Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 09, pp. 1–37.

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

EXISTENCE AND GLOBAL BEHAVIOR OF WEAK SOLUTIONS

TO A DOUBLY NONLINEAR EVOLUTION FRACTIONAL

p-LAPLACIAN EQUATION

JACQUES GIACOMONI, ABDELHAMID GOUASMIA, ABDELHAFID MOKRANE

Abstract. In this article, we study a class of doubly nonlinear parabolic prob-

lems involving the fractional p-Laplace operator. For this problem, we discuss

existence, uniqueness and regularity of the weak solutions by using the time-
discretization method and monotone arguments. For global weak solutions, we

also prove stabilization results by using the accretivity of a suitable associated

operator. This property is strongly linked to the Picone identity that provides
further a weak comparison principle, barrier estimates and uniqueness of the

stationary positive weak solution.

1. Introduction and statement of main results

Let 1 < q ≤ p <∞, 0 < s < 1, QT := (0, T )× Ω, where Ω ⊂ RN , with N > sp,
is an open bounded domain with C1,1 boundary. ΓT := (0, T ) × ∂Ω denotes the
lateral boundary of the cylinder QT . In this work, we deal with the existence,
uniqueness and other qualitative properties of the weak solution to the following
doubly nonlinear parabolic equation:

q

2q − 1
∂t(u

2q−1) + (−∆)spu = f(x, u) + h(t, x)uq−1 in QT ,

u > 0 in QT ,

u = 0 on ΓT ,

u(0, ·) = u0 in Ω.

(1.1)

Here (−∆)spu is the fractional p-Laplace operator, defined for 1 < p <∞, as

(−∆)spu(x) := 2 P.V.

∫
RN

|u(x)− u(y)|p−2(u(x)− u(y))

|x− y|N+sp
dy,

where P.V. denotes the Cauchy principal value. We refer to [21, 29, 38] for the
main properties of this nonlinear fractional elliptic operator.

Throughout this article we assume the following hypothesis:

(H1) f : Ω × R+ → R+ is a continuous function, such that f(x, 0) ≡ 0 and f is
positive on Ω× R+\{0}.

(H2) For a.e. x ∈ Ω, s 7→ f(x,s)
sq−1 is non-increasing in R+\{0}.

2010 Mathematics Subject Classification. 35B40, 35K59, 35K55, 35K10, 35R11.
Key words and phrases. Fractional p-Laplace equation; doubly nonlinear evolution equation;
Picone identity; stabilization; nonlinear semi-group theory.
c©2021 Texas State University.

Submitted June 6, 2020. Published February 23, 2021.

1



2 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

(H3) If q = p, s 7→ f(x,s)
sp−1 is decreasing in R+\{0} for a.e. x ∈ Ω and

limr→+∞
f(x,r)
rp−1 = 0 uniformly in x ∈ Ω.

(H4) There exists h ∈ L∞(Ω)\{0}, h ≥ 0 such that h(t, x) ≥ h(x) a.e. in QT .
(H5) If q = p,

‖h‖L∞(QT ) < λ1,s,p := inf
φ∈W s,p

0 (Ω)\{0}

||φ||p
W s,p

0 (Ω)

‖φ‖pLp(Ω)

.

(H6) If q = p, h, f fulfills the condition

inf
x∈Ω

(
h(x) + lim

s→0+

f(x, s)

sp−1

)
> λ1,s,p.

1.1. State of the art. The study of nonlocal elliptic operators arouse more and
more interest in mathematical modeling, see e.g. [8, 11, 12, 14, 27, 34, 42] and
the references cited therein. Concerning the investigation on parabolic equations
involving nonlocal operators, we refer to [1, 5, 15, 16, 18, 19, 24, 25, 30, 31, 32,
33, 35, 37, 38, 39, 41] without giving an exhaustive list. These types of operators
arise in several contexts: in finance, physics, fluid dynamics, image processing
and in various fields like continuum mechanics, stochastic processes of Lévy type,
phase transitions, population dynamics, optimal control and game theory, see for
further discussion [15, 17, 21, 29, 39] and the references therein. In particular [15]
shows some non-local diffusion models coming from game theory. In connection to
our doubly nonlinear problem (1.1), [37] shows different methods (entropy method
and contraction semi-group theory) two evolution models of flows in porous media
involving fractional operators:
• The first model is based on Darcy’s law and is given by

∂tu = ∇ · (u∇P ) in (0,∞)× RN ,

P = (−∆)−su in (0,∞)× RN ,

u(0, x) = u0(x) in RN ,

where u is the particle density of the fluid, P is the pressure and (−∆)−s is the
inverse of the fractional Laplace operator (i.e. p = 2). The initial data u0 is
a nonnegative, bounded and integrable function in RN (see also [13] for further
explanations).
• The second model in analogy to classical models of transport through porous
media (see [22]) is described in the non local case by

∂tu+ (−∆)s(um) = 0. (1.2)

For s → 1− and m = 1, the limiting model (1.2) is the well known heat equation.
Furthermore if m > 1, (1.2) is known as the porous media equation (PME for
short) whereas in case m < 1 it is referred as the fast diffusion equation (FDE
for short). Existence and global behaviour of solutions are described in [37] for
the two types of models. We refer again to [39] for further explanations about the
physical background and the adequacy of nonlocal diffusion operators (see also [19]
for related issues). The paper [18] deals with the problem (1.2) in the special case
s = 1

2 , and p = 2 and investigates the local existence, uniqueness and regularity
of the weak solution. We highlight here that few results are available about the
parabolic equation involving fractional p-Laplacian operator in contrast with the
stationary elliptic equation.



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 3

In [25], considering the more general case 1 < p < ∞, authors obtain the ex-
istence, uniqueness, and regularity of the weak solution to the fractional reaction
diffusion equation

∂tu+ (−∆)spu+ g(x, u) = f(x, u) in QT ;

u = 0 in RN \QT ;

u(0, ·) = u0 in RN ,
(1.3)

with f and g, satisfying suitable growth and homogeneity conditions. In addition,
the authors prove that global solutions converge to the unique positive stationary
solution as t → ∞. Previously, [1] has dealt with the case where the nonlinearity
f depends only on x and t and have established the existence and some properties
of nonnegative entropy solutions. In [24], the authors have studied (1.3), under
similar conditions about f and g(x, u) := −|u(t, x)|q−2u(t, x), with q ≥ 2. They
prove the existence of locally-defined strong solutions to the problem with any
initial data u0 ∈ Lr(Ω) and r ≥ 2. They also investigate the occurrence of finite
time blow up behavior. In [30, 38] the results about existence, uniqueness and
T -accretivity in L1 of strong solutions to the fractional p-Laplacian heat equation
with Dirichlet or Neumann boundary conditions, are obtained through the theory
of nonlinear accretive operators. The asymptotic decay of solutions and the study
of asymptotic models as p → 1+ are also investigated. In [26], authors extend the
results obtained in [4] in case of singular nonlinearities and fractional diffusion. We
refer the reader to [28, 33, 36, 40, 41] for further investigations of above issues.

The aim of this article is to discuss similar issues about local existence, unique-
ness, regularity and global behavior of solutions to the doubly nonlinear and non
local equation (1.1). Up to our knowledge, (1.1) which covers several PME and
FDE models in the fractional setting has not been investigated in the literature.
By using the semi-discretization in time method applied to an auxiliary evolution
problem, we prove the local existence of weak energy solutions. The uniqueness of
weak solutions are obtained via the fractional version of the Picone identity (see
below) which leads to a new comparison principle and T -accretivity of an associ-
ated operator in L2. Using the comparison principle, we also prove the existence
of barrier functions from which we derive that weak solutions are global. We then
show that weak solutions converge to the unique non trivial stationary solution as
t→∞. To achieve this goal, our approach borrows techniques from the contraction
semi-group theory.

1.2. Preliminaries and functional setting. First, we recall some notation which
will be used throughout the paper. Considering a measurable function u : RN → R,
we adopt
• Let p ∈ [1; +∞[, the norm in the space Lp(Ω) is denoted by

‖u‖Lp(Ω) :=
(∫

Ω

|u|pdx
)1/p

.

• Set 0 < s < 1 and p > 1, we recall that the fractional Sobolev space W s,p(RN ) is
defined as

W s,p(RN ) :=
{
u ∈ Lp(RN ) :

∫
RN

∫
RN

|u(x)− u(y)|p

|x− y|N+sp
dx dy <∞},



4 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

endowed with the norm

‖u‖W s,p(RN ) :=
(
‖u‖p

Lp(RN )
+

∫
RN

∫
RN

|u(x)− u(y)|p

|x− y|N+sp
dx dy

)1/p

.

• The space W s,p
0 (Ω) is the set of functions

W s,p
0 (Ω) := {u ∈W s,p(RN ) : u = 0 a.e. in RN \ Ω},

and the norm is given by the Gagliardo semi-norm

‖u‖W s,p
0 (Ω) :=

(∫
RN

∫
RN

|u(x)− u(y)|p

|x− y|N+sp
dx dy

)1/p

.

We recall that by the fractional Poincaré inequality (e.g., in [29, Theorem 6.5];
see also Theorem 1.3 below), ‖ · ‖W s,p(RN ) and ‖ · ‖W s,p

0 (Ω) are equivalent norms

on W s,p
0 (Ω). From the results in [21], [29], we have that W s,p

0 (Ω) is continuously

embedded in Lr(Ω) when 1 ≤ r ≤ Np
N−sp and compactly for 1 ≤ r < Np

N−sp .

• Let α ∈ (0, 1], we consider the space of Hölder continuous functions:

Cα(Ω) =
{
u ∈ C(Ω), sup

x,y∈Ω, x 6=y

|u(x)− u(y)|
|x− y|α

<∞
}
,

endowed with the norm

‖u‖Cα(Ω) = ‖u‖L∞(Ω) + sup
x,y∈Ω,x 6=y

|u(x)− u(y)|
|x− y|α

.

• Let T > 0, and consider a measurable function

u :]0, T [→W s,p
0 (Ω),

and we denote u(t)(x) := u(t, x). Let C([0, T ],W s,p
0 (Ω)) the space of continuous

functions in [0, T ] with vector values in W s,p
0 (Ω), endowed with the norm

‖u‖C([0,T ],W s,p
0 (Ω)) := sup

t∈[0,T ]

‖u(t)‖W s,p
0 (Ω).

• We denote by d(·) the distance function up to the boundary ∂Ω. That means

d(x) := dist(x, ∂Ω) = inf
y∈∂Ω

|x− y|.

• We define for r > 0, the sets

Mr
ds(Ω) :=

{
u : Ω→ R+ : u ∈ L∞(Ω) and ∃c > 0 s.t.

c−1ds(x) ≤ ur(x) ≤ cds(x)
}
,

V̇ r+ := {u : Ω→ (0,∞) : u1/r ∈W s,p
0 (Ω)}.

(1.4)

• We define the weighted space

L∞ds(Ω) := {u : Ω→ R : u ∈ L∞(Ω) s.t.
u

ds(·)
∈ L∞(Ω)}.

Let φ1,s,p be the positive normalized eigenfunction (‖φ1,s,p‖L∞(Ω) = 1 ) of (−∆)sp
in W s,p

0 (Ω) associated to the first eigenvalue λ1,s,p. We recall that φ1,s,p ∈ C0,α(Ω)
for some α ∈ (0, s] (see Theorem 1.1 in [27]) and φ1,s,p ∈M1

ds(Ω) (see [27, Theorem
4.4] and [20, Theorem 1.5]). Next, we recall some results that will be used in the
sequel.



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 5

Proposition 1.1 (Discrete hidden convexity [9, Proposition 4.1]). Let 1 < p <∞
and 1 < q ≤ p. For every u0, u1 ≥ 0, we define

σt(x) = [(1− t)uq0(x) + tuq1(x)]1/q, t ∈ [0, 1], x ∈ RN .

Then

|σt(x)−σt(y)|p ≤ (1− t)|u0(x)−u0(y)|p + t|u1(x)−u1(y)|p, t ∈ [0, 1], x, y ∈ RN .

Proposition 1.2 (Discrete Picone inequality [9, Proposition 4.2]). Let 1 < p <∞
and 1 < r ≤ p. Let u, v be two Lebesgue-measurable functions with v ≥ 0 and u > 0.
Then

|u(x)− u(y)|p−2(u(x)− u(y))
[ v(x)r

u(x)r−1
− v(y)r

u(y)r−1

]
≤ |v(x)− v(y)|r|u(x)− u(y)|p−r.

As we will see, Proposition 1.2 provides a comparison principle, barrier estimates
and uniqueness of weak solutions.

Theorem 1.3 ([21, Theorem 6.5]). Let s ∈ (0, 1), p ≥ 1 with N > sp. Then,
there exists a positive constant C = C(N, p, s) such that, for any measurable and
compactly supported u : RN → R function, we have

‖u‖p
Lp
∗
s (RN )

≤ C
∫
RN

∫
RN

|u(x)− u(y)|p

|x− y|N+sp
dx dy,

where p∗s = Np
N−sp . Consequently, the space W s,p(RN ) is continuously embedded in

Lq(RN ) for q ∈ [p, p∗s].

Theorem 1.4 (Aubin-Lions-Simon, [7, Theorem II.5.16]). Let B0 ⊂ B1 ⊂ B2 be
three Banach spaces. We assume that the embedding of B1 in B2 is continuous and
that the embedding of B0 in B1 is compact. Let p, r such that 1 ≤ p, r ≤ ∞. For
T > 0, we define

Ep,r = {v ∈ Lp(]0, T [;B0) :
dv

dt
∈ Lr(]0, T [;B2)}.

