−2met∆u0∥Mp2

h2

≲ t
1
2−

n
2p2 ∥et∆u0∥

Ṅ
n
p

− n
p2

+1−2m

p,h,1

≲ t
1
2−

n
2p2 t−

1− n
p2
2 ∥u0∥

Ṅ
−2m+n

p
p,h,∞

≲ ∥u0∥
Ṅ

−2m+n
p

p,h,∞

.

(3.7)

Finally, estimates (3.5)-(3.7) yield (3.2).



8 A. EL IDRISSI, H. SRHIR, B. EL BOUKARI, J. EL GHORDAF EJDE-2025/80

(2) Since n < p ≤ ρ and ℏ
ρ = h

p , by using Lemmas 2.7 and 2.8 with ε = 1− n
p > 0, we obtain

t
1
2−

n
2ρ ∥et∆u0∥Ṅ 1−2m

ρ,ℏ,1
≲ t

1
2−

n
2ρ t−

1−n
p

2 −n
2 (

1
p−

1
ρ )∥u0∥

Ṅ
−2m+n

p
p,h,∞

≲ ∥u0∥
Ṅ

−2m+n
p

p,h,∞

. (3.8)

Thus, estimates (3.5) and (3.8) yield (3.4). □

The second lemma is a bilinear estimation.

Lemma 3.2. Let 1/2 < m < 1 and let 1 ≤ ℏ ≤ ρ < ∞, 1 ≤ ℏ1 ≤ ρ2 < ∞ and 1 ≤ ℏ2 ≤ ρ2 < ∞
satisfying 1

ρ = 1
ρ1

+ 1
ρ2

and 1
ℏ = 1

ℏ1
+ 1

ℏ2
. Then for all f ∈ Ṅ 1−2m

ρ1,ℏ1,1
and g ∈ Ṅ 1−2m

ρ2,ℏ2,1
, one has

∥f∇(−∆)−mg + g∇(−∆)−mf∥Ṅ 1−2m
ρ,ℏ,∞

≲ ∥f∥Ṅ 1−2m
ρ1,ℏ1,1

∥g∥Ṅ 1−2m
ρ2,ℏ2,1

. (3.9)

Proof. Using the paraproduct decomposition due to Bony [5], we have

f∇(−∆)−mg + g∇(−∆)−mf := J1 + J2 + J3, (3.10)

where

J1 :=
∑
l∈Z

∆̇lf∇(−∆)−mṠl−1g + ∆̇lg∇(−∆)−mṠl−1f,

J2 :=
∑
l∈Z

Ṡl−1f∇(−∆)−m∆̇lg + Ṡl−1g∇(−∆)−m∆̇lf,

J3 :=
∑
l∈Z

∑
|l−k′|≤1

∆̇lf∇(−∆)−m∆̇k′g + ∆̇lg∇(−∆)−m∆̇k′f.

Below, we estimate J1, J2 and J3 separately. For J1, we consider the estimate of its first term
only, while the second one can be treated similarly. So, by (2.1) and (2.2), the embedding (2.3),
Hölder’s inequality (2.7), and Remake 2.6, we obtain

∥∆̇j

∑
l∈Z

∆̇lf∇(−∆)−mṠl−1g∥Mρ
ℏ

≲
∑

|l−j|≤4

∥∆̇lf∇(−∆)−mṠl−1g∥Mρ
ℏ

≲
∑

|l−j|≤4

∥∆̇lf∥Mρ1
ℏ1

∥∇(−∆)−mṠl−1g∥Mρ2
ℏ2

≲ 2−(1−2m)j
∑

|l−j|≤4

2−(1−2m)(l−j)2(1−2m)l∥∆̇lf∥Mρ1
ℏ1

∥∇(−∆)−mg∥Mρ2
ℏ2

≲ 2−(1−2m)j
(

sup
|l−j|≤4

2−(1−2m)(l−j)
)
∥f∥Ṅ 1−2m

ρ1,ℏ1,1
∥∇(−∆)−mg∥Ṅ 0

ρ2,ℏ2,1

≲ 2−(1−2m)j∥f∥Ṅ 1−2m
ρ1,ℏ1,1

∥g∥Ṅ 1−2m
ρ2,ℏ2,1

.

