Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 89, pp. 1–21.

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

DOI: 10.58997/ejde.2025.89

DECAY ESTIMATES AND EXTINCTION PROPERTIES OF PARABOLIC

EQUATIONS WITH CLASSICAL AND FRACTIONAL TIME DERIVATIVES

FANMENG MENG, XIAN-FENG ZHOU

Abstract. In this article, we study the decay estimates and extinction properties of weak

solutions to some parabolic equations with classical and fractional time derivatives. Firstly, we

establish a new comparison principle for parabolic equations with mixed time derivatives. Based
on this comparison principle and energy methods, we obtain the power-law decay estimates for

weak solutions of nonhomogeneous abstract parabolic problems with mixed time-derivatives.

Furthermore, we present three specific applications of the decay results for the abstract parabolic
problem. Finally, we discus the finite time extinction property of the weak solution for the 1-

Kirchhoff type parabolic problem with mixed time-derivatives.

1. Introduction

This work considers the decay estimates and extinction properties of weak solutions to the
abstract parabolic problem

λ1∂tu(x, t) + λ2∂
α
0,tu(x, t) +N [u] = f(x, t) in Ω× R+,

u(x, t) = 0 in (RN \ Ω)× R+,

u(x, 0) = u0(x) in Ω,

(1.1)

where 0 < α < 1, λ1, λ2 > 0, λ1 + λ2 = 1, Ω is a bounded subset of RN with smooth boundary,
u = u(x, t) is the unknown function, u0 ∈ L∞(Ω), f(t) ∈ Ls(Ω), s ≥ 2, N [u] is a possible nonlocal
operator, ∂α

0,tu(x, t) denotes the Caputo fractional derivative of u of order α, which is defined by
[17]

∂α
0,tu(x, t) :=

d

dt
(kα ∗ [u− u0]) (t) =

1

Γ(1− α)

d

dt

∫ t

0

u(x, ϱ)− u0(x)

(t− ϱ)α
dϱ, (1.2)

where Γ(·) is the Euler’s gamma function and kα(η) =
η−α

Γ(1−α) .

Fractional calculus has attracted much attention not only because it involves profound mathe-
matical theory, but also because it appears in a variety of real-world phenomena in different forms
[4, 7, 11, 14, 16, 25, 29]. Compared with classical derivatives, equations with fractional derivatives
can better describe some physical phenomena. In particular, time-fractional derivatives have been
applied in the fields of wave equations [3, 22], porous media equations [5], fluid dynamics [30],
quantum physics [15], and so on.

A large body of literature is devoted to studying the existence, uniqueness, regularity and
asymptotic properties of solutions [8, 10, 13, 21, 23, 24]. For example, Smadiyeva et al. [21]

2020 Mathematics Subject Classification. 35B40, 26A33, 35K90.
Key words and phrases. Abstract parabolic equation; Caputo derivative; decay estimates; extinction properties;

comparison principle.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted July 1, 2025. Published September 16, 2025.

1



2 F. MENG, X.-F. ZHOU EJDE-2025/89

studied the time fractional evolution equation

∂α
0,tu(t, x) + a(t)A(u(t, x)) = 0, (t, x) ∈ R+ × Ω,

u(0, x) = u0(x), x ∈ Ω,

u(t, x) = 0, t ≥ 0, x ∈ ∂Ω,

(1.3)

where 0 < α ≤ 1, a ∈ L1(R+), Ω ⊂ RN is a bounded domain with smooth boundary, ∂α
0,t is the

Caputo fractional derivative operator. They established the decay rate of the solution of problem
(1.3) when A(u) is one of the following operators:

• Laplace operator: A(u) := ∆u = div(∇u);
• p-Laplace operator: A(u) := ∆pu = div(|∇u|p−2∇u);
• Porous medium operator: A(u) := div(g(u)∇u);
• Kirchhoff operator: A(u) := M(∥∇u∥Lq )∆pu.

However, most of these results depend on a specific single time derivative or space operator.
This paper investigates the decay behavior over time t in the Lebesgue norm of weak solutions
on a bounded domain for problem (1.1). The problem (1.1) involves the parabolic evolution
of the function u under the action of the spatial diffusion operator N , which has an appropriate
“ellipticity” property, and can be either classical, fractional or nonlinear. We set it in a very general
framework, which is suitable for both local operators and non-local operators. This analysis also
includes the combination of fractional and classical time-derivatives. Therefore, the results of this
paper are more general. This is a novelty of our paper.

In this paper, we also give several specific examples of the general framework with mixed time-
derivatives. More specifically, the cases in which the operator N in problem (1.1) is defined as the
following spatial diffusion operators are studied:

(i) the case of space-fractional double nonlinear operator;
(ii) the case of the sum of space-fractional double nonlinear operators in different directions;
(iii) the case of fractional p-Kirchhoff operator.

These results generalize and include some cases in [9, 19, 20, 21, 26] which can be regarded as
special cases of our results.

As usual, we say that the solution u vanishes in finite time if there exists T > 0 such that
u(·, t) ≡ 0 for all t ≥ T . To the best of our knowledge, there are few papers to discuss the
extinction properties of solutions of parabolic problems with fractional Kirchhoff operators. In
[19], Pucci et al. studied the following initial-boundary value problem with fractional p-Kirchhoff:

∂tu+M
(∫∫

R2N

|u(x)− u(y)|p

|x− y|N+sp
dxdy

)
(−∆)

s
p u = f(x, t) in Ω× R+,

u(x, 0) = u0(x) in Ω,

u = 0 in RN \ Ω,

(1.4)

where M : R+
0 → R+

0 is a continuous and nondecreasing function, 1 < p < N
s . Under suitable

assumptions, the well-posedness and extinction properties of solutions of the time integer order
parabolic problem (1.4) are obtained by using the sub-differential method. In the past, most of
the p-Kirchhoff type parabolic problems were studied in the case of p > 1. So far, the extinction
properties of solutions for time-fractional parabolic problems like (1.4) when p = 1 have not been
studied, nor have such problems with mixed derivatives. Therefore, this paper is the first attempt
to study the extinction properties of weak solutions of parabolic 1-Kirchhoff problems with mixed
time-derivatives (see Problem (5.1)). This is also a novelty of our paper.

The remaining part of this paper is organized as follows. In Section 2, we introduce some
important definitions, lemmas and properties, and we also prove a new comparison principle
which is needed to obtain the main results of this paper. In Section 3, we prove a result on the
time decay estimate of weak solutions to the abstract parabolic problem (1.1). In Section 4, we
present three specific applications of the result in Section 3. In Section 5, we consider the finite
time extinction property of weak solutions for the 1-Kirchhoff type parabolic problem with mixed
time-derivatives. The conclusion is introduced in Section 6.



EJDE-2025/89 DECAY ESTIMATES AND EXTINCTION PROPERTIES 3

2. Preliminaries

In this section, we introduce some tools and important results that will be used in the proofs
of the main theorems in this paper.

We fix the fractional exponent δ in (0, 1). For any p ∈ [1,+∞), the fractional Sobolev space
W δ,p(RN ) is defined as follows [6]:

W δ,p(RN ) :=
{
u ∈ Lp(RN ) :

|u(x)− u(y)|
|x− y|

N
p +δ

∈ Lp(RN × RN )
}
. (2.1)

This is a Banach space endowed with the norm

∥u∥W δ,p(RN ) =
(
∥u∥p

Lp(RN )
+ [u]p

W δ,p(RN )

)1/p

,

where the term

[u]W δ,p(RN ) :=
(∫∫

R2N

|u(x)− u(y)|p

|x− y|N+δp
dxdy

)1/p

(2.2)

is the so-called the Gagliardo semi-norm of u.

Let Ω be a bounded open subset in RN , the spaces W δ,p(Ω) and W δ,p
0 (Ω) are defined by

W δ,p(Ω) :=
{
u ∈ Lp(Ω) :

|u(x)− u(y)|
|x− y|

N
p +δ

∈ Lp(Ω× Ω)
}

(2.3)

and
W δ,p

0 (Ω) := {u ∈ W δ,p(Ω) : u = 0 a.e. in RN\Ω}.
The spaces are also endowed with the norm

∥u∥W δ,p(Ω) =
(
∥u∥pLp(Ω) + [u]p

W δ,p(Ω)

)1/p

.

Definition 2.1 ([27]). Let k ∈ L1
loc(R+). When α ≥ 0 and β > 0, k is said to be of class K(α, β)

if the following conditions hold:

(1) k is a sub-exponential growth, that is,
∫∞
0

e−εt|k(t)|dt < +∞ for all ε > 0;

(2) k is 1-regular, that is, there exists a constant c > 0 such that |λk̂′(λ)| ≤ c|k̂(λ)| for all
Reλ > 0;

(3) k is θ-sectorial, that is, |arg(k̂)(λ)| ≤ θ for all Reλ > 0;

(4) satisfying lim supλ→+∞ |k̂(λ)|λβ < ∞, lim infλ→+∞ |k̂(λ)|λβ > 0 and lim infλ→0 |k̂(λ)| >
0.

Definition 2.2 ([27]). We say that the kernel k ∈ L1
loc(R+) is of the type PC if it is non-negative

and non-increasing, and there exists another non-negative and non-increasing kernel l ∈ L1
loc(R+)

such that k ∗ l = 1 on (0,+∞). Furthermore, we call (k, l) a PC pair.