Then the following holds:

(a) If p <∞, then the embedding of Ep,r in Lp(]0, T [;B1) is compact.
(b) If p =∞ and r > 1, then the embedding of Ep,r in C([0, T ];B1) is compact.

We now recall the definition of the strict ray-convexity.

Definition 1.5. Let X be a real vector space. Let C be a non empty convex cone
in X. A functional W : C → R will be called ray-strictly convex (strictly convex,
respectively) if it satisfies

W((1− t)v1 + tv2) ≤ (1− t)W(v1) + tW(v2),

for all v1, v2 ∈ C and for all t ∈ (0, 1), where the inequality is always strict unless
v1
v2
≡ c > 0 (always strict unless v1 ≡ v2, respectively).

Remark 1.6. We observe that by Proposition 1.1, the set V̇ r+ defined in (1.4) is a

convex cone, i.e. for λ ∈ (0,∞), f, g ∈ V̇ r+ implies λf + g ∈ V̇ r+.



6 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

Proposition 1.7 (Convexity). Let 1 < p < ∞ and 1 < r ≤ p. The functional

W : V̇ r+ → R+ defined by

W(w) :=
1

p

∫
RN

∫
RN

|w(x)1/r − w(y)1/r|p

|x− y|N+sp
dx dy,

is ray-strictly convex on V̇ r+. Furthermore, if p 6= r, then W is even strictly convex

on V̇ r+.

Proof. According to Definition 1.5, let us consider any w1, w2 ∈ V̇ r+ and t ∈ [0, 1].
Let us denote w = tw1 + (1− t)w2, we obtain by Proposition 1.1

W(w) ≤ tW(w1) + (1− t)W(w2). (1.5)

If the equality holds, then

|w(x)1/r − w(y)1/r|p = t |w1(x)1/r − w1(y)1/r|p + (1− t) |w2(x)1/r − w2(y)1/r|p

a.e. x, y ∈ RN .. If p = r, we obtain∣∣‖a‖`r − ‖b‖`r ∣∣r = ‖a− b‖r`r a.e. x, y ∈ RN ,

where ‖ · ‖`r denotes the `r-norm in R2, and

a =
(
(tw1(x))1/r, ((1− t)w2(x))1/r

)
, b =

(
(tw1(y))1/r, ((1− t)w2(y))1/r

)
.

Since r > 1, there exists a constant c > 0 such that w1 = cw2 a.e. x ∈ RN . Then,
W is ray-strictly convex on V̇ r+. On the other hand, if p 6= r thanks to the strict

convexity of τ 7→ τ
p
r on R+, we obtain w1 = w2 a.e. x ∈ RN and W is strictly

convex on V̇ r+. �

Lemma 1.8. Let 1 < p <∞. Then, for 1 < r ≤ p and for any u, v two measurable
and positive functions in Ω:

|u(x)− u(y)|p−2
(
u(x)− u(y)

)[u(x)r − v(x)r

u(x)r−1
− u(y)r − v(y)r

u(y)r−1

]
+ |v(x)− v(y)|p−2(v(x)− v(y))

[v(x)r − u(x)r

v(x)r−1
− v(y)r − u(y)r

v(y)r−1

]
≥ 0

(1.6)

for a.e. x, y ∈ Ω. Moreover, if u, v ∈ W s,p
0 (Ω) and if the equality occurs in (1.6)

for a.e. x, y ∈ Ω, then we have the following two statements:

(1) u/v ≡ const > 0 a.e. in Ω.
(2) If also p 6= r, then u ≡ v a.e. in Ω.

Proof. Let u, v be two measurable functions such that u, v > 0 in Ω and 1 < r ≤ p.
Then by using Proposition 1.2, we obtain for x, y ∈ Ω,

|u(x)− u(y)|p−2(u(x)− u(y))
[ v(x)r

u(x)r−1
− v(y)r

u(y)r−1

]
≤ |v(x)− v(y)|r|u(x)− u(y)|p−r.

(1.7)

Let us start with the case r = p. By using the above inequality,in this case, we
obtain

|u(x)− u(y)|p−2(u(x)− u(y))
[u(x)p − v(x)p

u(x)p−1
− u(y)p − v(y)p

u(y)p−1

]
≥ |u(x)− u(y)|p − |v(x)− v(y)|p.

(1.8)



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 7

By exchanging the roles of u and v, we obtain

|v(x)− v(y)|p−2(v(x)− v(y))
[v(x)p − u(x)p

v(x)p−1
− v(y)p − u(y)p

v(y)p−1

]
≥ |v(x)− v(y)|p − |u(x)− u(y)|p.

(1.9)

Combining (1.8) and (1.9), we obtain

|u(x)− u(y)|p−2(u(x)− u(y))[
u(x)p − v(x)p

u(x)p−1
− u(y)p − v(y)p

u(y)p−1
]

+ |v(x)− v(y)|p−2(v(x)− v(y))
[v(x)p − u(x)p

v(x)p−1
− v(y)p − u(y)p

v(y)p−1

]
≥ 0

which concludes the proof of (1.6) for r = p.
We deal finally with the case 1 < r < p. By using Young’s inequality, (1.7)

implies

|u(x)− u(y)|p−2(u(x)− u(y))
[u(x)r − v(x)r

u(x)r−1
− u(y)r − v(y)r

u(y)r−1

]
≥ r

p
[|u(x)− u(y)|p − |v(x)− v(y)|p].

(1.10)

Reversing the role of u and v:

|v(x)− v(y)|p−2(v(x)− v(y))
[v(x)r − u(x)r

v(x)r−1
− v(y)r − u(y)r

v(y)r−1

]
≥ r

p
[|v(x)− v(y)|p − |u(x)− u(y)|p].

(1.11)

Adding the above inequalities, we obtain (1.6).
Now, let us consider u, v ∈ W s,p

0 (Ω), such that u > 0, v > 0 a.e. in Ω and

θ ∈ (0, 1). Setting w := (1 − θ)ur + θvr, one can easily check that w ∈ V̇ r+. Thus,
by Proposition 1.7, it is easy to prove that the function, defined in [0, 1],

θ 7→ Φ(θ) :=W(w) =W((1− θ)ur + θvr)

is convex, differentiable and for θ ∈ (0, 1):

Φ′(θ) =

∫
R2N\(Ωc×Ωc)

|w(x)1/r − w(y)1/r|p−2(w(x)1/r − w(y)1/r)

|x− y|N+sp

×
(v(x)r − u(x)r

w(x)1− 1
r

− v(y)r − u(y)r

w(y)1− 1
r

)
dx dy.

Finally, let us assume that the equality in (1.6) holds. By the monotonicity of
Φ′ : (0, 1)→ R, we deduce that Φ′(θ) = const in (0, 1). It follows that Φ : [0, 1]→ R
must be linear, i.e.

Φ(θ) =W(w) = (1− θ)Φ(0) + θΦ(1) = (1− θ)W(ur) + θW(vr),

for all θ ∈ [0, 1]. We conclude that u ≡ const.v with const > 0 and if p 6= r, then
u ≡ v, thanks to Proposition 1.7. �



8 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

1.3. Main results. We consider the associated problem of (1.1),

vq−1∂t(v
q) + (−∆)spv = h(t, x)vq−1 + f(x, v) in QT ,

v > 0 in QT ,

v = 0 on ΓT ,

v(0, ·) = v0 in Ω.

(1.12)

Claim 1.9. Any bounded weak solution of the above problem is also a weak solution
to problem (1.1).

To this aim, we introduce the notion of the weak solution to problem (1.12) as
follows.

Definition 1.10. Let T > 0. A weak solution to problem (1.12) is any nonnegative
function v ∈ L∞(0, T ;W s,p

0 (Ω)) ∩ L∞(QT ) such that v > 0 in Ω, ∂t(v
q) ∈ L2(QT )

and satisfying for any t ∈ (0, T ]:∫ t

0

∫
Ω

∂t(v
q)vq−1ϕdx ds

+

∫ t

0

∫
RN

∫
RN

|v(s, x)− v(s, y)|p−2(v(s, x)− v(s, y))(ϕ(s, x)− ϕ(s, y))

|x− y|N+sp
dx dy ds

=

∫ t

0

∫
Ω

(h(s, x)vq−1 + f(x, v))ϕdx ds,

for any ϕ ∈ L2(QT ) ∩ L1(0, T ;W s,p
0 (Ω)), with v(0, .) = v0 a.e. in Ω.

Remark 1.11. According to Definition 1.10, a weak solution of (1.12) belongs to
L∞(QT ). Then, we obtain

q

2q − 1
∂t(v

2q−1) = vq−1∂t(v
q)

weakly, and we deduce that a weak solution to (1.12) is a weak solution to (1.1).

Our main result about existence and properties of solutions to (1.12) is as follows.

Theorem 1.12. Let T > 0 and q ∈ (1, p]. Assume that f satisfies (H1)–(H3), (H6)
and

(H7) The map x 7→ φ1−q
1,s,p(x)f(x, φ1,s,p(x)) belongs to L2(Ω).

Assume in addition that h ∈ L∞(QT ) satisfies (H4), (H5) and that v0 ∈M1
ds(Ω)∩

W s,p
0 (Ω). Then there exists a unique weak solution v to (1.12). Furthermore,

(i) v ∈ C([0, T ];W s,p
0 (Ω)) and satisfies for any t ∈ [0, T ] the energy estimate∫ t

0

∫
Ω

(
∂vq

∂t
)2 dx ds+

q

p
‖v(t)‖p

W s,p
0 (Ω)

=

∫ t

0

∫
Ω

h
(∂vq
∂t

)
dx ds+

∫ t

0

∫
Ω

f(x, v)

vq−1

∂vq

∂t
dx ds+

q

p
‖v0‖pW s,p

0 (Ω)
.

(ii) If w is a weak solution to (1.12) associated to the initial data w0 ∈M1
ds(Ω)∩

W s,p
0 (Ω) and the right hand side g ∈ L∞(QT ) satisfying (H4) and (H5),

then the following estimate (T -accretivity in L2(Ω)) holds:

‖(vq(t)−wq(t))+‖L2(Ω) ≤ ‖(vq0 −w
q
0)+‖L2(Ω) +

∫ t

0

‖(h(s)− g(s))+‖L2(Ω)ds (1.13)

for any t ∈ [0, T ].



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 9

The T -acretivity in L2 stated in (1.13) was proved for p-Laplace operators in [22]
with a different approach (by the study of properties of the associated subdiffer-
ential via the potential theory) and for quasilinear elliptic operators with variable
exponents in [2] (see also [6] and [3] for related issues). The uniqueness of the
solution in Theorem 1.12 can be also obtained by the following theorem under less
restrictive assumptions about v0 and h.

Theorem 1.13. Let v, w be two solutions of the problem (1.12) in sense of Def-
inition 1.10, with respect to the initial data v0, w0 ∈ L2q(Ω), v0, w0 ≥ 0 and

h, h̃ ∈ L2(QT ). Then, for any t ∈ [0, T ],

‖vq(t)− wq(t)‖L2(Ω) ≤ ‖vq0 − w
q
0‖L2(Ω) +

∫ t

0

‖h(s)− h̃(s)‖L2(Ω)ds. (1.14)

Using the theory of maximal accretive operators, we introduce the nonlinear
operator Tq : L2(Ω) ⊃ D(Tq)→ L2(Ω) defined by

Tqu = u
1−q
q

(
2P.V.

∫
RN

|u1/q(x)− u1/q(y)|p−2(u1/q(x)− u1/q(y))

|x− y|N+sp
dy

− f(x, u1/q)
) (1.15)

with

D(Tq) =
{
w : Ω→ R+, w1/q ∈W s,p

0 (Ω), w ∈ L2(Ω), Tqw ∈ L2(Ω)
}
.

Using the T -accretive property of Tq in L2(Ω) proved below and under additional
assumptions on regularity of initial data, we obtain the following stabilization result
for the weak solutions to the problem (1.12).

Theorem 1.14. Assume that the hypothesis in Theorem 1.12 hold for any T > 0.
Let v be the weak solution of the problem (1.12) with the initial data v0 ∈M1

ds(Ω)∩
W s,p

0 (Ω). Assume in addition that there exists h∞ ∈ L∞(Ω) such that

l(t) ‖h(t, ·)− h∞‖L2(Ω) = O(1) as t→∞ (1.16)

with l continuous and positive on ]s0; +∞[ and
∫ +∞
s

dt
l(t) < +∞, for some s > s0 ≥

0. Then, for any r ≥ 1,

‖vq(t, ·)− vq∞‖Lr(Ω) → 0 as t→∞,

where v∞ is the unique stationary solution to (1.12) associated to the potential h∞.