Similarly,

∥∆̇j

∑
l∈Z

∆̇lg∇(−∆)−mṠl−1f∥Mρ
ℏ
≲ 2−(1−2m)j∥f∥Ṅ 1−2m

ρ1,ℏ1,1
∥g∥Ṅ 1−2m

ρ2,ℏ2,1
.

Which implies that

∥∆̇jJ1∥Mρ
ℏ
≲ 2−(1−2m)j∥f∥Ṅ 1−2m

ρ1,ℏ1,1
∥g∥Ṅ 1−2m

ρ2,ℏ2,1
. (3.11)

Analogously for J2, applying the embedding (2.3), Hölder’s inequality (2.7), and Remark 2.6 again,
since m > 1/2, one has

∥∆̇j

∑
l∈Z

Ṡl−1f∇(−∆)−m∆̇lg∥Mρ
ℏ

≲
∑

|l−j|≤4

∥Ṡl−1f∇(−∆)−m∆̇lg∥Mρ
ℏ



EJDE-2025/80 FRACTIONAL POROUS MEDIUM EQUATIONS 9

≲
∑

|l−j|≤4

∑
k≤l−2

∥∆̇kf∥Mρ1
ℏ1

∥∇(−∆)−m∆̇lg∥Mρ2
ℏ2

≲
∑

|j−l|≤4

2−(1−2m)l
∑

k≤l−2

2−(1−2m)(k−l)2(1−2m)k∥∆̇kf∥Mρ1
ℏ1

∥∇(−∆)−mg∥Mρ2
ℏ2

≲
∑

|j−l|≤4

2−(1−2m)l
(

sup
k≤l−2

2−(1−2m)(k−l)
)
∥f∥Ṅ 1−2m

ρ1,ℏ1,1
∥∇(−∆)−mg∥Ṅ 0

ρ2,ℏ2,1

≲ 2−(1−2m)j
( ∑

|j−l|≤4

2−(1−2m)(l−j)
)
∥f∥Ṅ 1−2m

ρ1,ℏ1,1
∥g∥Ṅ 1−2m

ρ2,ℏ2,1

≲ 2−(1−2m)j∥f∥Ṅ 1−2m
ρ1,ℏ1,1

∥g∥Ṅ 1−2m
ρ2,ℏ2,1

.

Thus, when m > 1/2, we have

∥∆̇jJ2∥Mρ
ℏ
≲ 2−(1−2m)j∥f∥Ṅ 1−2m

ρ1,ℏ1,1
∥g∥Ṅ 1−2m

ρ2,ℏ2,1
. (3.12)

We are now moving on to the last term J3. We use the following formula, based on an analysis of
the algebraic structure of the system (1.1):

(J3)i =
∑
l∈Z

∑
|l−k′|≤1

∆̇lf∂i(−∆)−m∆̇k′g + ∆̇lg∂i(−∆)−m∆̇k′f = K1
i +K2

i +K3
i ,

for i = 1, 2, . . . , n. Where (J3)i is the ith exponent of (J3) and

K1
i :=

∑
l∈Z

∑
|l−k′|≤1

(−∆)m
[(

(−∆)−m∆̇lf
)(

∂i(−∆)−m∆̇k′g
)]

,

K2
i :=

∑
l∈Z

∑
|l−k′|≤1

2∇m ·
[(

(−∆)−m∆̇lf
)(

∂i∇m(−∆)−m∆̇k′g
)]

,

K3
i :=

∑
l∈Z

∑
|l−k′|≤1

∂i

[(
(−∆)−m∆̇lf

)
∆̇k′g

]
.