Lemma 2.3 ([27]). Let T > 0 and H be a real Hilbert space. If there exist k ∈ L1
loc(R+) and some

l ∈ K(α, β) with 0 < α < 1 and β < π such that k ∗ l = 1 on (0,+∞), then

u ∈ L2(0, T ;H) and k ∗ u ∈ W 1,2
0 (0, T ;H) =⇒ k ∗ ∥u∥2H ∈ W 1,1

0 (0, T ),

where W 1,2
0 (0, T ;H) =

{
u ∈ L2(0, T ;H) : du

dt ∈ L2(0, T ;H), u(0) = 0
}
.

Lemma 2.4 ([6]). Let 0 < δ < 1 and p ≥ 1. If δp < N , then there exists C⋆ > 0 depending on
N , δ and p such that

∥v∥p
L

Np
N−δp (RN )

≤ C⋆

∫∫
R2N

|v(x)− v(y)|p

|x− y|N+δp
dxdy, ∀v ∈ W δ,p(RN ).

Lemma 2.5 ([28]). Let p ∈ (1,+∞), T > 0, and Ω is a subset of RN with arbitrary measure.
If k ∈ W 1,1(0, T ) is non-negative and non-increasing, then for any u0 ∈ Lp(Ω) and any u ∈
Lp(0, T ;Lp(Ω)) it holds

∥u(t)∥p−1
p ∂t (k ∗ (∥u(t)∥p − ∥u0∥p)) ≤

∫
Ω

|u(t)|p−2u(t)∂t(k ∗ [u− u0])(t)dx

for almost everywhere t ∈ (0, T ), where ∥ · ∥p denotes the norm of Lp(Ω).



4 F. MENG, X.-F. ZHOU EJDE-2025/89

It is known that the Riemann-Liouville kernel [28]

kα(t) =
t−α

Γ(1− α)
and lα(t) =

tα−1

Γ(α)
, t > 0 (2.4)

for 0 < α < 1 belong to the PC defined in Definition 2.2. However, the Riemann-Liouville kernel
(2.4) does not belong to W 1,1(0, T ) for all T > 0, so in order to be able to apply Lemma 2.5,
we need to recall the Yosida approximation of the time fractional derivative operator proposed in
[2, 27] and its properties. For convenience, we only provide some important properties needed in
this paper as follows.

Property 2.6 ([2, 27]). Let p ≥ 1 and X be a real Banach space. For a fractional derivative
operator defined as Bu = d

dt (kα ∗ u), where

D(B) :=
{
u ∈ Lp(0, T ;X) : kα ∗ u ∈ W 1,p(0, T ;X), (kα ∗ u)(0) = 0

}
,

its Yosida approximation Bnu is defined as Bnu = nB(n+ B)−1u (Bnu can also be expressed as
Bnu = d

dt (kn,α ∗ u)), n ∈ N, where kn,α = nsn,α, with sn,α being the unique solution of the scalar
Volterra equation

sn,α(t) + n(l ∗ sn,α)(t) = 1, t > 0, n ∈ N.
Then

(1) Bnu → Bu in Lp(0, T ;X) as n → +∞ for any u ∈ D(B);
(2) the kernel sn,α, n ∈ N is nonnegative and nonincreasing in (0,∞) and sn,α ∈ W 1,1(0, T )

(see Prop. 4.5 in [18]). Hence, the kernel kn,α, n ∈ N possesses the same properties;
(3) kn,α → kα in L1(0, T ) as n → +∞.

To prove our main results, we need to state the following comparison principle involving mixed
time-derivatives.

Lemma 2.7. Let 0 < α < 1, λ1, λ2, T > 0, f ∈ C(R) and g ∈ L1([0, T )). Assume that f is
nondecreasing. Suppose that υ, ω ∈ W 1,1([0, T )) satisfy υ(0) ≤ ω(0) and

λ1∂tυ(t) + λ2∂t(kα ∗ [υ(t)− υ(0)]) + f(υ) ≤ g(t), a.e. t ∈ [0, T ), (2.5)

λ1∂tω(t) + λ2∂t(kα ∗ [ω(t)− ω(0)]) + f(ω) ≥ g(t), a.e. t ∈ [0, T ). (2.6)

Then υ(t) ≤ ω(t) for all t ∈ [0, T ).

Proof. Subtracting inequality (2.5) from inequality (2.6) yields

λ1∂t(υ(t)− ω(t)) + λ2∂t(kα ∗ [υ(t)− ω(t)]) + f(υ(t))− f(ω(t)) ≤ kα(t)(υ(0)− ω(0)) ≤ 0, (2.7)

thanks to υ(0) ≤ ω(0). Integrating (2.7) with respect to t, we obtain

λ1

∫ t

0

∂s(υ(s)− ω(s))ds+ λ2

∫ t

0

∂s(kα ∗ [υ(s)− ω(s)])ds+

∫ t

0

f(υ(s))− f(ω(s))ds ≤ 0,

which implies that

λ1(υ(t)− ω(t)) + λ2kα ∗ (υ(t)− ω(t))− λ2kα ∗ (υ(t)− ω(t))
∣∣
t=0

+

∫ t

0

f(υ(s))− f(ω(s))ds

≤ λ1(υ(0)− ω(0)),

that is,

λ1(υ(t)− ω(t)) + λ2kα ∗ (υ(t)− ω(t)) +

∫ t

0

f(υ(s))− f(ω(s))ds ≤ 0. (2.8)

Let (υ(t)− ω(t))+ = max{υ(t)− ω(t), 0}. Multiplying (υ(t)− ω(t))+ on both sides of (2.8) gives

λ1(υ(t)− ω(t))(υ(t)− ω(t))+ +
λ2

Γ(1− α)
(υ(t)− ω(t))+

∫ t

0

(t− s)−α(υ(s)− ω(s))ds

+ (υ(t)− ω(t))+

∫ t

0

f(υ(s))− f(ω(s))ds ≤ 0.

(2.9)



EJDE-2025/89 DECAY ESTIMATES AND EXTINCTION PROPERTIES 5

If (υ(t)− ω(t)) ≤ 0, then (υ(t)− ω(t))+ = 0, which implies that

(υ(t)− ω(t))(υ(t)− ω(t))+ = 0 = (υ(t)− ω(t))2+

and

(υ(t)− ω(t))+

∫ t

0

(t− s)−α(υ(s)− ω(s))ds

= 0

= (υ(t)− ω(t))+

∫ t

0

(t− s)−α(υ(s)− ω(s))+ds.

If (υ(t)− ω(t)) > 0, then (υ(t)− ω(t))+ = (υ(t)− ω(t)), which means that

(υ(t)− ω(t))(υ(t)− ω(t))+ = (υ(t)− ω(t))2+

and

(υ(t)− ω(t))+

∫ t

0

(t− s)−α(υ(s)− ω(s))ds

= (υ(t)− ω(t))+

∫ t

0

(t− s)−α(υ(s)− ω(s))+ds

still hold for all 0 < s < t. Therefore, (2.9) can be reduced to

λ1(υ(t)− ω(t))2+ +
λ2

Γ(1− α)
(υ(t)− ω(t))+

∫ t

0

(t− s)−α(υ(s)− ω(s))+ds

+ (υ(t)− ω(t))+

∫ t

0

f(υ(s))− f(ω(s))ds ≤ 0.

(2.10)

Since f is nondecreasing, then the third term in (2.10) is nonnegative, and then this term can be
removed to give

λ1(υ(t)− ω(t))2+ +
λ2

Γ(1− α)
(υ(t)− ω(t))+

∫ t

0

(t− s)−α(υ(s)− ω(s))+ds ≤ 0,

which implies that (υ(t)− ω(t))+ = 0, that is, υ(t) ≤ ω(t) in [0, T ]. □

Throughout this article, for u(x, t), we also define it by

u(t)(x) := u(x, t) (x ∈ RN , t ∈ R+
0 ).

3. Decay estimates for problem (1.1)

In this section, we utilize energy methods and a new comparison principle (Lemma 2.7) to
investigate the decay estimates of weak solutions for the abstract parabolic problem (1.1). Before
that, we first introduce the definition of a weak solution to (1.1).

Definition 3.1. Let s ≥ 2, f ∈ Ls(Ω) and u(·, 0) = u0(·). A function u ∈ W 1,2(0, T ;Ls(Ω)) is

said to be a weak solution of problem (1.1), if N [u] ∈ L2(Ω), kα ∗ (u − u0) ∈ W 1,2
0 (0, T ;L2(Ω)),

and for almost every t ∈ (0, T ), T > 0, it holds that

λ1

∫
Ω

∂tu(x, t)φ(x, t)dx+ λ2

∫
Ω

∂α
0,tu(x, t)φ(x, t)dx =

∫
Ω

(
f(x, t)−N [u](x, t)

)
φ(x, t)dx (3.1)

for all φ(t) ∈ Ls(Ω).