This article is organized as follows: In Section 2, we study the stationary non-
linear problem

v2q−1 + λ(−∆)spv = h0(x)vq−1 + λf(x, v) in Ω,

v > in Ω,

v = 0 in RN \ Ω,

related to the parabolic problem (1.12) and establish the existence and the unique-
ness results in case h0 ∈ L∞(Ω) [Theorem 2.2, Corollary 2.4] and in case h0 ∈ L2(Ω)
[Theorem 2.5, Corollary 2.6]. Section 3 is devoted to prove Theorem 1.12. The proof
is divided into three main steps. First, by using a semi-discretization in time with



10 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

implicit Euler method, we prove the existence of a weak solution in sense of Def-
inition1.10 (see Theorem 3.1). Next, we prove the contraction property given in
Theorem 1.13 which implies the uniqueness of the weak solution stated in Corollary
3.2. The regularity of weak solutions is established in Theorem 3.4 that brings the
completion of the proof of Theorem 1.12. In Section 4, we show the stabilization re-
sult (see Theorem 1.14) for problem (1.12) via classical arguments of the semi-group
theory. Finally in the appendix 5.1, we establish some new regularity results (L∞

bound) for a class of quasilinear elliptic equations involving fractional p-Laplace
operator. Via the Picone identity, we also obtain a new weak comparison principle
that provides existence of barrier functions for stationary problems of (1.12).

2. p-fractional elliptic equation associated with problem (1.1)

The aim of this section is to study the elliptic problem corresponding to (1.12).
For this, we have several cases.

2.1. Potential h0 ∈ L∞(Ω). We consider the elliptic problem

v2q−1 + λ(−∆)spv = h0(x)vq−1 + λf(x, v) in Ω,

v > 0 in Ω

v = 0 in RN \ Ω,

(2.1)

where λ is a positive parameter and h0 ∈ (L∞(Ω))+ satisfying the hypothesis

(H8) h0(x) ≥ λh(x) for a.e. in Ω, where h is defined in (H4).

We have the following notion of weak solutions.

Definition 2.1. A weak solution of the problem (2.1) is any nonnegative and
nontrivial function v ∈W := W s,p

0 (Ω) ∩ L2q(Ω) such that for any ϕ ∈W,∫
Ω

v2q−1ϕdx+ λ

∫
RN

∫
RN

|v(x)− v(y)|p−2(v(x)− v(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy

=

∫
Ω

h0v
q−1ϕdx+ λ

∫
Ω

f(x, v)ϕdx.

(2.2)

We first investigate the existence and uniqueness of the weak solution to (2.1).

Theorem 2.2. Assume that f satisfies (H1), (H2), (H6). In addition suppose that
h0 ∈ L∞(Ω) and satisfies (H8). Then, for any 1 < q ≤ p and λ > 0, there exists a
positive weak solution v ∈ C(Ω) ∩M1

ds(Ω) to (2.1).
Moreover, let v1, v2 be two weak solutions to (2.1) with h1, h2 ∈ L∞(Ω) satisfy

(H8), respectively, we have (with the notation t+ = max{0, t}),

‖(vq1 − v
q
2)+‖L2 ≤ ‖(h1 − h2)+‖L2 . (2.3)

Proof. We divided the proof into 3 steps.

Step 1: Existence of a weak solution. Consider the energy functional J
corresponding to the problem (2.1), defined on W equipped with the Cartesian
norm ‖ · ‖W = ‖ · ‖W s,p

0 (Ω) + ‖ · ‖L2q(Ω) by

J (v) =
1

2q

∫
Ω

v2qdx+
λ

p

∫
RN

∫
RN

|v(x)− v(y)|p

|x− y|N+ps
dx dy

− 1

q

∫
Ω

h0(v+)qdx− λ
∫

Ω

F (x, v)dx

(2.4)



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 11

where

F (x, t) =

{∫ t
0
f(x, s)ds if 0 ≤ t < +∞,

0 if −∞ < t < 0.

We extend accordingly the domain of f to all of Ω× R by setting

f(x, t) =
∂F

∂t
(x, t) = 0 for (x, t) ∈ Ω× (−∞, 0).

From (H1) and (H2) there exists C > 0 large enough such that for any (x, s) ∈
Ω× R+,

0 ≤ f(x, s) ≤ C(1 + sq−1). (2.5)

Thus, we infer that:
• J is well defined and weakly lower semi-continuous on W.
• From (2.5), the Hölder inequality and Theorem 1.3, we obtain

J (v) ≥ 1

2q
‖v‖2qL2q(Ω) +

λ

p
‖v‖p

W s,p
0 (Ω)

− 1

q
‖h0‖L2(Ω)‖v‖qL2q(Ω) − Cλ

∫
Ω

|v|dx

− λC
q

∫
Ω

|v|qdx

≥ ‖v‖qL2q(Ω)

(
c1‖v‖qL2q(Ω) − c2

)
+ ‖v‖W s,p

0 (Ω)

(
c3‖v‖p−1

W s,p
0 (Ω)

− c4
)
,

where the constants c1, c2, c3 and c4 do not depend on v. Therefore, we obtain that
J (v) is coercive on W. Therefore, J admits a global minimizer on W, denoted by
v0. Thus, adopting the notation t = t+ − t−, we have

J (v0) = J (v+
0 ) +

1

2q

∫
Ω

(v−)2qdx+
λ

p

∫
RN

∫
RN

|(v−)(x)− (v−)(y)|p

|x− y|N+ps
dx dy

+
2λ

p

∫
RN

∫
RN

|(v−)(x)− (v+)(y)|p

|x− y|N+ps
dx dy ≥ J (v+

0 ).

Therefore, v0 ≥ 0. In order to show that v0 6≡ 0 in Ω, we find a suitable function
v in W such that J (v) < 0 = J (0). For that, we start by dealing with the case
q < p. Let φ ∈ C1

c (Ω) be nonnegative and non trivial with supp(φ) ⊂ supp(h).
Then, for any t > 0,

J (tφ) ≤ c1 t2q + c2 t
p − c3tq,

where the constants c1, c2 and c3 are independent of t and c3 > 0 thanks to h0 ≥
λh 6≡ 0. Hence for t > 0 small enough, J (tφ) < 0. We now consider the remaining
case q = p. Assumption (H6) implies that for c > 0 small enough there exists
s0 = s0(c) > 0 such that

λh(x) sp−1 + λf(x, s) > λ (λ1,p,s + c) sp−1

for all s ≤ s0 and uniformly in x ∈ Ω. Hence, for ε small enough, we deduce that

J (εφ1,p,s) <
1

2p
‖φ1,p,s‖2pL2p(Ω)ε

2p +
λ

p
‖φ1,p,s‖pW s,p

0 (Ω)
εp

− λ

p
(λ1,p,s + c)‖φ1,p,s‖pLp(Ω)ε

p

= εp
( 1

2p
‖φ1,p,s‖2pL2p(Ω)ε

p − c λ

p
‖φ1,p,s‖pLp(Ω)

)
< 0.

Since J (0) = 0, we deduce v0 6≡ 0. From the Gâteaux differentiability of J , we
obtain that v0 satisfies (2.2).



12 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

Step 2: Regularity and positivity of weak solutions. We first claim that all
weak solutions to (2.1) belongs to L∞(Ω). To this aim, we adapt arguments from
[[23], Theorem 3.2]. Precisely, let v0 be a weak solution. Then, it is enough to
prove that

‖v0‖L∞(Ω) ≤ 1 if ‖v0‖Lp(Ω) ≤ δ for some δ > 0 small enough. (2.6)

For this purpose, we consider the function wk defined as follows

wk(x) := (v0(x)− (1− 2−k))+ for k ≥ 1.

We first state the following straightforward observations about wk(x),

wk ∈W s,p
0 (Ω) and wk = 0 a.e. in ∂Ω,

and
wk+1(x) ≤ wk(x) a.e. in RN ,

v0(x) < (2k+1 + 1)wk(x) for x ∈ {wk+1 > 0}.
(2.7)

Also the inclusion

{wk+1 > 0} ⊆ {wk > 2−(k+1)} (2.8)

holds for all k ∈ N.
Setting Vk := ‖wk‖pLp(Ω), using (2.5), (2.7) and the inequality

x+ − y+|p ≤ |x− y|p−2(x+ − y+)(x− y)

for any x, y ∈ R, we obtain

λ‖wk+1‖pW s,p
0 (Ω)

= λ

∫
RN

∫
RN

|wk+1(x)− wk+1(y)|p

|x− y|N+sp
dx dy

≤ λ
∫
RN

∫
RN

|v0(x)− v0(y)|p−2(wk+1(x)− wk+1(y))(v0(x)− v0(y))

|x− y|N+sp
dx dy

≤
∫

Ω

(h0(x)vq−1
0 + λf(x, v0))wk+1dx

≤ C1[

∫
{wk+1>0}

wk+1dx+

∫
{wk+1>0}

vq−1
0 wk+1dx]

≤ C1[|{wk+1 > 0}|1−
1
pV

1/p
k + (2k+1 + 1)q−1|{wk+1 > 0}|1−

q
pV

q
p

k ]

where C1 > 0 is a constant. Now, from (2.8) we have

Vk =

∫
Ω

wpkdx ≥
∫
{wk+1>0}

wpkdx ≥ 2−(k+1)p|{wk+1 > 0}|. (2.9)

Therefore,

‖wk+1‖pW s,p
0 (Ω)

≤ C2(2k+1 + 1)p−1Vk

where C2 > 0 is a constant. On the other hand, by the Hölder’s inequality, fractional
Sobolev imbeddings (Theorem 1.3) and (2.9), we obtain

Vk+1 =

∫
{wk+1>0}

wpk+1dx ≤ C3‖wk+1‖pW s,p
0 (Ω)

(
2(k+1)pVk

) sp
N ,

where C3 > 0 is a constant. Hence, the above inequality

Vk+1 ≤ CkV 1+α
k , for all k ∈ N



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 13

holds for a suitable constant C > 1 and α = sp
N . This implies that

lim
k→∞

Vk = 0 (2.10)

provided that

‖v0‖pLp(Ω) = V0 ≤ C−
1
α2 =: δp

as it can be easily checked. Since wk converges to (v0− 1)+ a.e. in RN , from (2.10)
we infer that (2.6) holds as desired. Then, we deduce that v0 ∈ L∞(Ω) and [27,
Theorem 1.1] provides the C0,α(Ω)-regularity of v0, for some α ∈ (0, s]. Now, we
show that v0 > 0 in Ω. We argue by contradiction: Suppose that there exists
x0 ∈ Ω, where v0(x0) = 0, then it follows that

0 > 2λ

∫
RN

|v0(x0)− v0(y)|p−2(v0(x0)− v0(y))

|x0 − y|N+sp
dy

= h0(x)v0(x0)q−1 + λf(x0, v0(x0))− v0(x0)2q−1 = 0

from which we obtain a contradiction. Thus v0 > 0 in Ω. Finally, starting with the
case q = p, the Hopf lemma (see [20, Theorem 1.5] implies that v0 ≥ k ds(x) for
some k > 0. Next, supposing q < p, we have that for ε > 0 small enough, εφ1,s,p is
a subsolution to problem (2.1). Indeed, for a constant ε > 0 small enough, we have

(εφ1,s,p)
2q−1 + λ(−∆)sp(εφ1,s,p) ≤ h0(x)(εφ1,s,p)

q−1 + λf(x, εφ1,s,p) in Ω.

From the comparison principle (Theorem 5.4), we obtain εφ1,s,p ≤ v0. Then, we
deduce that v0 ≥ kds(x) for some k > 0. Again by using [[27], Theorem 4.4], we
obtain that v0 ∈M1

ds(Ω).

Step 3: Contraction property (2.3) Let v1, v2 ∈M1
ds(Ω) be two weak solutions

of (2.1) associated to h1 and h2 respectively. Namely, for any Φ,Ψ ∈W we have∫
Ω

v2q−1
1 Φdx+ λ

∫
RN

∫
RN

|v1(x)− v1(y)|p−2(v1(x)− v1(y))(Φ(x)− Φ(y))

|x− y|N+sp
dx dy

=

∫
Ω

h1v
q−1
1 Φdx+ λ

∫
Ω

f(x, v1)Φdx

and∫
Ω

v2q−1
2 Ψ dx+ λ

∫
RN

∫
RN

|v2(x)− v2(y)|p−2(v2(x)− v2(y))(Ψ(x)−Ψ(y))

|x− y|N+sp
dx dy

=

∫
Ω

h2v
q−1
2 Ψ dx+ λ

∫
Ω

f(x, v2)Ψ dx.

Since v1, v2 ∈M1
ds(Ω) ∩W 1,s

0 (Ω), we obtain that

Φ =
(vq1 − v

q
2)+

vq−1
1

, Ψ =
(vq2 − v

q
1)−

vq−1
2

are well-defined and belong to W.
Subtracting the two expressions above and using (H2) and Lemma 1.8, we obtain∫

Ω

((vq1 − v
q
2)+)2dx ≤

∫
Ω

(h1 − h2)(vq1 − v
q
2)+dx.

Finally, applying the Hölder inequality we obtain (2.3). �

Remark 2.3. Inequality (2.3) implies the uniqueness of the weak solution to the
problem (2.1) in the sense of Definition 2.2 in M1

ds(Ω).



14 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

From Theorem 2.2, we deduce the T -accretivity of Tq (see (1.15)) as follows.

Corollary 2.4. Let λ > 0, q ∈ (1, p], f : Ω×R+ → R+ satisfies (H1), (H2), (H6).
Assume in addition that h0 ∈ L∞(Ω) satisfies (H8). Then, there exists a unique
solution u ∈ C(Ω) of the problem

u+ λTqu = h0 in Ω,

u > 0 in Ω,

u ≡ 0 in RN \ Ω.