To estimate the above three terms, we use Hölder’s inequality (2.7), Remark 2.6 and Cauchy-
Schwartz’s inequality in the following way: From (2.2), there is d0 ∈ N such that

∥∆̇jK
1
i ∥Mρ

ℏ
≲ 22mj

∑
|l−k′|≤1
l,k′≥j−d0

∥
(
(−∆)−m∆̇lf

)(
∂i(−∆)−m∆̇k′g

)
∥Mρ

ℏ

≲ 22mj
∑

|l−k′|≤1
l,k′≥j−d0

2−2ml∥∆̇lf∥Mρ1
ℏ1

2(1−2m)k′
∥∆̇k′g∥Mρ2

ℏ2

≲ 22mj
∑

|l−k′|≤1
l,k′≥j−d0

2−j
(

sup
l≥j−d0

2−(l−j)
)
2(1−2m)l∥∆̇lf∥Mρ1

ℏ1

2(1−2m)k′
∥∆̇k′g∥Mρ2

ℏ2

≲ 2−(1−2m)j∥f∥Ṅ 1−2m
ρ1,ℏ1,2

∥g∥Ṅ 1−2m
ρ2,ℏ2,2

.

(3.13)

∥∆̇jK
2
i ∥Mρ

ℏ
≲ 2mj

∑
|l−k′|≤1
l,k′≥j−d0

∥
(
(−∆)−m∆̇lf

)(
∂i∇m(−∆)−m∆̇k′g

)
∥Mρ

ℏ

≲ 2mj
∑

|l−k′|≤1
l,k′≥j−d0

2−2ml∥∆̇lf∥Mρ1
ℏ1

2(1−m)k′
∥∆̇k′g∥Mρ2

ℏ2

≲ 2mj
∑

|l−k′|≤1
l,k′≥j−d0

2−2ml∥∆̇lf∥Mρ1
ℏ1

2ml
(

sup
|l−k′|≤1

2m(k′−l)
)
2(1−2m)k′

∥∆̇k′g∥Mρ2
ℏ2



10 A. EL IDRISSI, H. SRHIR, B. EL BOUKARI, J. EL GHORDAF EJDE-2025/80

≲ 2−(1−2m)j
∑

|l−k′|≤1
l,k′≥j−d0

(
sup

l≥j−d0

2−(1−m)(l−j)
)
2(1−2m)l∥∆̇lf∥Mρ1

ℏ1

2(1−2m)k′
∥∆̇k′g∥Mρ2

ℏ2

.

Since m < 1, then

∥∆̇jK
2
i ∥Mρ

ℏ
≲ 2−(1−2m)j∥f∥Ṅ 1−2m

ρ1,ℏ1,2
∥g∥Ṅ 1−2m

ρ2,ℏ2,2
. (3.14)

∥∆̇jK
3
i ∥Mρ

ℏ

≲ 2j
∑

|l−k′|≤1
l,k′≥j−d0

∥
(
(−∆)−m∆̇lf

)
∆̇k′g∥Mρ

ℏ

≲ 2j
∑

|l−k′|≤1
l,k′≥j−d0

2−2ml∥∆̇lf∥Mρ1
ℏ1

∥∆̇k′g∥Mρ2
ℏ2

≲ 2j
∑

|l−k′|≤1
l,k′≥j−d0

2−2ml∥∆̇lf∥Mρ1
ℏ1

2−(1−2m)l
(

sup
|l−k′|≤1

2−(1−2m)(k′−l)
)
2(1−2m)k′

∥∆̇k′g∥Mρ2
ℏ2

≲ 2−(1−2m)j
∑

|l−k′|≤1
l,k′≥j−d0

(
sup

l≥j−d0

2−2(1−m)(l−j)
)
2(1−2m)l∥∆̇lf∥Mρ1

ℏ1

2(1−2m)k′
∥∆̇k′g∥Mρ2

ℏ2

.

Since m < 1, we have

∥∆̇jK
3
i ∥Mρ

ℏ
≲ 2−(1−2m)j∥f∥Ṅ 1−2m

ρ1,ℏ1,2
∥g∥Ṅ 1−2m

ρ2,ℏ2,2
. (3.15)

Thus, (3.13), (3.14), (3.15) and the embedding (2.4), give us

∥∆̇jJ3∥Mρ
ℏ
≤

n∑
i=1

3∑
k=1

∥∆̇jK
k
i ∥Mρ

ℏ
≲ 2−(1−2m)j∥f∥Ṅ 1−2m

ρ1,ℏ1,1
∥g∥Ṅ 1−2m

ρ2,ℏ2,1
. (3.16)

Eventually, we combine (3.11), (3.12) and (3.16), then multiply by 2(1−2m)j and take l∞−norm
on both sides of the resultant estimate to obtain (3.9). The proof of Lemma 3.2 is complete. □

The last estimate is a corollary of the following lemma.