Theorem 3.2. Let s ≥ 2 and u(·, 0) = u0(·). Suppose u is a weak solution of problem (1.1) in the

sense of Definition 3.1. If there exist γ > 0, C0 > 0 and C > 0 such that ∥u0∥Ls(Ω) ≥ 1
2 (CC0)

1/γ

and ∥f(t)∥Ls(Ω) ≤ C0

(t+1)α for all t ≥ 0, and the solution u and the nonlinear operator N of (1.1)

satisfy

∥u(t)∥s−1+γ
Ls(Ω) ≤ C

∫
Ω

|u(x, t)|s−2u(x, t)N [u](x, t)dx, t ≥ 0, (3.2)



6 F. MENG, X.-F. ZHOU EJDE-2025/89

then

(λ1∂t + λ2∂
α
0,t)∥u(t)∥Ls(Ω) ≤

C0

(t+ 1)α
(3.3)

for some C0 > 0. Furthermore,

∥u(t)∥Ls(Ω) ≤
C#

1 + t
α
γ

(3.4)

for some C# > 0, depending only on α, γ, C, C0 and u0.

Proof. Choosing φ(t) := |u(t)|s−2u(t) in (3.1), we have

λ1

∫
Ω

|u(x, t)|s−2u(x, t)∂tu(x, t)dx+ λ2

∫
Ω

|u(x, t)|s−2u(x, t)∂α
0,tu(x, t)dx

=

∫
Ω

|u(x, t)|s−2u(x, t)
(
f(x, t)−N [u](x, t)

)
dx.

(3.5)

It is easy to show that

1

s
∂t|u(x, t)|s = |u(x, t)|s−1 u(x, t)

|u(x, t)|
∂tu(x, t) = |u(x, t)|s−2u(x, t)∂tu(x, t). (3.6)

Integrating both sides of (3.6) with respect to x over Ω yields∫
Ω

|u(x, t)|s−2u(x, t)∂tu(x, t)dx =
1

s

∫
Ω

∂t|u(x, t)|sdx = ∥u(t)∥s−1
Ls(Ω)∂t∥u(t)∥Ls(Ω). (3.7)

It can be obtained from (1.2) that∫
Ω

|u(x, t)|s−2u(x, t)∂α
0,tu(x, t)dx

=

∫
Ω

|u(x, t)|s−2u(x, t)∂t (kα ∗ [u− u0]) (t)dx

=

∫
Ω

|u(x, t)|s−2u(x, t) [∂t (kα ∗ (u− u0))− ∂t (kn,α ∗ (u− u0))] (t)dx

+

∫
Ω

|u(x, t)|s−2u(x, t)∂t (kn,α ∗ [u− u0]) (t)dx,

(3.8)

where kn,α ∈ W 1,1(0, T ) is the approximation sequence of kα (see [27]). Then the equation (3.8)
and Lemma 2.5 imply that∫

Ω

|u(x, t)|s−2u(x, t)∂α
0,tu(x, t)dx

≥
∫
Ω

|u(x, t)|s−2u(x, t) [∂t (kα ∗ (u− u0))− ∂t (kn,α ∗ (u− u0))] (t)dx

+ ∥u(t)∥s−1
Ls(Ω)∂t

(
kn,α ∗

[
∥u(t)∥Ls(Ω) − ∥u0∥Ls(Ω)

])
.

(3.9)

Since s ≥ 2 and Ω is a bounded subset of RN , it follows that Ls(Ω) ↪→ L2(Ω). Taking the limit
on both sides of inequality (3.9) as n → +∞, applying Lemma 2.3 and Property 2.6 yields∫

Ω

|u(x, t)|s−2u(x, t)∂α
0,tu(x, t)dx ≥ ∥u(t)∥s−1

Ls(Ω)∂t

(
kα ∗

[
∥u(t)∥Ls(Ω) − ∥u0∥Ls(Ω)

])
,

which implies that∫
Ω

|u(x, t)|s−2u(x, t)∂α
0,tu(x, t)dx ≥ ∥u(t)∥s−1

Ls(Ω)∂
α
0,t∥u(t)∥Ls(Ω). (3.10)

Now, substituting (3.7) and (3.10) into (3.5), we obtain

∥u(t)∥s−1
Ls(Ω)

(
λ1∂t∥u(t)∥Ls(Ω) + λ2∂

α
0,t∥u(t)∥Ls(Ω)

)
≤

∫
Ω

|u(x, t)|s−2u(x, t) (f(x, t)−N [u](x, t)) dx.
(3.11)



EJDE-2025/89 DECAY ESTIMATES AND EXTINCTION PROPERTIES 7

By using (3.2), the decay estimate of f and Hölder’s inequality, (3.11) can be reduced to

∥u(t)∥s−1
Ls(Ω)

(
λ1∂t + λ2∂

α
0,t

)
∥u(t)∥Ls(Ω) ≤ −

∥u(t)∥s−1+γ
Ls(Ω)

C
+

∫
Ω

|u(x, t)|s−1|f(x, t)|dx

≤ −
∥u(t)∥s−1+γ

Ls(Ω)

C
+ ∥u(t)∥s−1

Ls(Ω)∥f(t)∥Ls(Ω)

≤ −
∥u(t)∥s−1+γ

Ls(Ω)

C
+

C0∥u(t)∥s−1
Ls(Ω)

(t+ 1)α
,

(3.12)

which implies that(
λ1∂t + λ2∂

α
0,t

)
∥u(t)∥Ls(Ω) ≤ −

∥u(t)∥γLs(Ω)

C
+

C0

(t+ 1)α
, t ≥ 0 (3.13)

provided that ∥u(t)∥Ls(Ω) ̸= 0. If ∥u(t)∥Ls(Ω) = 0, obviously (3.13) also holds. Therefore, the
estimation (3.3) holds for all t ≥ 0.

Next, we introduce the function

φ(t) =

{
2∥u0∥Ls(Ω), 0 ≤ t < t0,

2∥u0∥Ls(Ω)t
α
γ

0 t−
α
γ , t ≥ t0,

(3.14)

and prove the inequality

λ1∂tφ(t) + λ2∂
α
0,tφ(t) +

φγ(t)

C
≥ C0

(t+ 1)α
, t ≥ 0. (3.15)

Indeed, for the case 0 ≤ t ≤ t0, we have

λ1∂tφ(t) + λ2∂
α
0,tφ(t) +

1

C
φγ(t) = 2λ1∂t∥u0∥Ls(Ω) + 2λ2∂

α
0,t∥u0∥Ls(Ω) +

2γ

C
∥u0∥γLs(Ω)

≥ C0

(t+ 1)α

(3.16)

due to ∥u0∥Ls(Ω) ≥ 1
2 (CC0)

1/γ
.

For the case t ≥ t0, we obtain

λ1∂tφ(t) + λ2∂
α
t0,tφ(t)

= −
2αλ1∥u0∥Ls(Ω)t

α
γ

0 t−
α
γ −1

γ
−

2αλ2∥u0∥Ls(Ω)t
α
γ

0

γΓ(1− α)

∫ t

t0

ϱ−
α
γ −1

(t− ϱ)α
dϱ

= −
2αλ1∥u0∥Ls(Ω)t

α
γ

0 t−
α
γ −1

γ
−

2αλ2∥u0∥Ls(Ω)t
α
γ

0 t−α−α
γ

γΓ(1− α)

∫ 1

t0
t

s−
α
γ −1

(1− s)α
ds.

(3.17)

If t > 2t0, that is,
t0
t < 1

2 , then (3.17) can be reduced to

λ1∂tφ(t) + λ2∂
α
t0,tφ(t)

= −
2αλ1∥u0∥Ls(Ω)t

α
γ

0 t−
α
γ −1

γ

−
2αλ2∥u0∥Ls(Ω)t

α
γ

0 t−α−α
γ

γΓ(1− α)

(∫ 1/2

t0
t

s−
α
γ −1

(1− s)α
ds+

∫ 1

1
2

s−
α
γ −1

(1− s)α
ds

)
≥ −

2αλ1∥u0∥Ls(Ω)t
α
γ

0 t−
α
γ −1+αt−α

γ

−
2αλ2∥u0∥Ls(Ω)t

α
γ

0 t−α−α
γ

γΓ(1− α)

(
2α

∫ 1/2

t0
t

s−
α
γ −1ds+ 2

α
γ +1

∫ 1

1
2

(1− s)−αds
)



8 F. MENG, X.-F. ZHOU EJDE-2025/89

≥ −
2α−

α
γ αλ1∥u0∥Ls(Ω)t

−α

γt1−α
0

+
2α+

α
γ +1λ2∥u0∥Ls(Ω)t

α
γ

0 t−α−α
γ

Γ(1− α)

−
2α+1λ2∥u0∥Ls(Ω)t

−α

Γ(1− α)
−

2α+
α
γ +1αλ2∥u0∥Ls(Ω)t

α
γ

0 t−α−α
γ

γΓ(2− α)

≥
(
−

2α−
α
γ αλ1∥u0∥Ls(Ω)

γt1−α
0

−
2α+1λ2∥u0∥Ls(Ω)

Γ(1− α)
−

2α+1αλ2∥u0∥Ls(Ω)

γΓ(2− α)

)
t−α,

which implies that

λ1∂tφ(t) + λ2∂
α
t0,tφ(t) +

φγ(t)

C

≥
(
−

2α−
α
γ αλ1∥u0∥Ls(Ω)

