Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 29, pp. 1–11.

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

EXTINCTION IN FINITE TIME OF SOLUTIONS TO

FRACTIONAL PARABOLIC POROUS MEDIUM EQUATIONS

WITH STRONG ABSORPTION

NGUYEN ANH DAO

Dedicated to Prof. Jesus Ildefonso Dı́az on his 70th birthday

Abstract. In this article we study the solutions of a general fractional par-

abolic porous medium equation with a non-Lipschitz absorption term. We

obtain the existence of weak solutions, Lp-estimates, and decay estimates.
Also, we show that weak solutions must vanish after a finite time, even for

large initial data.

1. Introduction

In this article, we study the fractional parabolic porous medium equation with
a non-Lipschitz absorption term,

∂tu− div(|u|m1∇(−∆)−s[|u|m2−1u]) + |u|β−1u = 0 in RN × (0, T ),

u(x, 0) = u0(x) in RN ,
(1.1)

where m1,m2 > 0, s ∈ (0, 1), β ∈ (0, 1), and N ≥ 2. Equations of type (1.1) with
s = 0 and m2 = 1, without the absorption term, correspond to the well-known
porous medium equation ∂tu = div(um1∇u).

This equation appears in applications such as the standard model for gas flow
through a porous medium (Darcy-Leibenzon-Muskat), Boussinesq’s model of ground-
water flow, and a model of population dynamics (Gurtin-McCamy) (see [19]). These
applications have served as a motivation for many authors to study equation (1.1),
see for example [2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17], and the references therein.
Most of the known results concern the existence of weak solutions, decay estimates
and finite speed of propagation. This is the main feature of porous media equations
and gave rise to free boundary problems.

Now, we like to mention some recent results concerning equation (1.1). Biler et
al. [1] studied (1.1) with α = 2(1−s), m1 = 1, m = m2 + 1, without the absorption
term |u|β−1u:

∂tu− div(|u|∇α−1(|u|m−2u)) = 0 ,

2010 Mathematics Subject Classification. 35R11, 35K65.
Key words and phrases. Nonlocal nonlinear parabolic equation; fractional Laplacian;

finite time extinction.
c©2021 Texas State University.

Submitted December 11, 2020. Published April 13, 2021.

1



2 N. A. DAO EJDE-2021/29

They constructed nonnegative self-similar solutions of Barenblatt-Pattle-Zeldovich
type, and obtained an existence of weak solutions u satisfying the decay estimate

‖u(t)‖Lp ≤ Ct−
N(1− 1

p
)

N(m−1)+α ‖u0‖
N(m−1)/p+α
N(m−1)+α

L1 . (1.2)

Stan et al. [18] investigated (1.1) with s ∈ (0, 1), m2 = 1, m1 = m − 1 > 0,
without the absorption term. The authors studied the existence of nonnegative
weak solutions for all integrable initial data u0. They obtained the smoothing
effect Lp-L∞, for p ≥ 1:

‖u(t)‖L∞ ≤ Ct−
N

N(m−1)+2p(1−s) ‖u0‖
2p(1−s)

N(m−1)+2p(1−s)
Lp , (1.3)

with C = C(N, s,m, p) > 0. Moreover, the finite and infinite speed of propagation
have been also studied by the same authors in [17]. Very recently, Dao-Dı́az studied
(1.1), and obtained the following result.

Theorem 1.1 ([12]). Let m1,m2 > 0 and s ∈ (0, 1). Suppose that u0 ∈ L1(RN ) ∩
L∞(RN ). Then, there exists a weak solution u of (1.1) satisfying the following
properties:

(i) Lq-estimates: For any 1 ≤ q ≤ ∞, we have

‖u(t)‖Lq ≤ ‖u0‖Lq , for a.e. t ∈ (0, T ) . (1.4)

(ii) Decay estimates: Let p ≥ 1 be such that m1 +m2 > 1− 2p(1−s)
N . Then

‖u(t)‖L∞ ≤ Ct−
1

p(1−α0)+σ0 ‖u0‖
p(1−α0)

p(1−α0)+σ0

Lp , (1.5)

with α0 =
(
N − 2(1− s)

)
/N , and σ0 = m1 +m2 − 1.

(iii) Finite time extinction: If m1 +m2 < α0 then, there is a finite time T0 > 0
such that

u(x, t) = 0, for (x, t) ∈ RN × (T0,∞) . (1.6)

Inspired by the above results, we want to prove the existence of weak solutions
to equation (1.1), which satisfies estimates (1.4), (1.5). After that, we show that
such a weak solution must vanish after a finite time, even when beginning with
a large initial data u0. It is known that this phenomenon occurs because of the
strong absorption term |u|β−1u. see [11, 13, 14] for another strong absorption term
u−βχ{u>0}.