Lemma 3.3 ([25]). Let s ∈ R, β > 0, 1 ≤ h ≤ p < ∞, 1 ≤ h0 ≤ p0 < ∞ and 1 ≤ r ≤ ∞. If

p0 ≥ p and h0

p0
= h

p . Then for all u ∈ ˜̇N s
p,h,r, we have

t
β
2 +n

2

(
1
p−

1
p0

)
∥et∆u∥Ṅ s+β

p0,h0,r

t→0−−−→ 0.

Lemma 3.4. Let 0 < T < ∞, 1
2 < m < 1, 1 ≤ h ≤ p < ∞ and u0 ∈

˜
Ṅ−2m+n

p

p,h,∞ .

(1) Let n
2m < p ≤ n and let p1, p2, h1, h2 satisfy (2.12). Then, one has

tm− n
2p1 ∥et∆u0∥Mp1

h1

+ t
1
2−

n
2p2 ∥|∇|1−2met∆u0∥Mp2

h2

t→0−−−→ 0.

(2) Let p > n, the parameters ρ, ℏ satisfy (2.13) and 1 ≤ r ≤ ∞. Then, one has

t
1
2−

n
2ρ ∥et∆u0∥Ṅ 1−2m

ρ,ℏ,r

t→0−−−→ 0.

Proof. (1) By applying the embedding (2.3) and Lemma 3.3 for s = −2m+ n
p , β = 2m− n

p > 0,

p0 = p1 > p, h0 = h1 and r = 1, we obtain

tm− n
2p1 ∥et∆u0∥Mp1

h1

≲ tm− n
2p1 ∥et∆u0∥Ṅ 0

p1,h1,1

t→0−−−→ 0. (3.17)

For the second term, by the Adams theorem [1], we have

t
1
2−

n
2p2 ∥|∇|1−2met∆u0∥Mp2

h2

≲ t
1+ζ
2 − n

2p3 ∥|∇|ζ+1−2met∆u0∥Mp3
h3

,



EJDE-2025/80 FRACTIONAL POROUS MEDIUM EQUATIONS 11

where ζ > 0 and p3, h3 satisfy

1 ≤ h3 ≤ p3 < ∞, p3 > p,
h3

p3
=

h2

p2
,

1

p3
=

1

p2
+

ζ

d
.

Using again the embedding (2.3) and Lemma 3.3 for s = −2m+ n
p , β = ζ+1− n

p > 0, p0 = p3 > p,

h0 = h3 and r = 1, we obtain

t
1+ζ
2 − n

2p3 ∥|∇|ζ+1−2met∆u0∥Mp3
h3

≲ t
1+ζ
2 − n

2p3 ∥|∇|ζ+1−2met∆u0∥Ṅ 0
p3,h3,1

≲ t
1+ζ
2 − n

2p3 ∥|et∆u0∥Ṅ ζ+1−2m
p3,h3,1

t→0−−−→ 0.

Thus,

t
1
2−

n
2p2 ∥|∇|1−2met∆u0∥Mp2

h2

t→0−−−→ 0. (3.18)

Then (3.17) and (3.18) yield the desired result.
(2) By choosing s = −2m+ n

p , β = 1− n
p > 0, p0 = ρ ≥ p and h0 = ℏ, Lemma 3.3 gives us the

desired limit. □

4. Proof of main theorems

We find local solutions with any initial data u0 in the subspaces
˜

Ṅ−2m+n
p

p,h,∞ and global solutions

with small initial data u0 in the critical Besov-Morrey spaces Ṅ−2m+n
p

p,h,∞ , for the gathered n < p < ∞
and n

2m < p ≤ n in Subsection 4.1 and Subsection 4.2, respectively.