γt1−α
0

−
2α+1λ2∥u0∥Ls(Ω)

Γ(1− α)
−

2α+1αλ2∥u0∥Ls(Ω)

γΓ(2− α)

)
t−α

+
2γ∥u0∥γLs(Ω)t

α
0 t

−α

C

≥
(
−

2α−
α
γ αλ1∥u0∥Ls(Ω)

γt1−α
0

−
2α+1λ2∥u0∥Ls(Ω)

Γ(1− α)
−

2α+1αλ2∥u0∥Ls(Ω)

γΓ(2− α)

+
2γ−1∥u0∥γLs(Ω)t

α
0 t

−α

C

)
t−α +

2γ−1∥u0∥γLs(Ω)t
α
0

C(t+ 1)α

≥
(
−

2α−
α
γ αλ1∥u0∥Ls(Ω)

γt1−α
0

−
2α+1λ2∥u0∥Ls(Ω)

Γ(1− α)
−

2α+1αλ2∥u0∥Ls(Ω)

γΓ(2− α)

+
tα0C0

2

)
t−α +

tα0C0

2(t+ 1)α

≥ C0

(t+ 1)α

(3.18)

provided that we choose a large enough t0. If t0 ≤ t ≤ 2t0, then (3.17) can be reduced to

λ1∂tφ(t) + λ2∂
α
t0,tφ(t)

= −
2αλ1∥u0∥Ls(Ω)t

α
γ

0 t−
α
γ −1

γ
−

2αλ2∥u0∥Ls(Ω)t
α
γ

0 t−α−α
γ

γΓ(1− α)

∫ 1

t0
t

s−
α
γ −1

(1− s)α
ds

≥ −
2αλ1∥u0∥Ls(Ω)t

α
γ

0 t−
α
γ −1+αt−α

γ

−
2αλ2∥u0∥Ls(Ω)t

α
γ

0 t−α−α
γ

γΓ(1− α)

(
t0
t

)−α
γ −1 ∫ 1

t0
t

(1− s)−αds

≥ −
2αλ1∥u0∥Ls(Ω)t

−α

γtα−1
0

−
2αλ2∥u0∥Ls(Ω)t

−1
0 (t− t0)

1−α

γΓ(2− α)

≥ −
2αλ1∥u0∥Ls(Ω)t

−α

γtα−1
0

−
2αλ2∥u0∥Ls(Ω)t

−1
0 t1−α

γΓ(2− α)

≥
(
−

2αλ1∥u0∥Ls(Ω)

γtα−1
0

−
4αλ2∥u0∥Ls(Ω)

γΓ(2− α)

)
t−α,



EJDE-2025/89 DECAY ESTIMATES AND EXTINCTION PROPERTIES 9

which implies that

λ1∂tφ(t) + λ2∂
α
t0,tφ(t) +

φγ(t)

C

≥
(
−

2αλ1∥u0∥Ls(Ω)

γtα−1
0

−
4αλ2∥u0∥Ls(Ω)

γΓ(2− α)

)
t−α +

2γ∥u0∥γLs(Ω)t
α
0 t

−α

C

≥
(
−

2αλ1∥u0∥Ls(Ω)

γtα−1
0

−
4αλ2∥u0∥Ls(Ω)

γΓ(2− α)
+

2γ−1∥u0∥γLs(Ω)t
α
0

C

)
t−α +

2γ−1∥u0∥γLs(Ω)t
α
0

C(t+ 1)α

≥
(
−

2αλ1∥u0∥Ls(Ω)

γtα−1
0

−
4αλ2∥u0∥Ls(Ω)

γΓ(2− α)
+

tα0C0

2

)
t−α +

tα0C0

2(t+ 1)α

≥ C0

(t+ 1)α

(3.19)

provided that we choose a large enough t0. Therefore, combining (3.16)-(3.19), it can be concluded
that (3.15) holds for all t ≥ 0. Since ∥u0∥Ls(Ω) ≤ φ(0) by (3.14), combining inequalities (3.13) and
(3.15), and then applying the comparison principle (Lemma 2.7), we can deduce that ∥u(t)∥Ls(Ω) ≤
φ(t), which implies that

∥u(t)∥Ls(Ω) ≤
C#

1 + t
α
γ

for some C# > 0 and for all t ≥ 0. Thus, the estimate (3.4) is established. The proof is
complete. □

4. Applications of Theorem 3.2

In this section, we present three specific applications of Theorem 3.2 (see 4.1-4.3).

4.1. Space-fractional double nonlinear operator. The space-fractional double nonlinear op-
erator is defined (up to normalization factors) by

Nm,δ,p : u 7→ (−∆)δpu
m(x) = lim

ε→0+

∫
RN\Bε(x)

|um(x)− um(y)|p−2 (um(x)− um(y))

|x− y|N+δp
dy, (4.1)

where 0 < δ < 1, p > 1, m > 0, Bε(x) =
{
y ∈ RN : |x− y| < ε

}
and um is the m-th power of u.

Note that the multiplicative constant is also neglected in the definition of the operators below.

• When m = 1, the operator (4.1) is transformed into the fractional p-Laplacian [6]:

(−∆)δpu(x) := lim
ε→0+

∫
RN\Bε(x)

|u(x)− u(y)|p−2 (u(x)− u(y))

|x− y|N+δp
dy. (4.2)

• When p = 2, the operator (4.1) is transformed into the fractional porous medium operator
[12]:

(−∆)δum(x) := lim
ε→0+

∫
RN\Bε(x)

um(x)− um(y)

|x− y|N+2δ
dy. (4.3)

• Taking p = 2 in (4.2) or m = 1 in (4.3) can continue to be reduced to fractional Laplacian
[6].

In these settings, we have the following decay estimates.

Theorem 4.1. Let 0 < δ < 1, p > 1 and s ≥ 2. Suppose u is a weak nonnegative solution

of problem (1.1) with u ∈ W 1,2(0, T ;W δ,p
0 (Ω) ∩ Ls(Ω)). Further assume that the operator N in

problem (1.1) is defined by (4.1). Then, for any s ≥ 2, there exists a constant C# > 0 depending
on N , s, Ω and δ such that

∥u(t)∥Ls(Ω) ≤
C#

1 + t
α

mp−m
, t > 0. (4.4)



10 F. MENG, X.-F. ZHOU EJDE-2025/89

Proof. Let v := u
mp−m+s−1

p . We first prove that

|v(x, t)− v(y, t)|p ≤ C̃|um(x, t)− um(y, t)|p−2 (um(x, t)− um(y, t))
(
us−1(x, t)− us−1(y, t)

)
(4.5)

for some C̃ > 0. For this, we construct the auxiliary function

(1,+∞) ∋ ξ 7→ g(ξ) =

(
ξ

mp−m+s−1
p − 1

)p
(ξm − 1)

p−1
(ξs−1 − 1)

. (4.6)

Since mp−m+ s− 1 = m(p− 1) + (s− 1) > 0, it is not difficult to obtain

lim
ξ→1

g(ξ) = lim
ε→0

(
(1 + ε)

mp−m+s−1
p − 1

)p
((1 + ε)m − 1)

p−1
((1 + ε)s−1 − 1)

= lim
ε→0

( (mp−m+s−1)ε
p + o(ε)

)p
(mε+ o(ε))

p−1
((s− 1)ε+ o(ε))

=
(mp−m+ s− 1)p

p(mp)p−1(s− 1)

and

lim
ξ→+∞

g(ξ) = lim
ξ→+∞

(
1− 1

ξ
mp−m+s−1

p

)p
(
1− 1

ξm

)p−1(
1− 1

ξs−1

) = 1.

Thus, supξ∈(1,+∞) g(ξ) < +∞, then we may as well set

C̃ := sup
ξ∈(1,+∞)

g(ξ) < +∞. (4.7)

Obviously, when u(x, t) = u(y, t), and u(x, t) = 0 or u(y, t) = 0, the inequality (4.5) holds. When
u(x, t) ̸= u(y, t) ̸= 0, let u(x, t) > u(y, t) without loss of generality, then

g
(u(x, t)
u(y, t)

)
=

((
u(x,t)
u(y,t)

)mp−m+s−1
p − 1

)p

((
u(x,t)
u(y,t)

)m

− 1
)p−1((

u(x,t)
u(y,t)

)s−1

− 1
) (4.8)

=

(
u

mp−m+s−1
p (x, t)− u

mp−m+s−1
p (y, t)

)p

(um(x, t)− um(y, t))
p−1

(us−1(x, t)− us−1(y, t))
. (4.9)

Combining (4.7), (4.8) and v = u
mp−m+s−1

p , we can conclude that (4.5) holds.
Next, we proceed to prove that there exists C ′ > 0 such that

∥v(t)∥pLq(Ω) ≤ C ′
∫∫

R2N

|v(x, t)− v(y, t)|p

|x− y|N+δp
dxdy (4.10)

holds for all q ∈
[
1, Np

N−δp

]
when p ∈ (1, N

δ ) and for any q ∈ [1,+∞] when p ∈ [Nδ ,+∞). In fact,

if p ∈ (1, N
δ ), then (4.10) can be obtained by Lemma 2.4 and Hölder’s inequality. If p ∈ [Nδ ,+∞),

let a > max{p, q}, then 0 < N
p − N

a < N
p < δ < 1. For δ̂ ∈ (Np − N

a ,
N
p ) ⊂ (0, 1), using Lemma 2.4

again, we obtain

∥v(t)∥p
L

Np

N−δ̂p (Ω)