Let us define

Θ(u) = |u|m1∇(−∆)−s[|u|m2−1u], QT = RN × (0, T ).

Definition 1.2. Let u0 ∈ L1(RN ) ∩L∞(RN ). We say that u is a weak solution of
(1.1) if u ∈ L1(0, T ;L∞(RN )) ∩ L∞(QT ) satisfies div Θ(u) ∈ L2(0, T ;Y (BR)), and∫ T

0

∫
RN

(−uϕt + Θ(u) · ∇ϕ+ |u|β−1uϕ) dx dt = 0, ∀ϕ ∈ C∞c (QT ),

where

Y (BR) =

{
H−1(BR), if s ∈ [1/2, 1),

W−2,p(BR), p > 1 such that m2p
p−1 ≥ 1, if s ∈ (0, 1/2).

Note that H−1(BR) is the dual space of H1
0 (BR), and W−2,p(BR) the dual space

of W 2,p
0 (BR). Here BR is the ball in RN , with center at 0 and radius R.



EJDE-2021/29 FINITE TIME EXTINCTION OF SOLUTIONS 3

Remark 1.3. It follows from Definition 1.2 that u ∈ C([0, T ];Y (BR)), for any
R > 0. Thus, u(t) possesses an initial trace u0 in this sense. In particular, if either
s ∈ [1/2, 1) or m2 > m1, then u ∈ C([0, T ];H−1(BR)) for every R > 0.

Our main results read as follows.

Theorem 1.4. Let s ∈ (0, 1), β ∈ (0, 1), and m1,m2 > 0. Suppose that u0 ∈
L1(RN ) ∩ L∞(RN ). Then, there exists a weak solution of (1.1) satisfying (1.4),
(1.5), and (1.6) in Theorem 1.1.

Concerning the finite time extinction of solutions, it suffices to consider m1 +
m2 ≥ α0 in the following theorem since u vanishes after a finite time if provided
m1 +m2 < α0.

Theorem 1.5. Assume the hypotheses in Theorem 1.4. Suppose that m1+m2 ≥ α0.
Then, there exists a finite time T0 > 0 such that

u(x, t) = 0, for (x, t) ∈ RN × (T0,∞) . (1.7)

And T0 can be estimated as follows

T0 ≤ C‖u0‖p(1−γ0)
Lp , (1.8)

for some constant C > 0 (independent of u0), with

γ0 =
1

1 + 2(1−s)(1−β)
2(1−s)(p−1)+N(m1+m2)−β(N−2(1−s))

.

Note that γ0 ∈ (0, 1) since m1 +m2 ≥ α0.

Through this paper, the constant C may change value from step by step. More-
over, C = C(α, β, γ) means that the constant C merely depends on the parameters
α, β, γ. We denote ‖ · ‖X(RN ) = ‖ · ‖X , and

∫
RN f(x)dx =

∫
f(x)dx. Finally, A . B

means that there exists a positive constant c, independent of the data, such that
A ≤ cB.

2. Functional setting

Let p ≥ 1, and s ∈ (0, 1). For a given domain Ω ⊂ RN , we define the fractional
Sobolev space

W s,p(Ω) =
{
u ∈ Lp(Ω) :

∫
Ω

∫
Ω

|u(x)− u(y)|p

|x− y|N+sp
dx dy <∞

}
,

endowed with the norm

‖u‖W s,p(Ω) =
(
‖u‖pLp(Ω) +

∫
Ω

∫
Ω

|u(x)− u(y)|p

|x− y|N+sp
dx dy

)1/p

.

We also denote the homogeneous fractional Sobolev space by Ẇ s,p(Ω), endowed
with the seminorm

‖u‖Ẇ s,p(Ω) =
(∫

Ω

∫
Ω

|u(x)− u(y)|p

|x− y|N+sp
dx dy

)1/p

.

In particular, we denote W s,2(RN ) by Hs(RN ), which turns out to be a Hilbert
space. It is well-known that we have the equivalent characterization

Hs(RN ) =
{
u ∈ L2(RN ) :

∫
(1 + |ξ|2s)|F{u}(ξ)|2 dξ <∞

}
,



4 N. A. DAO EJDE-2021/29

where F denotes the Fourier transform, and that we have

‖u‖Hs(RN ) =
(∫

(1 + |ξ|2s)|F{u}(ξ)|2 dξ
)1/2

.

In addition, for u ∈ Hs(RN ), the fractional Laplacian is defined by

(−∆)su(x) = C(N, s)p.v.

∫
u(x)− u(y)

|x− y|N+2s
dy = F−1{|ξ|2sF(u)(ξ)} . (2.1)

Then
‖u‖2Hs(RN ) = ‖u‖2L2 + C‖(−∆)s/2u‖2L2 .