4.1. Case n < p < ∞: Proof Theorems 2.9 (2) and 2.10 (2). . We first prove the following
bilinear estimate.

Proposition 4.1. Let 0 < T ≤ ∞, 1
2 < m < 1, n < p < ∞, 1 ≤ h ≤ p, the parameters ρ, ℏ satisfy

(2.13), and let

∥u∥Y 1
T
:= sup

t∈(0,T )

t
1
2−

n
2ρ ∥u(t)∥Ṅ 1−2m

ρ,ℏ,1
. (4.1)

Then

∥
∫ t

0

e(t−t′)∆∇ ·
(
u∇(−∆)−mw + w∇(−∆)−mu

)
dt′∥YT

≲ ∥u∥Y 1
T
∥w∥Y 1

T
.

Proof. Recall that for 0 < T ≤ ∞, ∥ · ∥YT
is given by

∥u∥YT
:= sup

t∈(0,T )

∥u(t)∥
Ṅ

−2m+n
p

p,h,∞

+ sup
t∈(0,T )

t
1
2−

n
2ρ ∥u(t)∥Ṅ 1−2m

ρ,ℏ,1
.

We denote H := v∇(−∆)−mw + w∇(−∆)−mu. Applying Remark 2.6, the embedding (2.4) and
Lemma 2.8, we obtain

∥
∫ t

0

e(t−t′)∆∇ ·Hdt′∥
Ṅ

−2m+n
p

p,h,∞

≲
∫ t

0

∥e(t−t′)∆Hdt′∥
Ṅ

1−2m+n
p

p,h,1

dt′

≲
∫ t

0

(t− t′)−
n
2p−

n
2 (

2
ρ−

1
p )∥H∥Ṅ 1−2m

ρ
2
, ℏ
2
,∞

dt′.

From Lemma 3.2, we have

∥
∫ t

0

e(t−t′)∆∇ ·Hdt′∥
Ṅ

−2m+n
p

p,h,∞

≲
∫ t

0

(t− t′)−
n
ρ ∥u(t′)∥Ṅ 1−2m

ρ,ℏ,1
∥w(t′)∥Ṅ 1−2m

ρ,ℏ,1
dt′

≲
∫ t

0

(t− t′)−
n
ρ (t′)2(−

1
2+

n
2ρ )∥u∥Y 1

T
∥w∥Y 1

T
dt′

≲ ∥u∥Y 1
T
∥w∥Y 1

T

∫ t

0

(t− t′)−
n
ρ (t′)−1+n

ρ dt′

≲ ∥u∥Y 1
T
∥w∥Y 1

T
.

(4.2)



12 A. EL IDRISSI, H. SRHIR, B. EL BOUKARI, J. EL GHORDAF EJDE-2025/80

For the second term of ∥ · ∥YT
, we apply Remark 2.6 and Lemmas 2.8 and 3.2 again, to obtain

∥
∫ t

0

e(t−t′)∆∇ ·Hdt′∥Ṅ 1−2m
ρ,ℏ,1

≲
∫ t

0

∥e(t−t′)∆Hdt′∥Ṅ 2−2m
ρ,ℏ,1

≲
∫ t

0

(t− t′)−
1
2−

n
2 (

2
ρ−

1
ρ )∥H∥Ṅ 1−2m

ρ
2
, ℏ
2
,∞

dt′

≲
∫ t

0

(t− t′)−
1
2−

n
2ρ ∥u(t′)∥Ṅ 1−2m

ρ,ℏ,1
∥w(t′)∥Ṅ 1−2m

ρ,ℏ,1
dt′

≲ ∥u∥Y 1
T
∥w∥Y 1

T

∫ t

0

(t− t′)−
1
2−

n
2ρ (t′)−

1
2+

n
2ρ (t′)−

1
2+

n
2ρ dt′

≲ t−
1
2+

n
2ρ ∥u∥Y 1

T
∥w∥Y 1

T
.

Finally, we obtain the estimate of ∥ · ∥YT
by multiplying the two sides of the above inequality by

t
1
2−

n
2ρ , and combine the resultant estimate with (4.2). The proof of Proposition 4.1 is complete. □

Now we can prove the existence of local solutions of system (1.1) for any initial data u0 in

˜
Ṅ−2m+n

p

p,h,∞ with 1 ≤ h ≤ p < ∞, in the case n < p < ∞.