(2.11)

Namely, u belongs to V̇ q+ ∩M
1/q
ds (Ω), and satisfies∫

Ω

uΨ dx+ λ

∫
RN

∫
RN
|u1/q(x)− u1/q(y)|p−2

(
u1/q(x)− u1/q(y)

)
×
(
(u

1−q
q Ψ)(x)− (u

1−q
q Ψ)(y)

)/
|x− y|N+sp dx dy

=

∫
Ω

h0Ψ dx+ λ

∫
Ω

f(x, u1/q)u
1−q
q Ψ dx

(2.12)

for any Ψ such that

|Ψ|1/q ∈ L∞ds(Ω) ∩W s,p
0 (Ω). (2.13)

Moreover, if u1 and u2 are two solutions of (2.11), corresponding to h1 and h2

respectively, then

‖(u1 − u2)+‖L2 ≤ ‖
(
u1 − u2 + λ(Tq(u1)− Tq(u2))

)+‖L2 . (2.14)

Proof. We define the energy functional ξ on V̇ q+ ∩ L2(Ω) as ξ(u) = J (u1/q), where
J is defined in (2.4). Let v0 be the weak solution of (2.1) and the global minimizer
of (2.4). We set u0 = vq0. Then

u0 ∈ V̇ q+ ∩M
1/q
ds (Ω).

Let Ψ ≥ 0 satisfy (2.13), then there exists t0 = t0(Ψ) > 0 such that for t ∈
(0, t0), u0 + tΨ > 0. Hence, we have

0 ≤ ξ(u0 + tΨ)− ξ(u0)

=
1

2q

(∫
Ω

(tΨ)2dx+ 2t

∫
Ω

u0Ψ dx
)
− 1

q

∫
Ω

th0Ψ dx

+
λ

p

(∫
RN

∫
RN

|(u0 + tΨ)1/q(x)− (u0 + tΨ)1/q(y)|p

|x− y|N+ps
dx dy

−
∫
RN

∫
RN

|(u0)1/q(x)− (u0)1/q(y)|p

|x− y|N+ps
dx dy

)
− λ
(∫

Ω

F (x, (u0 + tΨ)1/q)dx−
∫

Ω

F (x, (u0)1/q)dx
)
.

Then dividing by t and passing to the limit t→ 0, we obtain that u0 satisfies (2.12).

On the other hand, consider u1 ∈ V̇ q+ ∩M
1/q
ds (Ω) a solution satisfying (2.12). Thus

v1 = u
1/q
1 satisfies (2.2), by Remark 2.3, we deduce v1 = v2. Finally, (2.14) follows

from (2.3). �



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 15

2.2. Potential h0 ∈ L2(Ω). In this subsection, we extend the existence results
above.

Theorem 2.5. Assume that f satisfies (H1), (H2), (H6). Then, for any 1 < q ≤
p, λ > 0 and h0 ∈ L2(Ω) satisfies (H8), there exists a positive weak solution v ∈W
to (2.1). Moreover assuming that h0 belongs to Lr(Ω) for some r > N

sp , v ∈ L∞(Ω).

Moreover, let v1, v2 be two weak solutions to (2.1) associated with h1, h2 ∈ L2(Ω),
respectively, satisfy (H8). Then, we have

‖(vq1 − v
q
2)+‖L2 ≤ ‖(h1 − h2)+‖L2 . (2.15)

Proof. Let h̃n ∈ C1
c (Ω), h̃n ≥ 0 with h̃n → h0 in L2(Ω), we take hn = max(h̃n, λh).

By Theorem 2.2, for any n ≥ n0, define vn ∈ C0,α(Ω) ∩ M1
ds(Ω) as the unique

positive weak solution of (2.1). Then, for any ϕ ∈W,∫
Ω

v2q−1
n ϕdx

+ λ

∫
RN

∫
RN

|vn(x)− vn(y)|p−2(vn(x)− vn(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy

=

∫
Ω

hnv
q−1
n ϕdx+ λ

∫
Ω

f(x, vn)ϕdx.

(2.16)

One has

(a− b)2r ≤ (ar − br)2 for any r ≥ 1, a, b ≥ 0 (2.17)

from which together with (2.3) it follows for any n,m ∈ N∗,

‖(vn − vm)+‖L2q ≤ ‖(vqn − vqm)+‖1/qL2 ≤ ‖(hn − hm)+‖1/qL2 .

Thus we deduce that (vn) converges to some v ∈ L2q(Ω). We infer that the limit

v does not depend on the choice of the sequence (hn). Indeed, consider h̃n 6= hn
such that h̃n → h0 in L2(Ω) and ṽn the positive solution to (2.1) corresponding to

h̃n which converges to ṽ. Then, for any n ∈ N, (2.3) implies

‖(vqn − ṽqn)+‖L2 ≤ ‖(hn − h̃n)+‖L2

and passing to the limit we obtain ṽ ≥ v and then by reversing the role of v and ṽ,
we obtain ṽ = v.

For n ∈ N∗, let hn = min{h0, nλh}. So, it is easy to check by (2.3), (vn)n∈N is
nondecreasing and for any n ∈ N∗, vn ≤ v a.e. in Ω which implies

v(x) ≥ v1(x) ≥ c ds(x) > 0 in Ω (2.18)

for some c independent of n. We choose ϕ = vn in (2.16), by the Hölder inequality
and (2.5), we obtain∫

RN

∫
RN

|vn(x)− vn(y)|p

|x− y|N+sp
dx dy ≤ C[‖vn‖qL2q(Ω)(‖hn‖L2(Ω) + 1) + ‖vn‖L2q(Ω)]

(2.19)
where C does not depend on n. Then, we deduce that (vn)n∈N is uniformly bounded
in W s,p

0 (Ω). Hence,{ |vn(x)− vn(y)|p−2(vn(x)− vn(y))

|x− y|
N+sp
p′

}
is bounded in Lp

′
(RN × RN )



16 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

where p′ = p
p−1 and by the pointwise convergence of vn to v, we obtain

|vn(x)− vn(y)|p−2(vn(x)− vn(y))

|x− y|
N+sp
p′

→ |v(x)− v(y)|p−2(v(x)− v(y))

|x− y|
N+sp
p′

a.e. in RN × RN . It follows that

|vn(x)− vn(y)|p−2(vn(x)− vn(y))

|x− y|
N+sp
p′

⇀
|v(x)− v(y)|p−2(v(x)− v(y))

|x− y|
N+sp
p′

weakly in Lp
′
(RN × RN ). Then, since ϕ ∈W = W s,p

0 (Ω) ∩ L2q(Ω), we obtain

lim
n→∞

∫
RN

∫
RN

|vn(x)− vn(y)|p−2(vn(x)− vn(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy

=

∫
RN

∫
RN

|v(x)− v(y)|p−2(v(x)− v(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy.

With similar arguments, by the Hölder inequality, (v2q−1
n )n∈N and (hnv

q−1
n )n∈N are

uniformly bounded in L
2q

2q−1 (Ω). By (2.5), we infer that f(x, vn) are uniformly

bounded in L
2q
q−1 (Ω) and f(x, vn) → f(x, v) a.e. in Ω. Since ϕ ∈W = W s,p

0 (Ω) ∩
L2q(Ω), we obtain

lim
n→∞

∫
Ω

v2q−1
n ϕdx =

∫
Ω

v2q−1ϕdx, lim
n→∞

∫
Ω

hnv
q−1
n ϕdx =

∫
Ω

hvq−1ϕdx,

lim
n→∞

∫
Ω

f(x, vn)ϕdx =

∫
Ω

f(x, v)ϕdx.

By passing to the limit in (2.16), v is a weak solution to (2.1). Finally, the fact
that v ∈ L∞(Ω) follows from Corollary 5.3. �

From Theorem 5.4, we obtain the following result.

Corollary 2.6. Let λ > 0, q ∈ (1, p], f : Ω × R+ → R+ satisfy (H1), (H2), (H6).
In addition suppose that h0 ∈ L2(Ω) ∩ Lr(Ω), for some r > N

sp and satisfies (H8).

Then, there exists a unique solution u of problem (2.11). Namely, u belongs to

V̇ q+∩L∞(Ω), satisfies (2.12) for any Ψ satisfying (2.13) and there exists c > 0 such
that u(x) ≥ cdsq(x) a.e. in Ω.

Moreover, if u1 and u2 are two solutions to the problem (2.11) associated with
h1, h2 ∈ L2(Ω) satisfy (H8), then

‖(u1 − u2)+‖L2 ≤ ‖(u1 − u2 + λ(Tq(u1)− Tq(u2)))+‖L2 . (2.20)

Proof. The existence of a solution v in Theorem 2.5 can be obtained by a global
minimization argument as in Step 1 of the proof of Theorem 2.2. Therefore, we
deduce from Theorem 5.4 that v is a global minimizer of J defined in (2.4).

As in the proof of Corollary 2.4, we can define the energy functional ξ on V̇ q+ ∩
L2(Ω) as ξ(u) = J (u1/q). We set u0 = vq0. Then, u0 belongs to V̇ q+ ∩ L∞(Ω). By
(2.18) we obtain u0(x) ≥ cdsq(x) a.e. in Ω. Let Ψ satisfy (2.13), then for t small
enough, ξ(u0 + tΨ)− ξ(u0) ≥ 0. By using the Taylor expansion, we deduce that u0

satisfies (2.12). Finally, (2.15) gives (2.20). �



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 17

3. Existence of a weak solution to parabolic problem (1.1)

In light of Remark 1.11, we consider problem (1.12) and establish the existence
of weak solution when v0 ∈M1

ds(Ω) ∩W s,p
0 (Ω). In this section, we prove Theorem

1.12. We begin the next subsection with some auxiliary results.

3.1. Existence and regularity of a weak solution. We divided the subsection
into three main parts concerning: existence, uniqueness, and regularity of solutions.

Existence of a weak solution.

Theorem 3.1. Under the assumptions of Theorem 1.12, there exists a weak solu-
tion v to the problem (1.12) (in sense of Definition 1.10). Furthermore, v belongs
to C([0, T ];Lr(Ω)) for any 1 ≤ r < ∞ and there exists C > 0 such that, for any
t ∈ [0, T ] :

C−1ds(x) ≤ v(t, x) ≤ Cds(x) a.e. in Ω. (3.1)

Proof. We use the time semi-discretization method: Let n? ∈ N∗ and T > 0. We
set ∆t = T

n? and for n ∈ {1, . . . , n?}, we define tn = n∆t.
We perform the proof along four main steps.

Step 1: Approximation of h. For n ∈ {1, . . . , n?}, we define for (t, x) ∈
[tn−1, tn)× Ω,

h∆t
(t, x) = hn(x) :=

1

∆t

∫ tn

tn−1

h(s, x)ds.

The Jensen’s inequality implies that

‖h∆t
‖L2(QT ) ≤ ‖h‖L2(QT ).

Hence h∆t
∈ L2(QT ), hn ∈ L2(Ω). It is easy to prove by density arguments that

h∆t
→ h in L2(QT ).

On the other hand, we obtain

‖h∆t
‖L∞(QT ) ≤ ‖h‖L∞(QT ).

Step 2: Time discretization of problem (1.12). We define the following im-
plicit Euler scheme: v0 = v0 and for n ≥ 1, vn is the weak solution of(vqn − vqn−1

∆t

)
vq−1
n + (−∆)spvn = hnvq−1

n + f(x, vn) in Ω,

vn > 0 in Ω,

vn = 0 in RN \ Ω.

(3.2)

The sequence (vn)n=1,2,...,n? is well-defined. Indeed, existence and uniqueness of

v1 ∈ C(Ω) ∩M1
ds(Ω) follow from Theorem 2.2 with h0 = ∆th

1 + vq0 ∈ L∞(Ω) and
∆th

1 +vq0 ≥ ∆th. Hence by induction we obtain in the same way the existence and

the uniqueness of the solution vn for any n = 2, 3, . . . , n? where vn ∈ C(Ω)∩M1
ds(Ω).

Step 3: Existence of subsolutions and supersolutions. In this step, we
establish the existence of a sub-solution w and a super-solution w such that vn ∈
[w,w] for all n ∈ {0, 1, 2, . . . , n?}. First, we rewrite (3.2) as

v2q−1
n + ∆t(−∆)spvn = (∆th

n + vqn−1)vq−1
n + ∆tf(x, vn). (3.3)



18 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

As in Theorem 2.2, we prove that for any µ ∈ (0, 1], the problem below admits a
unique weak solution wµ ∈ C(Ω) ∩M1

ds(Ω),

(−∆)spw = µ(hwq−1 + f(x,w)) in Ω,

w ≥ 0 in Ω,

w = 0 in RN \ Ω,

(3.4)

where h is defined in (H4).
Let µ1 < µ2 ≤ 1 and wµ1

, wµ2
∈ C(Ω)∩M1

ds(Ω) be two weak solutions of (3.4).
Then ∫

RN

∫
RN

|wµ1
(x)− wµ1

(y)|p−2(wµ1
(x)− wµ1

(y))(Φ(x)− Φ(y))

|x− y|N+sp
dx dy

= µ1

∫
Ω

(hwq−1
µ1

+ f(x,wµ1
))Φdx

and ∫
RN

∫
RN

|wµ2
(x)− wµ2

(y)|p−2(wµ2
(x)− wµ2

(y))(Ψ(x)−Ψ(y))

|x− y|N+sp
dx dy

= µ2

∫
Ω

(hwq−1
µ2

+ f(x,wµ2
))Ψ dx.