≤ C1

∫∫
R2N

|v(x, t)− v(y, t)|p

|x− y|N+δ̂p
dxdy. (4.11)

Since δ̂ > N
p − N

a , that is, a < Np

N−δ̂p
, then by using Hölder’s inequality and (4.11), we obtain that

∥v(t)∥pLa(Ω) ≤ |Ω|
Np−a(N−δ̂p)

aN ∥v(t)∥p
L

Np

N−δ̂p (Ω)

≤ C2

∫∫
R2N

|v(x, t)− v(y, t)|p

|x− y|N+δ̂p
dxdy



EJDE-2025/89 DECAY ESTIMATES AND EXTINCTION PROPERTIES 11

for some constant C2 > 0. We choose an appropriate r > 0, since v is zero outside Ω and p < a,
it follows that

∥v(t)∥pLa(Ω) ≤ C2

∫∫
R2N

|v(x, t)− v(y, t)|p

|x− y|N+δ̂p
dxdy

= C2

∫
B(x,r)

∫
B(x,r)

|v(x, t)− v(y, t)|p

|x− y|N+δ̂p
dxdy

+ C2

∫
Bc(x,r)

∫
Bc(x,r)

|v(x, t)− v(y, t)|p

|x− y|N+δ̂p
dxdy

≤ C2

∫
B(x,r)

∫
B(x,r)

|x− y|p(δ−δ̂) |v(x, t)− v(y, t)|p

|x− y|N+δp
dxdy

+ C ′
2

∫
Bc(x,r)

∫
Bc(x,r)

|v(x, t)|p

|x− y|N+δ̂p
dxdy

≤ C2r
p(δ−δ̂)

∫∫
R2N

|v(x, t)− v(y, t)|p

|x− y|N+δp
dxdy

+ C ′
2

∫
RN

|v(x, t)|pdx
∫
Bc(x,r)

1

|x− y|N+δ̂p
dy

= C2r
p(δ−δ̂)

∫∫
R2N

|v(x, t)− v(y, t)|p

|x− y|N+δp
dxdy

+ C ′
2

∫
Ω

|v(x, t)|pdx
∫ +∞

r

∫
∂B(x,τ)

1

τN+δ̂p
dSydτ

≤ C2r
p(δ−δ̂)

∫∫
R2N

|v(x, t)− v(y, t)|p

|x− y|N+δp
dxdy

+ C ′′
2 ∥v(t)∥

p
La(Ω) lim

h→+∞

∫ h

r

1

τ1+δ̂p
dτ

= C2r
p(δ−δ̂)

∫∫
R2N

|v(x, t)− v(y, t)|p

|x− y|N+δp
dxdy +

C ′′
2

(δ̂p)rδ̂p
∥v(t)∥pLa(Ω),

where B(x, r) := {y ∈ RN : |x − y| ≤ r}, Bc(x, r) := {y ∈ RN : |x − y| > r} and C ′
2, C

′′
2 are

appropriate positive constants. Therefore,

∥v(t)∥pLa(Ω) ≤
C2(δ̂p)r

δp

(δ̂p)rδ̂p − C ′′
2

∫∫
R2N

|v(x, t)− v(y, t)|p

|x− y|N+δp
dxdy (4.12)

provided that (δ̂p)rδ̂p > C ′′
2 . Since q < a, by using Hölder’s inequality, we obtain

∥v(t)∥pLq(Ω) ≤ |Ω|
a−q
a ∥v(t)∥pLa(Ω). (4.13)

Combining (4.12) and (4.13), we obtain

∥v(t)∥pLq(Ω) ≤ C ′
∫∫

R2N

|v(x, t)− v(y, t)|p

|x− y|N+δp
dxdy,



12 F. MENG, X.-F. ZHOU EJDE-2025/89

where C ′ := |Ω|
a−q
a C2(δ̂p)r

δp

(δ̂p)rδ̂p−C′′
2

, which implies that (4.10) is obtained. It is known that u is zero

outside Ω, then substituting v = u
mp−m+s−1

p and (4.5) into (4.10) yields(∫
Ω

|u(x, t)|
q(mp−m+s−1)

p dx
)p/q

≤ C̃C ′
∫∫

R2N

|um(x, t)− um(y, t)|p−2 (um(x, t)− um(y, t))

×
(
us−1(x, t)− us−1(y, t)

) dxdy

|x− y|N+δp

= 2C̃C ′
∫∫

R2N

|um(x, t)− um(y, t)|p−2 (um(x, t)− um(y, t))us−1(x, t)

|x− y|N+δp
dxdy

= C3

∫
Ω

us−1(x, t)(−∆)δpu
m(x, t)dx

(4.14)

for some C3 > 0. When p ∈ (1, N
δ ), it is not difficult to verify sp

mp−m+s−1 ∈ [1, Np
N−δp ] if s ≥ η,

where η := max{mp−m−1
p−1 , N(m−mp+1)

δp }. Therefore, if η ≤ 2, then for all s ≥ 2 or p ≥ N
δ , we can

choose q := sp
mp−m+s−1 , so that (4.14) is reduced to

∥u(t)∥s−1+mp−m
Ls(Ω) ≤ C3

∫
Ω

us−1(x, t)(−∆)δpu
m(x, t)dx, (4.15)

which implies that (3.2) in Theorem 3.2 is satisfied when γ := mp−m. Substituting γ = mp−m
into (3.4) of Theorem 3.2 leads to the conclusion that

∥u(t)∥Ls(Ω) ≤
C#

1 + t
α

mp−m
, t ≥ 0. (4.16)

If η > 2, then for s ≥ η or p ≥ N
δ , we can also choose q := sp

mp−m+s−1 such that the inequality

(4.15) holds. Thus, the estimate (4.16) can also be obtained. In addition, for 2 ≤ s < η, using
Hölder’s inequality, we can obtain ∥u(t)∥Ls(Ω) ≤ C4∥u(t)∥Lη(Ω). Since η ≥ η, then (4.15) is also
satisfied. In summary, we conclude that (4.16) holds for all s ≥ 2. The proof is complete. □

As special cases of Theorem 4.1, we can take m = 1 or p = 2, which correspond to the case
of fractional p-Laplacian defined in (4.2) and the case of fractional porous media defined in (4.3),
respectively. For the convenience of readers, we state these results as follows:

Corollary 4.2. Let 0 < δ < 1, p > 1 and s ≥ 2. Suppose u is a nonnegative solution of problem

(1.1) with u ∈ W 1,2(0, T ;W δ,p
0 (Ω)∩Ls(Ω)). Further assume that the operator N in problem (1.1)

is defined by (4.2). Then, for any s ≥ 2, there exists C# > 0 that may depend on N , s, Ω and δ
such that

∥u(t)∥Ls(Ω) ≤
C#

1 + t
α

p−1
, t ≥ 0.

Corollary 4.3. Let 0 < δ < 1, p > 1 and s ≥ 2. Suppose u is a nonnegative solution of problem

(1.1) with u ∈ W 1,2(0, T ;W δ,p
0 (Ω)∩Ls(Ω)). Further assume that the operator N in problem (1.1)

is defined by (4.3). Then, for any s ≥ 2, there exists C# > 0 that may depend on N , s, Ω and δ
such that

∥u(t)∥Ls(Ω) ≤
C#

1 + t
α
m
, t ≥ 0.

Next, we consider the more complex setting of the fractional double nonlinear operator.

4.2. Sum of space-fractional double nonlinear operators in different directions. For a
fixed i ∈ {1, 2, . . . , N}, let ei (the i-th element of the Euclidean basis {e1, . . . , eN} of RN ) be a
unit vector, representing different directions, then the fractional double nonlinear operator defined
in the ei direction is expressed as(

−∂2
xi

)δi
pi
um(x) :=

∫
R

|um(x)− um(x+ κei)|pi−2 (um(x)− um(x+ κei))

κ1+δipi

i

dκi, (4.17)



EJDE-2025/89 DECAY ESTIMATES AND EXTINCTION PROPERTIES 13

where pi > 1, 0 < δi < 1, m > 0, and um is the m-th power of u. Given β1, β2, . . . , βN > 0,
multiplying both sides of equation (4.17) by βi and summing over i from 1 to N , we obtain the
operator N that we are about to study, namely

Nm,δ,p,β : u 7→ (−∆β)
δ
p u

m(x) =

N∑
i=1

βi

(
−∂2

xi

)δi
pi
um(x), (4.18)

where β = (β1, . . . , βN ), δ = (δ1, . . . , δN ) and p = (p1, . . . pN ).
In our above settings, there is a decay estimate.

Theorem 4.4. Let m > 0 and s ≥ 2. Suppose u is a nonnegative weak solution of (1.1). And
assume that the operator N in (1.1) is defined by (4.18). Then, for any s ≥ 2, there exists a
constant C# > 0 depending on N , s, δ, Ω and β such that

∥u(t)∥Ls(Ω) ≤
C#

1 + t
α

mp⋆−m
, t ≥ 0, (4.19)

where p⋆ is the one pi that minimizes ∥u(t)∥s−1+mpi−m
Ls(Ω) , i ∈ {1, 2, . . . , N}.