Proof of Theorem 2.9 (2) For T > 0 and t ∈ [0, T ], we define the map

Ψ : u(t) → et∆u0 +

∫ t

0

e(t−t′)∆∇ ·
(
u∇(−∆)−mu

)
dt′, (4.3)

in the following metric space, for a small δ > 0,

Y 1
T := {u ∈ BC

(
0, T ; Ṅ−2m+n

p

p,h,∞
)
: ∥u∥Y 1

T
≤ δ},

equipped with the distance

d(u,w) := ∥u− w∥Y 1
T
.

According to Lemma 3.4 (2), we have the existence of T > 0 such that

∥et∆u0∥Y 1
T
≤ δ

2
,

and we choose T satisfying this. Then by Proposition 4.1, there exists C1 such that for every
u ∈ Y 1

T , we have

∥Ψ(u)∥Y 1
T
≤ ∥et∆u0∥Y 1

T
+ C1

(
∥u∥Y 1

T

)2 ≤ δ

2
+ C1δ

2,

and, by setting A(u,w) :=
∫ t

0
e(t−t′)∆∇ ·

(
u∇(−∆)−mw

)
dt′, there exists C2 such that for every

u,w ∈ Y 1
T , we obtain

d (Ψ(u),Ψ(w)) = ∥A(u, u)−A(w,w)∥Y 1
T

≤ ∥A(u, v − w)∥Y 1
T
+ ∥A(u− w,w)∥Y 1

T

≤ C2

(
∥u∥Y 1

T
+ ∥w∥Y 1

T

)
∥u− w∥Y 1

T

≤ 2C2δ∥u− w∥Y 1
T
.

Then, by choosing δ small enough such that

δ ≤ min
{ 1

2C1
,

1

4C2

}
,

we have

∥Ψ(u)∥Y 1
T
≤ δ, d (Ψ(u),Ψ(w)) ≤ 1

2
d(u,w).

Thus, according to Banach’s fixed point theorem, we obtain a unique fixed point u ∈ Y∞ of Ψ, that
is, the local solution of system (1.1). Furthermore, by setting ∥u∥Y 0

T
:= supt∈(0,T ) ∥u(t)∥Ṅ−2m+n

p
p,h,∞

,



EJDE-2025/80 FRACTIONAL POROUS MEDIUM EQUATIONS 13

and since Ψ(u) = v, then from Lemma 3.1 (2) and Proposition 4.1, we obtain

∥u∥Y 0
T
= ∥Ψ(u)∥Y 0

T
≲ ∥u0∥

Ṅ
−2m+n

p
p,h,∞

+
(
∥u∥Y 1

T

)2
≲ ∥u0∥

Ṅ
−2m+n

p
p,h,∞

+ δ2 < ∞.

The proof of the second statement of the Theorem 2.9 is thus complete.
Next, we demonstrate the existence of global solutions of Equation 1.1 for small initial data u0

in Ṅ−2m+n
p

p,h,∞ with 1 ≤ h ≤ p < ∞, in the case n < p < ∞.

Proof of Theorem 2.10 (2). For small C0 > 0, we define the metric space

Y∞ := {u ∈ BC
(
0,∞; Ṅ−2m+n

p

p,h,∞

)
: ∥u∥Y∞ ≤ C0},

equipped with the distance d(u,w) := ∥u− w∥Y∞ . Returning to the map (4.3), and according to
Lemma 3.1 (2) and Proposition 4.1, there exist C3, C4 > 0 such that for every u ∈ Y∞, we have

∥Ψ(u)∥Y∞ ≤ C3∥u0∥
Ṅ

−2m+n
p

p,h,∞

+ C4 (∥u∥Y∞)
2 ≤ C3∥u0∥

Ṅ
−2m+n

p
p,h,∞

+ C4C
2
0 ,

and, there exists C5 such that for every u,w ∈ Y∞, we obtain

d (Ψ(u),Ψ(w)) ≤ C5 (∥u∥Y∞ + ∥w∥Y∞) ∥u− w∥Y∞ ≤ 2C5C0∥u− w∥Y∞ .