We emphasize that if s > 0, then (−∆)−s = I2s is the Riesz potential. Moreover,
the fractional gradient ∇s can be written as ∇I1−s. And for any smooth bounded
function v : RN → R, we have

∇sv = C(N, s)

∫
RN

(v(x)− v(x+ z))
z

|z|N+1+s
dz ,

with a suitable constant C(N, s), see [1].

Approximation of the fractional Laplacian. For any s ∈ (0, 1), and for ε > 0,
let us define the operator

Lsε[f ](x) :=

∫
f(x)− f(y)

(|x− y|2 + ε2)
N+2s

2

dy, (2.2)

for x ∈ RN , and f ∈ S(RN ) (the Schwartz space). Note that Lsε can be considered
as a regularization of the fractional Laplacian (−∆)s (see [7]).
• Square root: By symmetry, we observe that

〈Lsε[f ], f〉L2 =
1

2

∫ ∫
|f(x)− f(y)|2

(|x− y|2 + ε2)
N+2s

2

dx dy .

Then, we denote Ls/2ε [f ] as a square root of Lsε[f ] in the Fourier transform sense,
and

‖Ls/2ε [f ]‖2L2 = 〈Lsε[f ], f〉L2 .

The following lemmas will be useful in proving Theorems 1.4 and 1.5. Their proof
can be found in [12].

Lemma 2.1. Let {fε}ε>0 be a sequence in L2(RN ) such that fε → f in L2(RN )
as ε→ 0. Then, for any s ∈ (0, 1), it holds

‖(−∆)−sLsε[fε]− f‖L2 → 0 . (2.3)

Next, we recall a generalized version of Stroock-Varopoulos’s inequality.

Lemma 2.2. Let s ∈ (0, 1), and let ψ, φ ∈ C1(R) be such that ψ′, φ′ ≥ 0. Then∫
ψ(f)Lsε[φ(f)]dx ≥ 0. (2.4)

If we take ψ(f) = f , then we obtain∫
fLsε[φ(f)] dx ≥

∫
|Ls/2ε Φ(f)|2 dx , (2.5)

where φ′ = (Φ′)2.

Finally, we have the following fundamental inequality.



EJDE-2021/29 FINITE TIME EXTINCTION OF SOLUTIONS 5

Lemma 2.3. Let α, β > 0, and θ = α+β
2 . Then, there is a constant C > 0 such

that∣∣|a|θ−1a− |b|θ−1b
∣∣2 ≤ C ∣∣|a|α−1a− |b|α−1b

∣∣ ∣∣|a|β−1a− |b|β−1b
∣∣ , ∀a, b ∈ R. (2.6)

3. Existence of solutions

In this section, we prove Theorem 1.4 using the Lemmas and the Propositions
below.

A regularized problem. We consider the following regularizing version of (1.1),

∂tu− δ1∆u+ δ2Ls0ε [Jκ(u)]− div Θε,ν(u) + |u|β−1uΦµ(u) = 0,

in RN × (0, T ) ,

u(0) = u0, in RN ,

(3.1)

where s0 = (1− 2s)+, Θε,ν(u) = Hν(u)∇(−∆)−1L1−s
ε [Gν(u)], and

Hν(u) =
|u|m1+2

ν2 + u2
, Gν(u) =

|u|m2+1u

ν2 + u2
, Jκ(u) =

|u|m0+1u

u2 + κ2
,

with m0 = 1
2 min{m1,

m2(N−2s0)
N }, and Φµ(u) is a cut-off function in a neighborhood

of u = 0, defined by Φµ(u) = Φ(uµ ), where Φ(s) ∈ C∞(R), 0 ≤ Φ(s) ≤ 1, for all

s ∈ R, and

Φ(s) =

{
0, if |s| ≤ 1 ,

1, if |s| ≥ 2 ,

for δ1, δ2, ε, κ, µ, ν ∈ (0, 1).
We shall prove the existence of solutions of (3.1) in a suitable functional space

by using the fixed-point theorem, and derive some energy estimates in order to pass
to the limit as ε, κ, ν, δ1, δ2, µ→ 0 alternatively. The proof is most likely to the one
in Section 3, [12]. Here, we just present some different points with the presence of
the absorption.

Let us put

X = L1(RN ) ∩ L∞(RN ) ,

with the associated norm ‖ · ‖X = ‖ · ‖L1(RN ) + ‖ · ‖L∞(RN ).