Proof. Let (κ1, κ2, . . . κi−1, κi+1, . . . , κN ) ∈ RN−1, i ∈ {1, 2, . . . , N}. Given a point x ∈ RN , we
use the notation x = (κ1, . . . , κN ) = κ1e1 + · · ·+ κNeN with κi ∈ R. We define the space

Ωi(κ1, κ2, . . . κi−1, κi+1, . . . , κN )

=
(
Ω ∩ {(κ1, κ2, . . . κi−1, 0, κi+1, . . . , κN ) + cei, c ∈ R}

)
i
⊂ R,

where Ω = (Ω1, . . . ,ΩN ) ⊂ RN . Further define the function

R ∋ κi 7→ u(κ1e1 + κ2e2 + · · ·+ κNeN , t) ∈ Ωi(κ1, κ2, . . . κi−1, κi+1, . . . , κN ).

It is known that u is zero outside Ωi, then by using the estimate (4.15) we obtain(∫
R
us(κ1e1 + κ2e2 + · · ·+ κNeN , t)dκi

) s−1+mpi−m

s

=
(∫

Ωi(κ1,κ2,...κi−1,κi+1,...,κN )

us(κ1e1 + κ2e2 + · · ·+ κNeN , t)dκi

) s−1+mpi−m

s

≤ C ′
5

∫
R
us−1(κ1e1 + κ2e2 + · · ·+ κNeN , t)

× (−∂2
xi
)δipi

um(κ1e1 + κ2e2 + · · ·+ κNeN , t)dκi

for some C ′
5 > 0. We integrate the above formula separately at coordinates (κ1, κ2, . . . κi−1, κi+1, . . . , κN )

to obtain (∫
RN

us(κ1e1 + κ2e2 + · · ·+ κNeN , t)dκ1dκ2 . . . dκN

) s−1+mpi−m

s

=
(∫

Ω

us(κ1e1 + κ2e2 + · · ·+ κNeN , t)dκ1dκ2 . . . dκN

) s−1+mpi−m

s

≤ C5

∫
RN

us−1(κ1e1 + κ2e2 + · · ·+ κNeN , t)

× (−∂2
xi
)δipi

um(κ1e1 + κ2e2 + · · ·+ κNeN , t)dκ1dκ2 . . . dκN ,

which implies that

∥u(t)∥s−1+mpi−m
Ls(Ω) =

(∫
Ω

us(x, t)dx
) s−1+mpi−m

s ≤ C5

∫
Ω

us−1(x, t)(−∂2
xi
)δipi

um(x, t)dx. (4.20)

Let ∥u(t)∥s−1+mp⋆−m
Ls(Ω) := min

{
∥u(t)∥s−1+mp1−m

Ls(Ω) , . . . , ∥u(t)∥s−1+mpN−m
Ls(Ω)

}
, which means that p⋆ is

the one pi that minimizes ∥u(t)∥s−1+mpi−m
Ls(Ω) , i ∈ {1, 2, . . . , N}. Thus, (4.20) can be reduced to

∥u(t)∥s−1+mp⋆−m
Ls(Ω) ≤ C5

∫
Ω

us−1(x, t)(−∂2
xi
)δipi

um(x, t)dx. (4.21)



14 F. MENG, X.-F. ZHOU EJDE-2025/89

By multiplying βi on both sides of (4.21) and summing i from 1 to N , and then combining (4.18)
yields

∥u(t)∥s−1+mp⋆−m
Ls(Ω) ≤ C6

∫
Ω

us−1(x, t)(−∆β)
δ
pu

m(x, t)dx

for some C6 > 0. This implies that the inequality (3.2) in Theorem 3.2 is satisfied when γ :=
mp⋆ − m, then the estimation (4.19) is established by substituting γ = mp⋆ − m into (3.4) of
Theorem 3.2. □

4.3. Fractional p-Kirchhoff operator. The function M : R+
0 → R+

0 in Kirchhoff operator is
continuous and nondecreasing. A typical example is

M(η) = M0 + kη, (4.22)

where M0 ≥ 0 and k > 0. Next, we will consider the cases where M is degenerate (M0 = 0 in
(4.22)) and where M is non-degenerate (M(0) > 0 in (4.22)).

Let 0 < δ < 1 and p > 1, the definition of the fractional p-Kirchhoff operator is given (up to
normalization factors) as

Nδ,p : u 7→ lim
ε→0+

M
(
[u]p

W δ,p(RN )

)∫
RN\Bε(x)

|u(x)− u(y)|p−2 (u(x)− u(y))

|x− y|N+δp
dy, (4.23)

where Bε(x) =
{
y ∈ RN : |x− y| < ε

}
, the definition of [u]W δ,p(RN ) is shown in (2.2) and the

definition of space W δ,p(RN ) is shown in (2.1).
In these settings, we have the following results.

Theorem 4.5. Let 0 < δ < 1, p > 1 and s ≥ 2. Suppose u is a weak solution of (1.1). Further
assume that the operator N in (1.1) is defined by (4.23). Then, we have the following statements:

(i) If the function M(·) is non-degenerate, then for any s ≥ 2, there exists a constant C# > 0
depending on N , p, δ, Ω and m0 such that

∥u(t)∥Ls(Ω) ≤
C#

1 + t
α

p−1
, t ≥ 0. (4.24)

(ii) If the function M(·) is degenerate, then for any s ≥ 2 when N ≤ 2δp, or for every

s ≤ N(2p−2)
N−2δp when N > 2δp, there exists a constant C# > 0 depending on N , p, δ and Ω

such that

∥u(t)∥Ls(Ω) ≤
C#

1 + t
α

2p−1
, t ≥ 0. (4.25)

Proof. (i) Firstly, we consider the case that M(·) is non-degenerate. Since M(·) is non-degenerate,
M(·) has a positive minimum, that is, there exists m0 > 0 such that

m0 = inf
η∈R+

0

M(η). (4.26)

Thus, ∫
Ω

|u(x, t)|s−2u(x, t)M
(
[u]p

W δ,p(RN )

)
(−∆)

δ
p u(x, t)dx

≥ m0

∫
Ω

|u(x, t)|s−2u(x, t) (−∆)
δ
p u(x, t)dx.

(4.27)

Letting v := |u|
p−2+s

p , we can prove that

|v(x, t)− v(y, t)|p

≤ C̃|u(x, t)− u(y, t)|p−2 (u(x, t)− u(y, t))
(
|u(x, t)|s−2u(x, t)− |u(y, t)|s−2u(y, t)

)
.

(4.28)

We construct the auxiliary function

(−1, 1) ∋ ζ 7→ g(ζ) =

(
1− |ζ|

p−2+s
p

)p
(1− |ζ|)p−2

(1− ζ) (1− |ζ|s−2ζ)
.



EJDE-2025/89 DECAY ESTIMATES AND EXTINCTION PROPERTIES 15

Note that p− 2 + s > p− 1 > 0, so there is

lim
ζ→1

g(ζ) = lim
ε→0

(
1− (1− ε)

p−2+s
p

)p

(1− (1− ε))
p−2

(1− (1− ε)) (1− (1− ε)s−1)

= lim
ε→0

( (p−2+s)ε
p + o(ε)

)p
(ε+ o(ε))

p−1
((q − 1)ε+ o(ε))

=
(p− 2 + s)p

pp(s− 1)
,

lim
ζ→−1

g(ζ) = lim
ε→0

(
1− (1− ε)

p−2+s
p

)p

4 (1− (1− ε))
p−2 = lim

ε→0

(
(p−2+s)

p

)p

εp

4εp−2
= 0.

Therefore, we can take

C̃ := sup
ζ∈(−1,1)

g(ζ) < +∞. (4.29)

Obviously, when u(x, t) = u(y, t), and u(x, t) = 0 or u(y, t) = 0, the inequality (4.28) holds. When
u(x, t) ̸= u(y, t) ̸= 0, let |u(x, t)| > |u(y, t)| without loss of generality, then

g
(u(y, t)
u(x, t)

)

=

(
1− |u(y,t)u(x,t) |

p−2+s
p

)p

(
1− |u(y,t)u(x,t) |

)p−2(
1− u(y,t)

u(x,t)

)(
1− |u(y,t)u(x,t) |s−2 u(y,t)

u(x,t)

)
=

(
|u(x, t)|

p−2+s
p − |u(y, t)|

p−2+s
p

)p

(|u(x, t)| − |u(y, t)|)p−2
(u(x, t)− u(y, t)) (|u(x, t)|s−2u(x, t)− |u(y, t)|s−2u(y, t))

.