Subtracting the above expressions and taking

Φ =
(wqµ1

− wqµ2
)+

wq−1
µ1

, Ψ =
(wqµ2

− wqµ1
)−

wq−1
µ2

,

we deduce that (wµ)µ is nondecreasing. From [27, Corollary 4.2 and Theorem 1.1],
we obtain for some µ0 > 0 and 0 < α ≤ s that

‖wµ‖C0,α(Ω) ≤ C(µ0) for any µ ≤ µ0 and ‖wµ‖L∞(Ω) → 0 as µ→ 0.

Furthermore, by using [27, Theorem 4.4], we can choose µ < 1 small enough such
that there exists w ∈ C(Ω) ∩M1

ds(Ω) satisfies 0 < w := w µ ≤ v0. We infer that w
is the subsolution of the problem (3.3) for n = 1, i.e.∫

Ω

w2q−1ϕdx+ ∆t

∫
RN

∫
RN

|w(x)− w(y)|p−2(w(x)− w(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy

≤ ∆t

∫
Ω

(h1wq−1 + f(x,w))ϕdx+

∫
Ω

vq0w
q−1ϕdx

for all ϕ ∈W and ϕ ≥ 0. We also recall that v1 satisfies∫
Ω

v2q−1
1 ψ dx+ ∆t

∫
RN

∫
RN

|v1(x)− v1(y)|p−2(v1(x)− v1(y))(ψ(x)− ψ(y))

|x− y|N+sp
dx dy

= ∆t

∫
Ω

(h1vq−1
1 + f(x, v1))ψ dx+

∫
Ω

vq0v
q−1
1 ψ dx

for all ψ ∈W. By Theorem 5.4, we obtain w ≤ v1 and then by induction 0 < w ≤
vn in Ω for n = 0, 1, 2, . . . , n?.

Next, we construct a uniform supersolution. We start with the case q < p for
which we consider the problem

(−∆)spw = 1 in Ω,

w = 0 in RN \ Ω.
(3.5)



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 19

As above, we can prove that there exists a unique weak solution w ∈ C(Ω)∩M1
ds(Ω)

to (3.5). We easily check that for some K > 0 fixed, wK = K
1
p−1w is the unique

weak solution of the problem

(−∆)spwK = K in Ω,

wK = 0 in RN \ Ω

and

c−1 d(x)sK
1
p−1 ≤ wK(x) ≤ c d(x)sK

1
p−1 , (3.6)

where c > 0 is a constant. Again by using [27, Theorem 4.4], we obtain w =
wK ≥ v0 for K large enough. By (2.5) and (3.6), it is easy to prove that w is the
supersolution of the problem

(−∆)spw = ‖h‖L∞(Ω)w
q−1 + f(x,w) in Ω,

w > 0 in Ω,

w = 0 in RN \ Ω.

(3.7)

We now study the case q = p. Using (H3), we can choose for any ε > 0,
r0 = r0(ε) > 0 large enough, such that for r ≥ r0,

f(x, r) ≤ εrp−1. (3.8)

Let w be the solution of the problem

(−∆)spw = C + βwp−1 in Ω,

w > 0 in Ω,

w = 0 in RN \ Ω

with C > 0 and β < λ1,p,s. Then, by a similar proof as in Theorem 2.2 step 2,
we obtain w ∈ L∞(Ω). On the other hand, by [20, Theorems 1.4 and 1.5, p. 768],
we obtain that w > 0 in Ω and satisfies w ≥ kds(x), for some k = k(C, β) > 0.
Finally, using [27, Theorem 4.4], we obtain that w ∈ M1

ds(Ω). By (3.8), (H5) and
for C > 0 large enough and β close enough to λ1,p,s, we obtain

(−∆)sp(w) = C + βwp−1 ≥ ‖h‖L∞(Ω)w
p−1 + f(x,w).

Hence, w = w is supersolution of (3.7). Again using [27, Theorem 4.4]] and taking
C > 0 large enough, we obtain v0 ≤ w.

Then, since w ≥ v0, w is the supersolution to (3.3) for n = 1, i.e.∫
Ω

w2q−1ϕdx+ ∆t

∫
Rn

∫
Rn

|w(x)− w(y)|p−2(w(x)− w(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy

≥ ∆t

∫
Ω

(h1wq−1 + f(x,w))ϕdx+

∫
Ω

vq0w
q−1ϕdx

for all ϕ ∈ W and ϕ ≥ 0. From Theorem 5.4, we obtain w ≥ v1 and then by
induction we have w ≥ vn for all n = 1, 2, 3, . . . , n?. Finally, we conclude that
w ≤ vn ≤ w for n = 0, 1, 2, 3, . . . , n?, i.e. c1d

s(x) ≤ vn(x) ≤ c2d
s(x) in Ω, where

c1, c2 are positive constants independent of n.

Step 3:A priori estimates. For n ∈ {1, 2, 3, . . . , n?} and t ∈ [tn−1, tn) let the
functions v∆t(t) and ṽ∆t(t) be as follows:

v∆t
(t) = vn,



20 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

ṽ∆t(t) =
(t− tn−1)

∆t
(vqn − v

q
n−1) + vqn−1.

One can easily check that

vq−1
∆t

∂ṽ∆t

∂t
+ (−∆)spv∆t = hnvq−1

∆t
+ f(x, v∆t). (3.9)

We observe now that as ∆t → 0, the discrete equation (3.9) converges to (1.12).
We further point out that there exists c > 0 independent of ∆t such that for any
(t, x) ∈ QT ,

c−1ds(x) ≤ v∆t
, ṽ

1/q
∆t
≤ cds(x). (3.10)

Now, multiplying (3.2) by
vqn−v

q
n−1

vq−1
n

∈ W s,p
0 (Ω) ∩ L∞(Ω) and summing from n = 1

to n′ ≤ n?, we obtain

n′∑
n=1

∫
Ω

∆t

(vqn − vqn−1

∆t
)2dx+

n′∑
n=1

∫
RN

∫
RN

|vn(x)− vn(
)
|p−2(vn(x)− vn(y))

|x− y|N+sp

×
[(vqn − vqn−1

vq−1
n

)
(x)−

(vqn − vqn−1

vq−1
n

)
(y)
]
dx dy

=

n′∑
n=1

∫
Ω

hn(vqn − v
q
n−1) dx+

n′∑
n=1

∫
Ω

f(x, vn)

vq−1
n

(vqn − v
q
n−1) dx.

Since vn ∈ [w,w] ⊂ M1
ds(Ω), we have that

( f(x,vn)

vq−1
n

(vqn − vqn−1)
)

is uniformly

bounded. By Young’s inequality, we have

n′∑
n=1

∫
Ω

∆t

(vqn − vqn−1

∆t

)2
dx+

n′∑
n=1

∫
RN

∫
RN

|vn(x)− vn(y)|p−2(vn(x)− vn(y))

|x− y|N+sp

×
[(vqn − vqn−1

vq−1
n

)
(x)−

(vqn − vqn−1

vq−1
n

)
(y)
]
dx dy

≤ 1

2

n′∑
n=1

∆t‖hn‖2L2(Ω) +
1

2

n′∑
n=1

∫
Ω

∆t

(vqn − vqn−1

∆t

)2

dx+ C,

i.e.

1

2

n′∑
n=1

∫
Ω

∆t(
vqn − v

q
n−1

∆t
)2dx+

n′∑
n=1

∫
RN

∫
RN

|vn(x)− vn(y)|p−2(vn(x)− vn(y))

|x− y|N+sp

×
[(vqn − vqn−1

vq−1
n

)
(x)−

(vqn − vqn−1

vq−1
n

)
(y)
]
dx dy

≤ 1

2

n′∑
n=1

∆t‖hn‖2L2(Ω) + C,

where C is independent of n′. Then by step 1, we obtain(∂ṽ∆t

∂t

)
is bounded in L2(QT ) uniformly in ∆t. (3.11)



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 21

Now, from Proposition 1.2 and by Young’s inequality in the case q < p, we have

|vn(x)− vn(y)|p−2(vn(x)− vn(y))
[vn−1(x)q

vn(x)q−1
− vn−1(y)q

vn(y)q−1

]
≤ |vn−1(x)− vn−1(y)|q|vn(x)− vn(y)|p−q

≤ q

p
|vn−1(x)− vn−1(y)|p +

p− q
p
|vn(x)− vn(y)|p.

(3.12)

Next, for p = q we obtain

|vn(x)− vn(y)|p−2(vn(x)− vn(x))
[vn−1(x)p

vn(x)p−1
− vn−1(y)p

vn(y)p−1

]
≤ |vn−1(x)− vn−1(y)|p.

(3.13)

Then, for any n′ ≥ 1 and p 6= q we obtain

1

2

n′∑
n=1

∆t‖hn‖2L2(Ω) + C

≥
n′∑
n=1

[ ∫
RN

∫
RN

|vn(x)− vn(y)|p

|x− y|N+sp
dx dy − q

p

∫
RN

∫
RN

|vn−1(x)− vn−1(y)|p

|x− y|N+sp
dx dy

− p− q
p

∫
RN

∫
RN

|vn(x)− vn(y)|p

|x− y|N+sp
dx dy

]
.

For p = q, we have

1

2

n′∑
n=1

∆t‖hn‖2L2(Ω) + C ≥
n′∑
n=1

[ ∫
RN

∫
RN

|vn(x)− vn(y)|p

|x− y|N+sp
dx dy

−
∫
RN

∫
RN

|vn−1(x)− vn−1(y)|p

|x− y|N+sp
dx dy

]
.

Finally, we obtain

1

2

n′∑
n=1

∆t‖hn‖2L2(Ω) + C ≥ q

p

[ ∫
RN

∫
RN

|vn′(x)− vn′(y)|p

|x− y|N+sp
dx dy

−
∫
RN

∫
RN

|v0(x)− v0(y)|p

|x− y|N+sp
dx dy

]
which implies that

(v∆t) is bounded in L∞(0, T ;W s,p
0 (Ω)) uniformly in ∆t. (3.14)

Since ṽ∆t = ξvqn + (1− ξ)vqn−1, where ξ = t−tn−1

∆t , by Proposition 1.1, we obtain∫
RN

∫
RN

|ṽ1/q
∆t (x)− ṽ1/q

∆t (y)|p

|x− y|N+sp
dx dy

≤ ξ
∫
RN

∫
RN

|vn(x)− vn(y)|p

|x− y|N+sp
+ (1− ξ)

∫
RN

∫
RN

|vn−1(x)− vn−1(y)|p

|x− y|N+sp
.

Then, we conclude that

(ṽ
1/q
∆t ) is bounded in L∞(0, T ;W s,p

0 (Ω)) uniformly in ∆t. (3.15)



22 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

Thus, v∆t
?
⇀ v and ṽ

1/q
∆t

?
⇀ ṽ in L∞(0, T ;W s,p

0 (Ω)). Furthermore using (2.17),
(3.11),

sup
t∈[0,T ]

‖ṽ1/q
∆t − v∆t‖2qL2q(Ω) ≤ sup

t∈[0,T ]

‖ṽ∆t − vq∆t‖
2
L2(Ω) ≤ C∆t→ 0 as ∆t→ 0.

(3.16)
It follows that v = ṽ.

Now, from (3.11), (3.15) and since W s,p
0 (Ω) ↪→ Lr(Ω) compactly for all 1 ≤ r <

p∗s, using Theorem 1.4 we obtain that (ṽ∆t) is compact in C([0, T ];Lr(Ω)). Then
from (3.16),

ṽ∆t → vq in C([0, T ];Lr(Ω)), for 1 ≤ r < p∗s.

Using ṽ∆t ∈ L∞(Ω) with the interpolation inequality with p∗s ≤ r <∞,

‖ · ‖r ≤ ‖ · ‖α∞‖ · ‖1−αp∗s
, with α ∈ [0, 1],

we obtain that

ṽ∆t → vq in C([0, T ];Lr(Ω)), for all r ≥ 1. (3.17)

Hence, from the estimate

sup
t∈[0,T ]

‖vq∆t
− ṽ∆t

‖L2(Ω) ≤ C(∆t)
1/2, (3.18)

we have

v∆t → v in L∞([0, T ];Lr(Ω)), for all r ≥ 1. (3.19)

Hence, (3.10) implies (3.1). From (3.11) and (3.17), we obtain

∂ṽ∆t

∂t
⇀

∂vq

∂t
in L2(QT ). (3.20)

Step 4: v satisfies (1.12). • First, from (3.14), we have{ |v∆t
(t, x)− v∆t

(t, y)|p−2(v∆t
(t, x)− v∆t

(t, y))

|x− y|
N+sp
p′

}
is bounded in L∞(0, T ;Lp

′
(RN × RN )), where p′ = p

p−1 , and by the pointwise

convergence of v∆t to v, we obtain as ∆t → 0+ and for a.e. t ∈ [0, T ],

|v∆t(t, x)− v∆t(t, y)|p−2(v∆t(t, x)− v∆t(t, y))

|x− y|
N+sp
p′

→ |v(t, x)− v(t, y)|p−2(v(t, x)− v(t, y))

|x− y|
N+sp
p′

a.e. in RN × RN , it follows that as ∆t → 0+,

|v∆t(t, x)− v∆t(t, y)|p−2(v∆t(t, x)− v∆t(t, y))

|x− y|
N+sp
p′

⇀
|v(t, x)− v(t, y)|p−2(v(t, x)− v(t, y))

|x− y|
N+sp
p′



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 23

weakly in Lp
′
((0, T )× R2N ). Then, we conclude that for any φ ∈ C∞c (QT ),∫ T

0

∫
RN

∫
RN

(
|v∆t

(t, x)− v∆t
(t, y)|p−2

(
v∆t

(t, x)− v∆t
(t, y)

)
× (ϕ(t, x)− ϕ(t, y))

)
/|x− y|N+sp dx dy dt

→
∫ T

0

∫
RN

∫
RN

(
|v(t, x)− v(t, y)|p−2

(
v(t, x)− v(t, y)

)
× (ϕ(t, x)− ϕ(t, y))

)
/|x− y|N+sp dx dy dt.