Assume that ∥u0∥
Ṅ

−2m+n
p

p,h,∞

≤ δ, for small δ > 0 satisfy

δ ≤ 1

2C3
C0,

and choosing C0 small enough such that

C0 ≤ min
{ 1

2C4
,

1

4C5

}
.

Then, for every u,w ∈ Y∞, one has

∥Ψ(u)∥Y∞ ≤ C0, d (Ψ(u),Ψ(w)) ≤ 1

2
d(u,w).

Eventually, using the Banach fixed point theorem, we obtain a unique fixed point u ∈ Y∞ of Ψ,
that is, the global solution of the system (1.1). This completes the proof.

4.2. The case n
2m < p ≤ n: Proof of Theorems 2.9 (1) and 2.10 (1). To finish the proofs of

Theorem 2.9 and Theorem 2.10, it suffices to prove the following crucial bilinear estimate. With
this estimate, by using Lemma 3.4 (1), Lemma 3.1 (1) and the same argument as in Subsection
4.1, we can obtain the local and global solutions for the case n

2m < p ≤ n.

Proposition 4.2. Let 0 < T ≤ ∞, 1
2 < m < 1, max{1, n

2m} < p ≤ n, 1 ≤ h ≤ p the parameters
p1, p2, h1, h2 satisfy (2.12), and let ∥ · ∥X1

T
be given as

∥u∥X1
T
:= sup

t∈(0,T )

tm− n
2p1 ∥u(t)∥Mp1

h1

+ sup
t∈(0,T )

t
1
2−

n
2p2 ∥|∇|1−2mu(t)∥Mp2

h2

. (4.4)

Then, we have

∥
∫ t

0

e(t−t′)∆∇ ·
(
u∇(−∆)−mw

)
dt′∥XT

≲ ∥u∥X1
T
∥w∥X1

T
.

Proof. We have

∥u∥XT
:= sup

t∈(0,T )

∥u(t)∥
Ṅ

−2m+n
p

p,h,∞

+ sup
t∈(0,T )

tm− n
2p1 ∥u(t)∥Mp1

h1

+ sup
t∈(0,T )

t
1
2−

n
2p2 ∥|∇|1−2mu(t)∥Mp2

h2

.

Let ϑ be defined by 1
ϑ := 1

p1
+ 1

p2
, so that ϑ < p and n

1+2m < ϑ < d. Applying Remark 2.6, the

embeddings (2.9), (2.5), (2.4) and (2.3), and Lemma 2.8, we obtain

∥
∫ t

0

e(t−t′)∆∇ ·
(
u∇(−∆)−mw

)
dt′∥

Ṅ
−2m+n

p
p,h,∞



14 A. EL IDRISSI, H. SRHIR, B. EL BOUKARI, J. EL GHORDAF EJDE-2025/80

≲
∫ t

0

∥e(t−t′)∆
(
u∇(−∆)−mw

)
dt′∥

Ṅ
1−2m+n

ϑ
ϑ,h,∞

≲
∫ t

0

(t− t′)−
1−2m+n

ϑ
2 ∥v∇(−∆)−mw∥Mϑ

h
dt′.

Using Hölder’s inequality (2.7) and since m < 1, n
1+2m < ϑ < d, then we have

∥
∫ t

0

e(t−t′)∆∇ ·
(
u∇(−∆)−mw

)
dt′∥

Ṅ
−2m+n

p
p,h,∞

≲
∫ t

0

(t− t′)−
1
2+m− n

2ϑ ∥u(t′)∥Mp1
h1

∥∇(−∆)−mw(t′)∥Mp2
h2

dt′

≲
∫ t

0

(t− t′)−
1
2+m− n

2ϑ (t′)−m+ n
2p1

− 1
2+

n
2p2 ∥u∥X1

T
∥w∥X1

T
dt′

≲ ∥u∥X1
T
∥w∥X1

T

∫ t

0

(t− t′)−
1
2+m− n

2ϑ (t′)−
1
2−m+ n

2ϑ dt′

≲ ∥u∥X1
T
∥w∥X1

T
.