(4.30)

Combining equations (4.29) and (4.30) with v = |u|
p−2+s

p , the inequality (4.28) can be obtained.
Since it has been proved at (4.10) that

∥v(t)∥pLq(Ω) ≤ C ′
∫∫

R2N

|v(x, t)− v(y, t)|p

|x− y|N+δp
dxdy

holds for all q ∈
[
1, Np

N−δp

]
when p ∈ (1, N

δ ) and for any q ∈ [1,+∞] when p ∈ [Nδ ,+∞). Therefore,

it can be inferred from v = |u|
p−2+s

p , (4.10) and (4.28) that(∫
Ω

|u(x, t)|
q(p+s−2)

p dx
)p/q

≤ C̃C ′
∫∫

R2N

|u(x, t)− u(y, t)|p−2

× (u(x, t)− u(y, t))
(
|u(x, t)|s−2u(x, t)− |u(y, t)|s−2u(y, t)

) dxdy

|x− y|N+δp

= 2C̃C ′
∫∫

R2N

|u(x, t)− u(y, t)|p−2 (u(x, t)− u(y, t)) |u(x, t)|s−2u(x, t)

|x− y|N+δp
dxdy

= 2C̃C ′
∫
Ω

|u(x, t)|s−2u(x, t)(−∆)δpu(x, t)dx.

(4.31)

When p ∈ (1, N
δ ), it is not difficult to verify sp

p+s−2 ≥ 1, and sp
p+s−2 ≤ Np

N−δp if s ≥ N(2−p)
δp .

Therefore, if N(2−p)
δp ≤ 2, then for all s ≥ 2 or p ≥ N

δ , we can choose q := sp
p+s−2 and substitute it

into inequality (4.31) to obtain that

∥u(t)∥s+p−2
Ls(Ω) ≤ 2C̃C ′

∫
Ω

|u(x, t)|s−2u(x, t)(−∆)δpu(x, t)dx. (4.32)



16 F. MENG, X.-F. ZHOU EJDE-2025/89

Combining (4.27) and (4.32), it can be concluded that

∥u(t)∥s+p−2
Ls(Ω) ≤ 2C̃C ′

m0

∫
Ω

|u(x, t)|s−2u(x, t)M
(
[u]p

W δ,p(RN )

)
(−∆)

δ
p u(x, t)dx, (4.33)

which implies that the condition (3.2) in Theorem 3.2 is satisfied when γ := p − 1. Substituting

γ = p− 1 into (3.4) of Theorem 3.2 yields the estimate (4.24). If N(2−p)
δp > 2, then for s ≥ N(2−p)

δp

or p ≥ N
δ , we can also choose q := sp

p+s−2 such that (4.33) holds. Thus, the estimate (4.24) can

also be obtained. In addition, for 2 ≤ s < N(2−p)
δp , by using Hölder’s inequality, we can obtain

∥u(t)∥Ls(Ω) ≤ C ′
4∥u(t)∥

L
N(2−p)

δp (Ω)
. Since N(2−p)

δp ≥ N(2−p)
δp , it follows that (4.33) is also satisfied.

In summary, we conclude that (4.24) holds for all s ≥ 2.

(ii) Next, we consider that M(·) is a degenerate case. Let v := |u|
2p+s−2

2p , we need to prove that

|v(x, t)− v(y, t)|p

≤ C̃|u(x, t)− u(y, t)|p−1
√
(u(x, t)− u(y, t))(|u(x, t)|s−2u(x, t)− |u(y, t)|s−2u(y, t))

(4.34)

for some C̃ > 0. For this, we construct the auxiliary function

(−1, 1) ∋ ς 7→ g(ς) =

(
1− |ς|

2p+s−2
2p

)2p
(1− |ς|)2p−2

(1− ς) (1− |ς|s−2ς)
.

Similar to the proof of inequality (4.28), the inequality (4.34) can be obtained (this proof is omitted
here). Note that inequality (4.10) is known as

∥v(t)∥pLq(Ω) ≤ C ′
∫∫

R2N

|v(x, t)− v(y, t)|p

|x− y|N+δp
dxdy

which holds for all q ∈ [1, Np
N−δp ] when p ∈ (1, N

δ ) and for any q ∈ [1,+∞] when p ∈ [Nδ ,+∞).

Combining (4.10) with (4.34) yields(∫
Ω

|u(x, t)|
q(2p+s−2)

2p dx
)p/q

≤ C ′′
∫∫

R2N

|u(x, t)− u(y, t)|p−1 (u(x, t)− u(y, t))
1/2

×
(
|u(x, t)|s−2u(x, t)− |u(y, t)|s−2u(y, t)

)1/2 dxdy

|x− y|N+δp
.

(4.35)

When p ∈ (1, N
δ ), if either N ≤ 2δp, s ≥ 2 or N > 2δp, s ≤ N(2p−2)

N−2δp holds, then it is not difficult to

prove that 2ps
2p+s−2 ∈

[
1, Np

N−δp

]
. Therefore, for s ≤ N(2p−2)

N−2δp or p ≥ N
δ , we can choose q := 2ps

2p+s−2

such that (4.35) is reduced to(∫
Ω

|u(x, t)|sdx
) 2p+s−2

2s

≤ C ′′
∫∫

R2N

|u(x, t)− u(y, t)|p−1 (u(x, t)− u(y, t))
1/2

×
(
|u(x, t)|s−2u(x, t)− |u(y, t)|s−2u(y, t)

)1/2 dxdy

|x− y|N+δp

≤ C ′′′
(∫∫

R2N

|u(x, t)− u(y, t)|p

|x− y|N+δp
dxdy

)1/2(∫∫
R2N

|u(x, t)− u(y, t)|p−2

× (u(x, t)− u(y, t))
(
|u(x, t)|s−2u(x, t)− |u(y, t)|s−2u(y, t)

) dxdy

|x− y|N+δp

)1/2

,

(4.36)

thanks to Hölder’s inequality. Since M(·) defined in (4.22) is degenerate, it follows that

M
(
[u]p

W δ,p(RN )

)
= k[u]p

W δ,p(RN )
= k

∫∫
R2N

|u(x, t)− u(y, t)|p

|x− y|N+δp
dxdy. (4.37)



EJDE-2025/89 DECAY ESTIMATES AND EXTINCTION PROPERTIES 17

It is known that u is zero outside Ω. Substituting (4.37) into (4.36) and squared on both sides,
we obtain

∥u(t)∥2p+s−2
Ls(Ω) ≤ 2(C ′′′)2

k
M

(
[u]p

W δ,p(RN )

)
×
∫∫

R2N

|u(x, t)− u(y, t)|p−2(u(x, t)− u(y, t))|u(x, t)|s−2u(x, t)

|x− y|N+δp
dxdy

= C7

∫
Ω

|u(x, t)|s−2u(x, t)M
(
[u]p

W δ,p(RN )

)
(−∆)

δ
p u(x, t)dx

for some C7 > 0. This implies that (3.2) in Theorem 3.2 is satisfied when γ := 2p−1. Substituting
γ = 2p− 1 into (3.4) of Theorem 3.2 yields the decay estimate (4.25). □

5. Finite time extinction

In this section, we discuss the finite time extinction properties of weak solutions of problem

(1.1) when N [u] := M
(
[u]W δ,1(RN )

)
(−∆)

δ
1 u and f = 0, as follows

λ1∂tu(x, t) + λ2∂
α
0,tu(x, t) +M

(
[u]W δ,1(RN )

)
(−∆)δ1u(x, t) = 0 in Ω× R+,

u(x, t) = 0 in (RN \ Ω)× R+,

u(x, 0) = u0(x) in Ω,

(5.1)

where M
(
[u]W δ,1(RN )

)
(−∆)

δ
1 u is the fractional 1-Kirchhoff operator (defined in (5.2) below)

In Theorem 4.5, we proved that for any p ∈ (1,+∞), the solution of the mixed time-fractional

nonlocal p-Kirchhoff type parabolic equation decays with behavior of t
−α
p−1 and t

−α
2p−1 . Now, we

show that this behavior does not occur when p = 1, but vanishes in finite time, which is defined
as follows.

Definition 5.1. Let u be a weak solution of (1.1). We say u(x, t) vanishes in finite time if there
exists a constant T > 0 such that u(x, t) ≡ 0 in Ω for t ≥ T .

For p ∈ (1,+∞), the p-Kirchhoff operator has been defined in (4.23). Now let us turn our
attention to the case of p = 1. Formally, the fractional 1-Laplacian operator of order δ ∈ (0, 1) of
a function u ∈ W δ,1(RN ) is defined as

M
(
[u]W δ,1(RN )

)
(−△)

δ
1 u := lim

ε→0+
M

(
[u]W δ,1(RN )

) ∫
RN\Bε(x)

1

|x− y|N+δ

u(x)− u(y)

|u(x)− u(y)|
dy, (5.2)

where Bε(x) =
{
y ∈ RN : |x− y| < ε

}
and the definition of [u]W δ,1(RN ) is shown in (2.2). Note

that in this formula one has to give a meaning to u(x)−u(y)
|u(x)−u(y)| when u(x) = u(y). To solve this

difficulty, we follow the idea of studying similar problems in [1]. More specifically, we replace
u(x)−u(y)
|u(x)−u(y)| by an antisymmetric L∞-function ρ(x, y) that satisfies ∥ρ(·, ·)∥L∞(RN×RN ) ≤ 1 such

that
ρ(x, y) ∈ sign(u(x)− u(y)) a.e. (x, y) ∈ RN × RN ,

where sign(ξ) is the multivalued sign of ξ. With this setting, we can give the definition of the
weak solution of problem (5.1) as follows.