(3.21)

• Next, from (2.17), (3.16) and (3.18) we have

‖vq−1
∆t
− vq−1‖L2(QT )

≤ T 1
2 ‖vq−1

∆t
− vq−1‖L∞(0,T ;L2(Ω))

≤ T 1
2 |Ω|

1
2q ‖vq−1

∆t
− vq−1‖

L∞(0,T ;L
2q
q−1 (Ω))

≤ T 1
2 |Ω|

1
2q ‖vq∆t

− vq‖
q−1
q

L∞(0,T ;L2(Ω))

≤ T 1
2 |Ω|

1
2q
[
‖vq∆t

− ṽ∆t
‖L∞(0,T ;L2(Ω)) + ‖ṽ∆t

− vq‖L∞(0,T ;L2(Ω))

] q−1
q → 0

(3.22)

as ∆t → 0. By the Hölder inequality, for all ϕ ∈ C∞c (QT ) we have

∣∣ ∫ T

0

∫
Ω

(vq−1
∆t

∂ṽ∆t

∂t
− vq−1 ∂v

q

∂t
)ϕ(t, x) dx dt

∣∣
≤
∣∣ ∫ T

0

∫
Ω

vq−1(
∂ṽ∆t

∂t
− ∂vq

∂t
)ϕ(t, x) dx dt

∣∣+ ‖vq−1
∆t
− vq−1‖L2(QT )‖

∂ṽ∆t

∂t
ϕ‖L2(QT )

and ∫ T

0

∫
Ω

(hnvq−1
∆t
− hvq−1)ϕdx dt

=

∫ T

0

∫
Ω

hn(vq−1
∆t
− vq−1)ϕdx dt+

∫ T

0

∫
Ω

(hn − h)vq−1ϕdx dt

≤ ‖hnϕ‖L2(QT )‖vq−1
∆t
− vq−1‖L2(QT ) + ‖vq−1ϕ‖L2(QT )‖hn − h‖L2(QT ).

Then from (3.11), (3.19), (3.20), (3.22) and Step 1, we obtain∫ T

0

∫
Ω

(vq−1
∆t

∂ṽ∆t

∂t
− vq−1 ∂v

q

∂t
)ϕ(t, x) dx dt→ 0, (3.23)∫ T

0

∫
Ω

(hnvq−1
∆t
− hvq−1)ϕ(t, x) dx dt→ 0 (3.24)

as ∆t → 0. From (3.19), we have f(x, v∆t)ϕ → f(x, v)ϕ a.e. in QT , (up to a
subsequence). Furthermore from (2.5) and (3.10), (f(x, v∆t)) is bounded in L2(QT )
uniformly in ∆t. Then, by the dominated convergence Theorem, we obtain∫ T

0

∫
Ω

f(x, v∆t)ϕdx dt→
∫ T

0

∫
Ω

f(x, v)ϕdx dt, as ∆t → 0. (3.25)



24 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

Finally, gathering (3.21), (3.23), (3.24),(3.25) and passing to the limit in (3.9) as
∆t → 0+, we conclude that v satisfies (1.12), i.e.∫ T

0

∫
Ω

∂t(v
q)vq−1ϕdx ds

+

∫ T

0

∫
RN

∫
RN

|v(s, x)− v(s, y)|p−2(v(s, x)− v(s, y))(ϕ(s, x)− ϕ(s, y))

|x− y|N+sp
dx dyds

=

∫ T

0

∫
Ω

(h(s, x)vq−1 + f(x, v))ϕdx ds

(3.26)
for any ϕ ∈ C∞c (QT ). Since C∞c (QT ) is dense in L2(QT )∩L1(0, T ;W s,p

0 (Ω)). Hence,
we conclude that (3.26) is satisfied for any ϕ ∈ L2(QT ) ∩ L1(0, T ;W s,p

0 (Ω)). �

Uniqueness.

Proof of Theorem 1.13. We again use the Picone identity. Let v and w be two weak
solutions to (1.12) with h and h̃ respectively. For ε ∈ (0; 1), we set

Φ :=
(v + ε)q − (w + ε)q

(v + ε)q−1
, Ψ :=

(w + ε)q − (v + ε)q

(w + ε)q−1
. (3.27)

Φ and Ψ belong to L2(QT ) ∩ L1(0, T ;W s,p
0 (Ω)) and for any t ∈ (0, T ],∫ t

0

∫
Ω

∂t(v
q)vq−1Φ dx ds

+

∫ t

0

∫
RN

∫
RN

|v(s, x)− v(s, y)|p−2(v(s, x)− v(s, y))(Φ(s, x)− Φ(s, y))

|x− y|N+sp
dx dy ds

=

∫ t

0

∫
Ω

(h(s, x)vq−1 + f(x, v))Φ dx ds

and∫ t

0

∫
Ω

∂t(w
q)wq−1Ψ dx ds

+

∫ t

0

∫
RN

∫
RN

|w(s, x)− w(s, y)|p−2(w(s, x)− w(s, y))(Ψ(s, x)−Ψ(s, y))

|x− y|N+sp
dx dy ds

=

∫ t

0

∫
Ω

(h̃(s, x)wq−1 + f(x,w))Ψ dx ds.

Summing the above equalities, we obtain Iε = Jε where

Iε =

∫ t

0

∫
Ω

(∂t(vq)vq−1

(v + ε)q−1
− ∂t(w

q)wq−1

(w + ε)q−1

)
((v + ε)q − (w + ε)q) dx ds

+

∫ t

0

∫
RN

∫
RN

|v(s, x)− v(s, y)|p−2(v(s, x)− v(s, y))

|x− y|N+sp

×
[ (v + ε)q(s, x)− (w + ε)q(s, x)

(v + ε)q−1(s, x)
− (v + ε)q(s, y)− (w + ε)q(s, y)

(v + ε)q−1(s, y)

]
dx dy ds

+

∫ t

0

∫
RN

∫
RN

|w(s, x)− w(s, y)|p−2(w(s, x)− w(s, y))

|x− y|N+sp

×
[ (w + ε)q(s, x)− (v + ε)q(s, x)

(w + ε)q−1(s, x)
− (w + ε)q(s, y)− (v + ε)q(s, y)

(w + ε)q−1(s, y)

]
dx dyds



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 25

and

Jε =

∫ t

0

∫
Ω

( hvq−1

(v + ε)q−1
− h̃wq−1

(w + ε)q−1

)
((v + ε)q − (w + ε)q) dx ds

+

∫ t

0

∫
Ω

( f(x, v)

(v + ε)q−1
− f(x,w)

(w + ε)q−1

)
((v + ε)q − (w + ε)q) dx ds.

First, we deal with Iε. Since v
v+ε ,

w
w+ε < 1 and v, w ∈ L∞(QT ),∣∣∂t(vq)vq−1

(v + ε)q−1
− ∂t(w

q)wq−1

(w + ε)q−1

∣∣|(v + ε)q − (w + ε)q| ≤ C (|∂t(vq)|+ |∂t(wq)|)

where C does not depend on ε. Moreover as ε→ 0+,(∂t(vq)vq−1

(v + ε)q−1
− ∂t(w

q)wq−1

(w + ε)q−1

)
((v + ε)q − (w + ε)q)→ 1

2
∂t(v

q − wq)2.

Therefore, by the dominated convergence Theorem and Lemma 1.8, we obtain

lim inf
ε→0

Iε ≥
1

2

∫ t

0

∫
Ω

∂t(v
q − wq)2 dx ds.

Next, dealing with Jε, dominated convergence Theorem implies

lim
ε→0

∫ t

0

∫
Ω

( hvq−1

(v + ε)q−1
− h̃wq−1

(w + ε)q−1

)
((v + ε)q − (w + ε)q) dx ds

=

∫ t

0

∫
Ω

(h− h̃)(vq − wq) dx ds.

Moreover, by using Fatou’s Lemma, we have

lim inf
ε→0

∫ t

0

∫
Ω

f(x, v)

(v + ε)q−1
(w + ε)q dx ds ≥

∫ t

0

∫
Ω

f(x, v)

vq−1
wq dx ds,

lim inf
ε→0

∫ t

0

∫
Ω

f(x,w)

(w + ε)q−1
(v + ε)q dx ds ≥

∫ t

0

∫
Ω

f(x,w)

wq−1
vq dx ds.

Hence gathering the three limits above and from (H2), we obtain

lim inf
ε→0

Jε ≤
∫ t

0

∫
Ω

(h− h̃)(vq − wq) dx ds.

Since Iε = Jε, using Hölder inequality we conclude that for any t ∈ [0, T ],

1

2

∫ t

0

∫
Ω

∂t(v
q − wq)2 dx ds ≤

∫ t

0

‖h− h̃‖L2(Ω)‖vq − wq‖L2(Ω)ds

and by Grönwall Lemma [10, Lemma A.5], we deduce (1.14). �

The uniqueness of the weak solution in sense of Definition 1.10 in Theorem 1.12
is a consequence of Theorem 1.13. Precisely, we have the following Corollary.

Corollary 3.2. Let v, w be weak solutions of (1.12) in sense of Definition 1.10
with the initial data v0 ∈ L2q(Ω), v0 ≥ 0 and h ∈ L2(QT ). Then, v ≡ w.

We use Theorem 3.1 and Corollary 3.2 to infer the existence result concerning
the parabolic problem involving the operator Tq.



26 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

Theorem 3.3. Under the assumptions of Theorem 1.12, for any the initial data

u0 such that u
1/q
0 ∈ M1

ds(Ω) ∩W s,p
0 (Ω), there exists a unique weak solution u ∈

L∞(QT ) of the problem

∂tu+ Tqu = h in QT ,

u > 0 in QT ,

u = 0 on ΓT ,

u(0, ·) = u0 in Ω.

(3.28)

In particular,

• u1/q ∈ L∞(0, T ;W s,p
0 (Ω)), ∂tu ∈ L2(QT );

• there exists c > 0 such that for any t ∈ [0, T ];

c−1ds(x) ≤ u1/q(t, x) ≤ cds(x) a.e. in Ω;

• for any t ∈ [0, T ], u satisfies∫ t

0

∫
Ω

∂tuΨ dx ds+

∫ t

0

∫
R2N

|u1/q(s, x)− u1/q(s, y)|p−2
(
u1/q(s, x)− u1/q(s, y)

)
×
(
(u

1−q
q Ψ)(s, x)− (u

1−q
q Ψ)(s, y)

)/
|x− y|N+sp dx dy ds

=

∫ t

0

∫
Ω

h(s, x)Ψ dx ds+

∫ t

0

∫
Ω

f(x, u1/q)u
1−q
q Ψ dx ds

for any Ψ ∈ L2(QT ) such that

|Ψ|1/q ∈ L1(0, T ;W s,p
0 (Ω)) ∩ L∞(0, T ;L∞ds(Ω)).

Moreover, for any 1 ≤ r <∞, u belongs to C([0, T ];Lr(Ω)).

The proof of the above theorem follows straightforward from Theorem 3.1 and
corollary 3.2.

Regularity of weak solutions.

Theorem 3.4. Under the assumptions of Theorem 1.12, the weak solution v, of
(1.12) obtained by Theorem 3.1, belongs to C(0, T ;W s,p

0 (Ω)) and for any t ∈ [0, T ]
satisfies ∫ t

0

∫
Ω

(
∂vq

∂t
)2 dx ds+

q

p
‖v(t)‖p

W s,p
0 (Ω)

=

∫ t

0

∫
Ω

h(
∂vq

∂t
) dx ds+

∫ t

0

∫
Ω

f(x, v)

vq−1

∂vq

∂t
dx ds+

q

p
‖v0‖pW s,p

0 (Ω)
.

Proof. Since v ∈ L∞(0, T ;W s,p
0 (Ω)) ∩ L∞(QT ) and ∂tv

q ∈ L2(QT ), by Theorem
1.4, we obtain that v belongs to C([0, T ];Lr(Ω)) for any r ≥ 1. From the Sobolev
embedding (Theorem 1.3), we have that W s,p

0 (Ω) is compactly embedded in Lp(Ω).
So we deduce that v : [0, T ] → W s,p

0 (Ω) is weakly continuous. Therefore, for any
t0 ∈ [0, T ],

‖v(t0)‖W s,p
0 (Ω) ≤ lim inf

t→t0
‖v(t)‖W s,p

0 (Ω). (3.29)



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 27

Multiplying (3.2) by
vqn−v

q
n−1

vq−1
n

∈ W, integrating over RN and summing from 1 ≤
n = N ′ to N ′′ ≤ n?, we obtain

N ′′∑
n=N ′

∫
Ω

∆t

(vqn − vqn−1

∆t
Big)2dx+

N ′′∑
n=N ′

∫
RN

∫
RN

|vn(x)− vn(y)|p−2(vn(x)− vn(y))

|x− y|N+sp

×
[(vqn − vqn−1

vq−1
n

)
(x)−

(vqn − vqn−1

vq−1
n

)
(y)
]
dx dy

=

N ′′∑
n=N ′

∆t

∫
Ω

hn
(vqn − vqn−1

∆t

)
dx+ ∆t

N ′′∑
n=N ′

∫
Ω

f(x, vn)

vq−1
n

(vqn − vqn−1

∆t

)
dx.