(4.5)

For the other terms of ∥ · ∥XT
, we have

∥
∫ t

0

e(t−t′)∆∇ ·
(
u∇(−∆)−mw

)
dt′∥Mp1

h1

≲
∫ t

0

∥e(t−t′)∆
(
u∇(−∆)−mw

)
dt′∥Ṅ 1

p1,h1,1

≲
∫ t

0

(t− t′)
− 1

2−
n
2

(
1
ϑ− 1

p1

)
∥v∇(−∆)−mw∥Ṅ 0

ϑ,
h1
p1

ϑ,∞

dt′

≲
∫ t

0

(t− t′)−
1
2−

n
2p2 ∥v∇(−∆)−mw∥Mϑ

h1
p1

ϑ

dt′

≲
∫ t

0

(t− t′)−
1
2−

n
2p2 ∥u(t′)∥Mp1

h1

∥∇(−∆)−mw(t′)∥Mp2
h2

dt′

≲
∫ t

0

(t− t′)−
1
2−

n
2p2 (t′)−

1
2+

n
2p2

−m+ n
2p1 ∥u∥X1

T
∥w∥X1

T
dt′

≲ t−m+ n
2p1 ∥u∥X1

T
∥w∥X1

T
,

and

∥|∇|1−2m

∫ t

0

e(t−t′)∆∇ ·
(
u∇(−∆)−mw

)
dt′∥Mp2

h2

≲
∫ t

0

∥e(t−t′)∆
(
u∇(−∆)−mw

)
dt′∥Ṅ 2−2m

p2,h2,1

≲
∫ t

0

(t− t′)
− 1

2 (2−2m)−n
2

(
1
ϑ− 1

p2

)
∥v∇(−∆)−mw∥Mϑ

h2
p2

ϑ

dt′

≲
∫ t

0

(t− t′)−1+m− n
2p1 ∥u(t′)∥Mp1

h1

∥∇(−∆)−mw(t′)∥Mp2
h2

dt′

≲
∫ t

0

(t− t′)−1+m− n
2p1 (t′)−m+ n

2p1
− 1

2+
n

2p2 ∥u∥X1
T
∥w∥X1

T
dt′

≲ t−
1
2+

n
2p2 ∥u∥X1

T
∥w∥X1

T
.

Thus, we arrive at

tm− n
2p1 ∥

∫ t

0

e(t−t′)∆∇ ·
(
u∇(−∆)−mw

)
dt′∥Mp1

h1

≲ ∥u∥X1
T
∥w∥X1

T
, (4.6)



EJDE-2025/80 FRACTIONAL POROUS MEDIUM EQUATIONS 15

t
1
2−

n
2p2 ∥|∇|1−2m

∫ t

0

e(t−t′)∆∇ ·
(
u∇(−∆)−mw

)
dt′∥Mp2

h2

≲ ∥u∥X1
T
∥w∥X1

T
. (4.7)

Therefore, we obtain Proposition 4.2. □

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Ahmed El Idrissi

LMACS Laboratory, Faculty of Science and Technology, Sultan Moulay Slimane University, Beni

Mellal, 23000, Morocco
Email address: ahmed.elidrissi@usms.ma

Halima Srhiri
LMACS Laboratory, Faculty of Science and Technology, Sultan Moulay Slimane University, Beni

Mellal, 23000, Morocco

Email address: halima.srhiri.1998@gmail.com

Brahim El Boukari

LMACS Laboratory, Faculty of Science and Technology, Sultan Moulay Slimane University, Beni

Mellal, 23000, Morocco
Email address: elboukaribrahim@yahoo.fr

Jalila El Ghordaf
LMACS Laboratory, Faculty of Science and Technology, Sultan Moulay Slimane University, Beni

Mellal, 23000, Morocco

Email address: elg jalila@yahoo.fr


	1. Introduction
	2. Preliminaries and main results
	2.1. Littlewood-Paley theory and Besov-Morrey spaces
	2.2. Essential lemmas
	2.3. Main results

	3. Key estimates
	4. Proof of main theorems
	4.1. Case n<p<: Proof Theorems 2.9 (2) and 2.10 (2)
	4.2. The case n2m<pn: Proof of Theorems 2.9 (1) and 2.10 (1)

	References