Definition 5.2. Let 0 < δ < 1, u(·, 0) = u0(·) and u0 ∈ L2(Ω). We say that the function

u ∈ W 1,2(0, T ;W δ,1
0 (Ω) ∩ L2(Ω)) is a weak solution of (5.1), if kα ∗ [u − u0] ∈ W 1,2

0 (0, T ;L2(Ω)),
and for almost all t ∈ (0, T ) there exists ρ(·, ·, t) ∈ L∞(RN ×RN ), ρ(x, y, t) = −ρ(y, x, t) for almost
all (x, y) ∈ RN × RN , ∥ρ(·, ·, t)∥L∞(RN×RN ) ≤ 1 such that

ρ(x, y, t) ∈ sign(u(x, t)− u(y, t)) a.e. (x, y, t) ∈ RN × RN × R+

and

λ1

∫
Ω

φ(x, t)∂tu(x, t)dx+ λ2

∫
Ω

φ(x, t)∂α
0,tu(x, t)dx

+M
(
[u]W δ,1(RN )

) ∫∫
R2N

(φ(x, t)− φ(y, t))

|x− y|N+δ
ρ(x, y, t)dxdy = 0

(5.3)



18 F. MENG, X.-F. ZHOU EJDE-2025/89

for all φ(t) ∈ W δ,1
0 (Ω) ∩ L2(Ω).

The main result of this section is as follows.

Theorem 5.3. Let N ≥ 2, 0 < α, δ < 1, and C⋆,m0 > 0. Assume that the function M : R+
0 → R+

0

is non-degenerate and u0 ∈ L
N
δ (Ω)\{0}. If the weak solution u of problem (5.1) in the sense of

Definition 5.2 is globally bounded, and

∥u0∥
L

N
δ (Ω)

≤ Y (0) ≤ m0

2C⋆

(λα
1Γ(2− α)

λ2

) 1
1−α

, (5.4)

where Y (·) denotes the supersolution of the equation λ1∂tg(t)+λ2∂
α
0,tg(t) = −m0

C⋆
, then u vanishes

in finite time.

Proof. For any q ≥ 2, taking φ(t) := |u(t)|q−2u(t) in (5.3) yields

λ1

∫
Ω

|u(x, t)|q−2u(x, t)∂tu(x, t)dx+ λ2

∫
Ω

|u(x, t)|q−2u(x, t)∂α
0,tu(x, t)dx

+M
(
[u]W δ,1(RN )

) ∫∫
R2N

(
|u(x, t)|q−2u(x, t)− |u(y, t)|q−2u(y, t)

)
|x− y|N+δ

ρ(x, y, t)dxdy = 0.

(5.5)

From equation (3.7) when s := q, we obtain∫
Ω

|u(x, t)|q−2u(x, t)∂tu(x, t)dx = ∥u(t)∥q−1
Lq(Ω)∂t∥u(t)∥Lq(Ω). (5.6)

Furthermore, from inequality (3.10) when s := q, we obtain∫
Ω

|u(x, t)|q−2u(x, t)∂α
0,tu(x, t)dx ≥ ∥u(t)∥q−1

Lq(Ω)∂
α
0,t∥u(t)∥Lq(Ω). (5.7)

Since M(·) is non-degenerate, then it can be inferred from (4.26) that

M
(
[u]W δ,1(RN )

)
≥ m0. (5.8)

It follows from

||m|q−2m− |n|q−2n| = sign(m− n)
(
|m|q−2m− |n|q−2n

)
, ∀m,n ∈ R

and

ρ(x, y, t) ∈ sign(u(x, t)− u(y, t)) a.e. (x, y, t) ∈ RN × RN × R+

that ∫∫
R2N

(
|u(x, t)|q−2u(x, t)− |u(y, t)|q−2u(y, t)

)
|x− y|N+δ

ρ(x, y, t)dxdy

=

∫∫
R2N

||u(x, t)|q−2u(x, t)− |u(y, t)|q−2u(y, t)|
|x− y|N+δ

dxdy.

(5.9)

Since p = 1, 0 < δ < 1, it follows that δp < N . Applying Lemma 2.4 to (5.9), we obtain that∫∫
R2N

||u(x, t)|q−2u(x, t)− |u(y, t)|q−2u(y, t)|
|x− y|N+δ

dxdy ≥ 1

C⋆

(∫
Ω

|u(t)|
N(q−1)
N−δ dx

)N−δ
N

. (5.10)

Substituting (5.6)-(5.10) into (5.5), it can be concluded that

λ1∥u(t)∥q−1
Lq(Ω)∂t∥u(t)∥Lq(Ω) + λ2∥u(t)∥q−1

Lq(Ω)∂
α
0,t∥u(t)∥Lq(Ω) +

m0

C⋆
∥u(t)∥q−1

L
N(q−1)
N−δ (Ω)

≤ 0. (5.11)

Since N ≥ 2, we have N
δ ≥ 2. Let q := N

δ , then (5.11) can be reduced to

λ1∂t∥u(t)∥
L

N
δ (Ω)

+ λ2∂
α
0,t∥u(t)∥LN

δ (Ω)
+

m0

C⋆
≤ 0. (5.12)

Now we define an energy functional

y(t) := ∥u(t)∥
L

N
δ (Ω)

.



EJDE-2025/89 DECAY ESTIMATES AND EXTINCTION PROPERTIES 19

This means that (5.12) can be expressed as

λ1
dy(t)

dt
+ λ2∂

α
0,ty(t) +

m0

C⋆
≤ 0, t ≥ 0. (5.13)

Next, we exhibit a supersolution Y (t) of the equation λ1
dg(t)
dt + λ2∂

α
0,tg(t) = −m0

C⋆
, namely

λ1
dY (t)

dt
+ λ2∂

α
0,tY (t) +

m0

C⋆
≥ 0, t ≥ 0. (5.14)

To achieve this goal, we first find a function Y (t) satisfying

λ1
dY (t)

dt
+

m0

2C⋆
= 0, (5.15)

It is not difficult to conclude that Y (t) satisfying equation (5.15) is

Y (t) =
(
Y (0)− m0

2λ1C∗
t
)
> 0, 0 < t < T,

Y (t) ≡ 0, t ≥ T,
(5.16)

where T = 2λ1C⋆Y (0)
m0

. For 0 < t < T , since

∂α
0,tY (t) = − m0t

1−α

2λ1C⋆Γ(2− α)
≥ − m0T

1−α

2λ1C⋆Γ(2− α)
= − mα

0Y
1−α(0)

(2λ1C⋆)αΓ(2− α)
. (5.17)

Combining (5.15) and (5.17) yields

λ1
dY (t)

dt
+ λ2∂

α
0,tY (t) +

m0

C⋆
= λ2∂

α
0,tY (t) +

m0

2C⋆

≥ − mα
0Y

1−α(0)

(2λ1C⋆)αΓ(2− α)
+

m0

2C⋆
≥ 0

thanks to Y (0) ≤ m0

2C⋆

(λα
1 Γ(2−α)

λ2

) 1
1−α , which implies that (5.14) holds. For t ≥ T , there is Y (t) ≡ 0,

obviously (5.14) also holds. Since ∥u0∥
L

N
δ (Ω)

≤ Y (0), combining (5.13) and (5.14), and applying

the comparison principle (Lemma 2.7), we can obtain that

∥u(t)∥
L

N
δ (Ω)

≤ Y (0)− m0

2λ1C∗
t, 0 < t < T,

∥u(t)∥
L

N
δ (Ω)

≡ 0, t ≥ T.

Therefore, the weak solution u vanishes in finite time. This proof is complete. □

6. Conclusion

This paper mainly studied the decay estimates and extinction properties of weak solutions for
some parabolic equations with mixed time-derivatives. By using the energy method and a new
comparison principle, the power-law decay estimates of weak solutions for the abstract parabolic
problem (1.1) were obtained. In addition, we also provided three specific applications of the decay
results of problem (1.1). For the p-Kirchhoff problem with mixed time-derivatives, when p > 1,

the weak solution decays according to the behavior of t
−α
p−1 and t

−α
2p−1 (see Theorem 4.5). However,

when p = 1, the weak solution no longer decays but vanishes in finite time. Therefore, at the
end of the article, the finite time extinction property of the weak solution of the 1-Kirchhoff type
parabolic problem with mixed time-derivatives was also obtained.

Acknowledgements. This work was supported by the National Natural Science Foundation of
China (11471015) and by the Postdoctoral Scientific Research Project for Anhui Jianzhu University
(2024QDHZ04).



20 F. MENG, X.-F. ZHOU EJDE-2025/89

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Fanmeng Meng

School of Mathematical Sciences, Anhui University, Hefei 230601, China
Email address: mengfanmeng96@163.com



EJDE-2025/89 DECAY ESTIMATES AND EXTINCTION PROPERTIES 21

Xian-Feng Zhou (corresponding author)

School of Mathematical Sciences, Anhui University, Hefei 230601, China

Email address: zhouxf@ahu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Decay estimates for problem (1.1)
	4. Applications of Theorem 3.2
	4.1. Space-fractional double nonlinear operator
	4.2. Sum of space-fractional double nonlinear operators in different directions
	4.3. Fractional p-Kirchhoff operator

	5. Finite time extinction
	6. Conclusion
	Acknowledgements

	References