Now, from (3.12) and (3.13), we obtain

N ′′∑
n=N ′

∫
Ω

∆t(
vqn − v

q
n−1

∆t
)2dx+

q

p
(‖vN ′‖pW s,p

0 (Ω)
− ‖vN ′′−1‖pW s,p

0 (Ω)
)

≤
N ′∑

n=N ′′

∆t

∫
Ω

hn(
vqn − v

q
n−1

∆t
)dx+ ∆t

N ′′∑
n=N ′

∫
Ω

f(x, vn)

vq−1
n

(
vqn − v

q
n−1

∆t
)dx.

(3.30)

For any t ∈ [t0, T ], we choose N ′ and N ′′ such that N ′∆t → t and N ′′∆t → t0. By
(H7), then (3.30) gives∫ t

t0

∫
Ω

(
∂vq

∂t
)2 dx ds+

q

p
‖v(t)‖p

W s,p
0 (Ω)

≤
∫ t

t0

∫
Ω

h(
∂vq

∂t
) dx ds+

∫ t

t0

∫
Ω

f(x, v)

vq−1

∂vq

∂t
dx ds+

q

p
‖v(t0)‖p

W s,p
0 (Ω)

.

(3.31)

Taking lim sup in (3.31) as t→ t+0 and by (3.29), we obtain

‖v(t0)‖W s,p
0 (Ω) = lim

t→t+0
‖v(t)‖W s,p

0 (Ω)

and hence the right-continuity of v : [0, T ]→W s,p
0 (Ω) follows.

Now, for proving the left continuity, consider 0 < η ≤ t− t0, multiply (1.12) by

τηv =
vq(·+ η, ·)− vq(·, ·)

ηvq−1
∈ L2(QT ) ∩ L1(0, T ;W s,p

0 (Ω))

and integrate over (t0, t)×Ω. Using Proposition 1.2 and Young’s inequality again,
we obtain∫ t

t0

∫
Ω

vq−1∂t(v
q)τηv dx ds+

q

pη

∫ t

t0

(‖v(s+ η)‖p
W s,p

0 (Ω)
− ‖v(s)‖p

W s,p
0 (Ω)

)ds

≥
∫ t

t0

∫
Ω

hvq−1τηv dx ds+

∫ t

t0

∫
Ω

f(x, v)τηv dx ds.



28 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

It follows that∫ t

t0

∫
Ω

vq−1∂t(v
q)τηv dx ds

+
q

pη

(∫ t+η

t

‖v(s)‖p
W s,p

0 (Ω)
ds−

∫ t0+η

t0

‖v(s)‖p
W s,p

0 (Ω)
ds
)

≥
∫ t

t0

∫
Ω

hvq−1τηv dx ds+

∫ t

t0

∫
Ω

f(x, v)τηv dx ds.

(3.32)

By the right continuity of v : [0, T ] → W s,p
0 (Ω) and by dominated convergence

Theorem, as η → 0+ we have

q

pη

∫ t+η

t

‖v(s)‖p
W s,p

0 (Ω)
ds→ q

p
‖v(t)‖p

W s,p
0 (Ω)

,

q

pη

∫ t0+η

t0

‖v(s)‖p
W s,p

0 (Ω)
ds→ q

p
‖v(t0)‖p

W s,p
0 (Ω)

.

Hence as η → 0+, (3.32) yields∫ t

t0

∫
Ω

(
∂vq

∂t
)2 dx ds+

q

p
‖v(t)‖p

W s,p
0 (Ω)

≥
∫ t

t0

∫
Ω

h(
∂vq

∂t
) dx ds+

∫ t

t0

∫
Ω

f(x, v)

vq−1

∂vq

∂t
dx ds+

q

p
‖v(t0)‖p

W s,p
0 (Ω)

.

From the above inequality, we deduce that the equality in (3.31) holds and the
left-continuity of v : [0, T ]→W s,p

0 (Ω) follows. �

4. Stabilization

4.1. Existence and uniqueness of the solution of the stationary problem
related to (1.12). In this subsection we deal with the stationary problem in order
to determine the asymptotic behavior of trajectories to (1.1). Precisely, we consider
the problem

(−∆)spv = b(x)vq−1 + f(x, v) in Ω,

v > 0 in Ω,

v = 0 in RN \ Ω

(4.1)

where b ∈ L∞(Ω) and nonnegative. We define the notion of a weak solution as
follows.

Definition 4.1. A positive function v ∈W s,p
0 (Ω)∩L∞(Ω) is called a weak solution

to problem (4.1) if∫
RN

∫
RN

|v(x)− v(y)|p−2(v(x)− v(y))(ϕ(x)− ϕ(y))

|x− y|N+sp
dx dy

=

∫
Ω

(b(x)vq−1 + f(x, v))ϕdx

(4.2)

for any ϕ ∈W s,p
0 (Ω).

Theorem 4.2. Assume that f satisfies (H1)–(H3). Let q ∈ (1, p]. In addition
if q = p suppose that ‖b‖∞ < λ1,p,s. Then, there exists a unique weak solution

v ∈ C(Ω) ∩M1
ds(Ω) to problem (4.1).



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 29

Proof. By following the same arguments as in Theorem 2.2, we deduce the existence
of a nonnegative global minimizer to the following energy functional

L(v) =
1

p

∫
RN

∫
RN

|v(x)− v(y)|p

|x− y|N+ps
dx dy − 1

q

∫
Ω

b(v+)qdx−
∫

Ω

F (x, v)dx

where F is defined in (2.4). Then, as in the proof of Theorem 2.2 step 2, we infer
that v ∈ L∞(Ω). Furthermore, by using [27, Theorem 1.1], there is α ∈ (0, s] such
that v ∈ C0,α(Ω). Next, by [20, Theorems 1.4 and 1.5, p. 768], we obtain that
v > 0 in Ω and satisfies v ≥ k ds(x) for some k > 0. Finally, [[27], Theorem 4.4]
implies that v ∈M1

ds(Ω).

Let v1, v2 ∈ C(Ω) ∩M1
ds(Ω) be two solutions of (4.1), we choose

vq1−v
q
2

vq−1
1

and
vq2−v

q
1

vq−1
2

as test functions in (4.1) satisfied by v1, v2 respectively. Then adding the equations,
we deduce from Lemma 1.8 and (H2),∫

RN

∫
RN

( |v1(x)− v1(y)|p−2(v1(x)− v1(y))

|x− y|N+sp

[(vq1 − vq2
vq−1

1

)
(x)−

(vq1 − vq2
vq1

)
(y)
]

+
|v2(x)− v2(y)|p−2(v2(x)− v2(y))

|x− y|N+sp

[(vq2 − vq1
vq−1

2

)
(x)−

(vq2 − vq1
vq−1

2

)
(y)
])
dx dy = 0.

Again by Lemma 1.8, for 1 < q < p, we obtain v1 ≡ v2 in RN . While for q = p, we
have v1(x) = k v2(x) a.e. in RN , for some k > 0. Without loss of generality, we can
assume that k ≤ 1. Then from (H2) we obtain

(−∆)sp(kv2) = kp−1(−∆)sp(v2) = kp−1(b(x)vp−1
2 + f(x, v2))

< b(x)(kv2)p−1 + f(x, kv2)

= (−∆)sp(v1)

which yields a contradiction. Hence k = 1 and v1 ≡ v2. �

Next, as in the proof of Corollary 2.4, we obtain the following result.

Corollary 4.3. Under the conditions of Theorem 4.2, there exists one and only

one weak solution u ∈ V̇ q+ ∩M
1/q
ds (Ω) to the problem

Tqu = b in Ω,

u > 0 in Ω,

u = 0 in RN \ Ω.

(4.3)

Furthermore,∫
Rn

∫
Rn

|u1/q(x)− u1/q(y)|p−2(u1/q(x)− u1/q(y))((u
1−q
q Ψ)(x)− (u

1−q
q Ψ)(y))

|x− y|N+sp
dx dy

=

∫
Ω

bΨ dx+

∫
Ω

f(x, u1/q)u
1−q
q Ψ dx

for all Ψ satisfies (2.13).

Proof of Theorem 1.14. We are ready now to prove our stabilization result by
using the same approach as in the proof of [25, Theorem 3.10].

Proof of Theorem 1.14. We consider two cases.



30 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

Case 1: h = h∞. We introduce the family of operators {S(t) : t ≥ 0} defined

on V̇ q+ ∩M
1/q
ds (Ω) as w(t) = S(t)w0 where w is the unique solution (obtained in

Theorem 3.3) to
∂tw + Tqw = h∞ in QT ,

w > 0 in QT ,

w = 0 on Γ,

w(0, ·) = w0 in Ω.

(4.4)

Thus, we claim that {S(t) : t ≥ 0} defines a semi-group of contractions in L2(Ω).
Indeed, from the uniqueness and above properties of solutions to problem (4.4) we

infer that for any w0 ∈ V̇ q+ ∩M
1/q
ds (Ω),

S(t+ s)w0 = S(t)S(s)w0,

S(0)w0 = w0.
(4.5)

From (3.17) and (3.19) the map [0,∞) 3 t 7→ S(t)w0 is continuous and T -accretive
L2(Ω). Note that ṽ = (S(t)w0)1/q is the solution of (1.12) with h = h∞ and the

initial data w
1/q
0 .

Let us denote v the solution of (1.12) with h = h∞ and the initial data v0. Hence
we obtain u(t) = v(t)q = S(t)u0 with u0 = vq0. Let w = wµ be the solution of (3.4)
and w = wK or the solution to (3.7) if q = p. Then, w,w ∈M1

ds(Ω) and for µ small
enough and K large enough, w is a subsolution and w a supersolution to (4.1) with
b = h∞ such that w ≤ v0 ≤ w. We then define u(t) = S(t)wq and u(t) = S(t)wq the
solutions to (4.4). Therefore, u := (v)q and u := (v)q are obtained by the iterative
scheme (3.2) with v0 = w and v0 = w. Hence, by comparison principle the maps
t 7→ u(t) and t 7→ u(t) are respectively nondecreasing and non-increasing. In the
other hand, (1.13) ensures that for any t ≥ 0,

w ≤ u(t) ≤ u(t) ≤ u(t) ≤ w. (4.6)

We set u∞ = lim
t→∞

u(t) and u∞ = lim
t→∞

u(t). Then from (4.5), we obtain

u∞ = lim
s→∞

S(t+ s)wq = S(t) lim
s→∞

(S(s)(wq)) = S(t)u∞,

u∞ = lim
s→∞

S(t+ s)wq = S(t) lim
s→∞

(S(s)(wq)) = S(t)u∞.

This implies that u∞ and u∞ are the stationary solutions to (4.3) with b = h∞.
By uniqueness, we have ustat := u∞ = u∞ where ustat is the stationary solution to
(4.4). Therefore from (4.6) and by dominated convergence Theorem, we obtain

‖u(t)− ustat‖L2(Ω) → 0 as t→∞.

Thus using (4.6) and the interpolation inequality with 2 < r <∞,

‖ · ‖r ≤ ‖ · ‖α∞‖ · ‖1−α2 ,

we obtain, the above convergence for any r ≥ 1.

Case 2: h 6≡ h∞. From (1.16), for any ε > 0 there exists t0 > 0 large enough such

that
∫ +∞
t0

1
l(t)dt < ε and for any t ≥ t0,

l(t)‖h(t, ·)− h∞‖L2(Ω) ≤M for some M > 0.

Let T > 0 and v be the solution of the problem (1.12) obtained by Theorem 3.1

with h and the initial data v0 = u
1/q
0 and set u = vq. Since v satisfies (3.1), we can



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 31

define ũ(t) = S(t + t0)u0 = S(t)u(t0). Then, by (1.13) and uniqueness argument,
we have for any t > 0,

‖u(t+ t0, ·)− ũ(t, ·)‖L2(Ω) ≤
∫ t

0

‖h(s+ t0, ·)− h∞‖L2(Ω)ds

≤M
∫ +∞

t0

1

l(s)
ds ≤Mε.

By Case 1, we have ũ(t)→ ustat in L2(Ω) as t→∞. Therefore, we obtain

‖u(t)− ustat‖L2(Ω) → 0 as t→∞.
Using again the interpolation inequality above, we conclude the proof of Theorem
1.14. �

5. Appendix

5.1. Regularity results. The first one is obtained by a similar proof as in [23]
(see also [25]).

Proposition 5.1. Let u ∈W s,p
0 (Ω) satisfying∫

RN

∫
RN

|u(x)− u(y)|p−2(u(x)− u(y))(Ψ(x)−Ψ(y))

|x− y|N+sp
dx dy =

∫
Ω

f(x, u)Ψ dx

(5.1)
for all Ψ ∈W s,p

0 (Ω), where f satisfies for all (x, t) ∈ Ω× R,

|f(x, t)| ≤ C (1 + |t|r−1), ∀x ∈ Ω, 1 < r ≤ p.
Then u ∈ L∞(Ω).

Proposition 5.2. Let 1 < q ≤ p. Assume that u ∈W and nonnegative satisfying
for any Ψ ∈W,∫

RN

∫
RN

|u(x)− u(y)|p−2(u(x)− u(y))(Ψ(x)−Ψ(y))

|x− y|N+sp
dx dy =

∫
Ω

huq−1Ψ dx (5.2)

where h ∈ L2(Ω) ∩ Lr(Ω) with r > max{1, Nsp} and h ≥ 0 a.e. in Ω. Then

u ∈ L∞(Ω).

Proof. We follow the main steps in the proof of [8, Theorem 3.1]. For every δ > 0,

we define uδ = u+ δ. Given β ≥ 1, we insert the test function ψ = uβδ − δβ in (5.2),
then we obtain∫

RN

∫
RN

|u(x)− u(x)|p−2(u(x)− u(x))(uδ(x)β − uδ(y)β)

|x− y|N+sp
dx dy ≤

∫
Ω

huq−1uβδ dx.

By using the inequality in [8, Lemma A.2], we obtain

βpp

(β + p− 1)p

∫
RN

∫
RN

|uδ(x)
β+p−1
p − uδ(y)

β+p−1
p |p

|x− y|N+sp
dx dy ≤

∫
Ω

huq−1uβδ dx.

By Theorem 1.3, we obtain(∫
Ω

(uδ(x)
β+p−1
p − δ

β+p−1
p )p

∗
sdx
)p/p∗s

≤ CN,s,p
∫
RN

∫
RN

|uδ(x)
β+p−1
p − uδ(y)

β+p−1
p |p

|x− y|N+sp
dx dy.



32 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

By the triangle inequality, the left-hand side of the bovd inequality, can be estimated
as (∫

Ω

(
u
β+p−1
p

δ

)p∗s dx)p/p∗s ≤ (∫
Ω

(u
β+p−1
p

δ − δ
β+p−1
p )p

∗
sdx
)p/p∗s

+ δβ+p−1|Ω|p/p
∗
s .

On the other hand, we use the inequality uβ+p−1
δ ≥ δp−quβ+q−1

δ , Hölder and inter-

polation inequalities, for r > N
sp and with the observation that p < pr′ < p∗s, where

r′ = r
r−1 to obtain∫

Ω

huq−1uβδ dx ≤ δ
q−p

∫
Ω

hup+β−1
δ dx ≤ δq−p‖h‖Lr

(∫
Ω

(
u
p+β−1
p

δ

)pr′
dx
)1/r′

≤ δq−p‖h‖Lr
(∫

Ω

(u
p+β−1
p

δ )p
∗
sdx
) pα
p∗s
(∫

Ω

up+β−1
δ dx

)1−α
(5.3)

where 1
pr′ = α

p + 1−α
p∗s

and 0 ≤ α ≤ 1. Using Young’s inequality,∫
Ω

huq−1uβδ dx ≤ δ
q−p‖h‖Lr

[
ε
(∫

Ω

(u
p+β−1
p

δ )p
∗
s

)p/p∗s
dx+ Cε

∫
Ω

up+β−1
δ dx

]
with Cε = ε−

1
α−1 , it is easy to see that

δp+β−1|Ω|p/p
∗
s ≤ 1

β

(p+ β − 1

p

)p
|Ω|

p
p∗s
−1
∫

Ω

up+β−1
δ dx.

Taking

ε =
β δp−q

2CN,s,p ‖h‖Lr

( p

p+ β − 1

)p
> 0,

we obtain (∫
Ω

(u
β+p−1
p

δ )p
∗
sdx
)p/p∗s

≤ CN,s,p
β

(p+ β − 1

p

)p
[δq−p‖h‖LrCε + |Ω|

p
p∗s
−1

]

∫
Ω

up+β−1
δ dx.

We then choose

δ = (Cε‖h‖Lr )
−1
q−p |Ω|

1
q−p ( p

p∗s
−1)

> 0

and set υ = β + p− 1. Then the previous inequality can be written as(∫
Ω

u
(
p∗s
p )υ

δ dx
) 1

(
p∗s
p

)υ ≤ [C|Ω|
p
p∗s
−1

]1/υ(
υ

p
)p/υ

(∫
Ω

uυδdx
)1/υ

with C = C(N, s, p) > 0. We now iterate the previous inequality, by taking the
sequence of exponents

υ0 = 1 and υn+1 = (
p∗s
p

)υn = (
p∗s
p

)n+1.

We have
∞∑
n=0

1

υn
=

∞∑
n=0

( p
p∗s

)n
=

p∗s
p∗s − p

,

∞∏
n=0

(υn
p

) p
υn <∞.



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 33

By starting from 0 at the step n we have

‖uδ‖Lυn+1 (Ω) ≤ [C |Ω|
p
p∗s
−1

]
∑n
i=0

1
υi

n∏
i=0

(
υi
p

)
p
υi ‖uδ‖L1(Ω).

By taking the limit as n appraoches ∞, we finally obtain

‖uδ‖L∞(Ω) ≤
C ′

|Ω|
‖uδ‖L1(Ω) ≤

C ′

|Ω|
(‖u‖L1(Ω) + δ|Ω|)

for some constant C ′ = C ′(N, p, s) > 0. �

Combining Proposition 5.1 with Proposition 5.2, we have the following corollary:

Corollary 5.3. Let 1 < q ≤ p. Assume u ∈ W, nonnegative and satisfying for
any nonnegative Ψ ∈W∫

Ω

u2q−1Ψ dx+

∫
RN

∫
RN

|u(x)− u(y)|p−2(u(x)− u(y))(Ψ(x)−Ψ(y))

|x− y|N+sp
dx dy

≤
∫

Ω

(f(x, u) + huq−1)Ψ dx

where f satisfies for all (x, t) ∈ Ω×R, |f(x, t)| ≤ C(1+tq−1) and h ∈ L2(Ω)∩Lr(Ω)
with r > max{1, Nsp}. Then u ∈ L∞(Ω).

5.2. Comparison principle. Following the proof of [2, Theorem 4.3] and using
Lemma 1.8, we have the following new comparison principle.

Theorem 5.4. Assume f satisfies (H1), (H2). Let v, v ∈W ∩ L∞(Ω) be nonneg-
ative functions respectively subsolution and supersolution to (2.1) for some h0 ∈
(Lr(Ω))+ with r ≥ 2. Then v ≤ v.

Proof. For any nonnegative pair Φ,Ψ ∈W we have∫
Ω

v2q−1Φdx+ λ

∫
RN

∫
RN

|v(x)− v(y)|p−2(v(x)− v(y))(Φ(x)− Φ(y))

|x− y|N+sp
dx dy

≤
∫

Ω

h0v
q−1Φ dx+ λ

∫
Ω

f(x, v)Φ dx

and∫
Ω

v2q−1Ψ dx+ λ

∫
RN

∫
RN

|v(x)− v(y)|p−2(v(x)− v(y))(Ψ(x)−Ψ(y))

|x− y|N+sp
dx dy

≥
∫

Ω

h0v
q−1Ψ dx+ λ

∫
Ω

f(x, v)Ψ dx.

Subtracting the above inequalities with test functions

Φ =
( (v + ε)q − (v + ε)q

(v + ε)q−1

)+

, Ψ =
( (v + ε)q − (v + ε)q

(v + ε)q−1

)−
∈W,



34 J. GIACOMONI, A. GOUASMIA, A. MOKRANE EJDE-2021/09

with ε ∈ (0, 1), we obtain∫
{v>v}

( v2q−1

(v + ε)q−1
− v2q−1

(v + ε)q−1

)
((v + ε)q − (v + ε)q) dx

+ λ

∫
{v>v}

∫
{v>v}

|v(x)− v(y)|p−2(v(x)− v(y))

|x− y|N+sp

×
[ (v(x) + ε)q − (v(x) + ε)q

(v(x) + ε)q−1
− (v(y) + ε)q − (v(y) + ε)q

(v(y) + ε)q−1

]
dx dy

+ λ

∫
{v>v}

∫
{v>v}

|v(x)− v(y)|p−2(v(x)− v(y))

|x− y|N+sp

×
[ (v(x) + ε)q − (v(x) + ε)q

(v(x) + ε)q−1
− (v(y) + ε)q − (v(y) + ε)q

(v(y) + ε)q−1

]
dx dy

≤
∫
{v>v}

h0

( vq−1

(v + ε)q−1
− vq−1

(v + ε)q−1

)
((v + ε)q − (v + ε)q)dx

+ λ

∫
{v>v}

( f(x, v)

(v + ε)q−1
− f(x, v)

(v + ε)q−1

)
((v + ε)q − (v + ε)q)dx.

(5.4)

Since v
v+ε ≤

v
v+ε < 1 in {v > v}, we obtain( v2q−1

(v + ε)q−1
− v2q−1

(v + ε)q−1

)
((v + ε)q − (v + ε)q)

=
(
vq
( v

(v + ε)

)q−1

− vq
( v

(v + ε)

)q−1)
((v + ε)q − (v + ε)q)

≤ vq((v + ε)q − (v + ε)q) ≤ vq(v + 1)q.

In the same spirit, we infer that

0 ≤ h0

( vq−1

(v + ε)q−1
− vq−1

(v + ε)q−1

)
((v + ε)q − (v + ε)q) ≤ h0(v + 1)q.

Moreover, as ε→ 0, we have( v2q−1

(v + ε)q−1
− v2q−1

(v + ε)q−1

)(
(v + ε)q − (v + ε)q

)
→ (vq − vq)2,

h0

( vq−1

(v + ε)q−1
− vq−1

(v + ε)q−1

)
((v + ε)q − (v + ε)q)→ 0.

Then, by the dominated convergence Theorem, we have

lim
ε→0

∫
{v>v}

( v2q−1

(v + ε)q−1
− v2q−1

(v + ε)q−1

)
((v + ε)q − (v + ε)q)dx

=

∫
{v>v}

(vq − vq)2 dx

(5.5)

and

lim
ε→0

∫
{v>v}

h0

( vq−1

(v + ε)q−1
− vq−1

(v + ε)q−1

)
((v + ε)q − (v + ε)q)dx = 0. (5.6)

Then by Fatou’s Lemma and (H1), we have

− lim
ε→0

inf

∫
{v>v}

f(x, v)

(v + ε)q−1
(v + ε)q dx ≤ −

∫
{v>v}

f(x, v)

vq−1
vq dx, (5.7)



EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 35

− lim
ε→0

inf

∫
{v>v}

f(x, v)

(v + ε)q−1
(v + ε)q dx ≤ −

∫
{v>v}

f(x, v)

vq−1 vq dx (5.8)

lim
ε→0

∫
{v>v}

f(x, v)(v + ε) dx =

∫
{v>v}

f(x, v)v dx, (5.9)

lim
ε→0

∫
{v>v}

f(x, v)(v + ε) dx =

∫
{v>v}

f(x, v)v dx. (5.10)

From Lemma 1.8, we have∫
{v>v}

∫
{v>v}

|v(x)− v(y)|p−2(v(x)− v(y))

|x− y|N+sp

×
[ (v(x) + ε)q − (v(x) + ε)q

(v(x) + ε)q−1
− (v(y) + ε)q − (v(y) + ε)q

(v(y) + ε)q−1

]
dx dy

+

∫
{v>v}

∫
{v>v}

|v(x)− v(y)|p−2(v(x)− v(y))

|x− y|N+sp

×
[ (v(x) + ε)q − (v(x) + ε)q

(v(x) + ε)q−1
− (v(y) + ε)q − (v(y) + ε)q

(v(y) + ε)q−1

]
dx dy ≥ 0.

(5.11)

Therefore, plugging (5.5)-(5.11) and taking lim sup
ε→0

in (5.4), we obtain from (H2),

0 ≤
∫
{v>v}

(vq − vq)2dx ≤ λ
∫
{v>v}

(f(x, v)

vq−1
− f(x, v)

vq−1

)
(vq − vq)dx ≤ 0

from which v ≤ v follows. �

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EJDE-2021/09 DOUBLY NONLINEAR FRACTIONAL EVOLUTION EQUATION 37

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bounded domains, Nonlinear Anal., 135 (2016), 129–157.
[42] M. Xiang, B. Zhang, V. D. Rǎdulescu; Existence of solutions for perturbed fractional p-

Laplacian equations, J. Differential Equations, 260 (2016), no. 2, 1392-1413.

Jacques Giacomoni

Université de Pau et des Pays de l’Adour, CNRS, E2S LMAP (UMR 5142), IPRA, Avenue
de l’Université-F-64013 Pau, France

Email address: jacques.giacomoni@univ-pau.fr

Abdelhamid Gouasmia

Laboratoire d’équations aux dérivées partielles non linéaires et histoire des

mathématiques, École Nationale Supérieure, B.P. 92, Vieux Kouba, 16050 Algiers, Al-
geria

Email address: gouasmia.abdelhamid@gmail.com

Abdelhafid Mokrane

Laboratoire d’équations aux dérivées partielles non linéaires et histoire des

mathématiques, École Nationale Supérieure, B.P. 92, Vieux Kouba, 16050 Algiers, Al-
geria

Email address: abdelhafid.mokrane@ens-kouba.dz


	1. Introduction and statement of main results
	1.1. State of art
	1.2. Preliminaries and functional setting
	1.3. Main results

	2. p-fractional elliptic equation associated with problem (1.1)
	2.1. Potential  h0 L()
	2.2. Potential  h0 L2()

	3. Existence of a weak solution to parabolic problem (1.1)
	3.1. Existence and regularity of a weak solution

	4. Stabilization
	4.1. Existence and uniqueness of the solution of the stationary problem related to (1.12)
	Proof of Theorem 1.14

	5. Appendix
	5.1. Regularity results
	5.2. Comparison principle

	References