Lemma 3.1. Let u0 ∈ X and f ∈ L1(QT ) ∩ L∞(QT ). Then, there exists a weak
solution u ∈ C([0, T ];X) satisfying problem (3.1) in the weak sense, i.e.∫ T

0

∫ (
−uϕt+δ1∇u·∇ϕ+δ2Ls0ε [Jκ(u)]ϕ−Θε,ν(u)·∇ϕ+|u|β−1uΦµ(u)ϕ

)
dx dt = 0 ,

for all ϕ ∈ C∞c (QT ).

Proof. We look for a mild solution u ∈ C([0, T ];X) as a fixed point of the map

T (u) = etδ1∆u0 +

∫ t

0

∇e(t−τ)δ1∆Θε,ν(u) dτ

−
∫ t

0

e(t−τ)δ1∆(δ2Ls0ε [Jκ(u)] + |u|β−1uΦµ(u)) dτ,

where et∆ is the semigroup corresponding to the heat kernel (4πt)−N/2 exp(−|x|
2

4t ).



6 N. A. DAO EJDE-2021/29

We note that |u|β−1uΦµ(u) is a locally Lipschitz function, then we can mimic the
proof of [12, Theorem 4] to obtain that T maps C([0, T ];X) into itself. Moreover,
there is a real number γ ∈ (0, 1) such that

‖T (u)− T (v)‖C([0,T ];X) ≤ C(R)T γ‖u− v‖C([0,T ];X) ,

for all u, v ∈ B(0, R) ⊂ C([0, T ];X). This implies that

‖T (u)− T (v)‖C([0,T ];X) ≤
1

2
‖u− v‖C([0,T ];X) ,

if T > 0 is chosen small enough. Thanks to the contraction mapping theorem, we
obtain a unique mild solution u to equation T (u) = u.

Finally, since the terms in (3.1) are regular, then it follows from the standard
regularity theory that u is smooth in RN × (0, T ). The proof is complete. �

Now, we prove an Lq-estimate of u.

Proposition 3.2. Let u be a solution of (3.1) in QT . Then, for every q ∈ [1,∞],
we have

‖u(t)‖Lq(RN ) ≤ ‖u0‖Lq(RN ), ∀t ∈ (0, T ) . (3.2)

Proof. For every q > 1, by testing |u|q−2u to (3.1), we obtain

1

q

d

dt

∫
|u(t)|q dx+ (q − 1)

∫
|u|q−2Hν(u)∇(−∆)−1L1−s

ε [Gν(u)] · ∇u dx

+ δ1(q − 1)

∫
|u|q−2|∇u|2 dx+ δ2

∫
Ls0ε [Jκ(u)]|u|q−2u dx

+

∫
|u|β+q−1Φµ(u) dx = 0 .

(3.3)

By applying Lemma 2.2 to ψ(u) = |u|q−2u, and φ(u) = Jκ(u), we obtain∫
|u|q−2uLs0ε [Jκ(u)] dx ≥ 0 . (3.4)

Next, we observe that∫
|u|q−2Hν(u)∇(−∆)−1L1−s

ε [Gν(u)] · ∇u dx

=

∫
∇(−∆)−1L1−s

ε [Gν(u)] · ∇H̃ν(u) dx

=

∫
H̃ν(u)(−∆)(−∆)−1L1−s

ε [Gν(u)] dx

=

∫
H̃ν(u)L1−s

ε [Gν(u)] dx ≥ 0 ,

(3.5)

with

H̃ν(u) =

∫ u

0

|s|q−2Hν(s) ds .

Note that the inequality in (3.5) was also obtained by Lemma 2.2. Thus,

d

dt

∫
|u(t)|q dx ≤ 0 .

This implies (3.2) for any q ∈ [1,∞). Finally, passing to the limit as q → ∞, we
also obtain (3.2) for the L∞-estimate.



EJDE-2021/29 FINITE TIME EXTINCTION OF SOLUTIONS 7

It remains to prove the L1-estimate of u. For every η > 0, let us put

χη(r) =

{
sign(r), if |r| > η,

r/η, if |r| ≤ η,

Testing (3.1) with χη(u) yields∫ (
utχη(u) + δ1∇u · ∇χη(u) + δ2Ls0ε [Jκ(u)]χη(u) + Θ(u) · ∇χη(u)

)
dx

+

∫
|u|β−1uΦµ(u)χη(u) dx = 0 .

(3.6)

Since χ′η(u) ≥ 0, it is clear that∫
∇u · ∇χη(u) dx =

∫
|∇u|2χ′η(u) dx ≥ 0 ,

and by Lemma 2.2, we have∫
Ls0ε [Jκ(u)]χη(u) dx ≥ 0,

∫
Θ(u) · ∇χη(u) dx ≥ 0 .

From (3.6) after integrating on (0, t) it follows that∫
Sη(u(t)) dx ≤

∫
Sη(u0) dx ,

with

Sη(u) =

∫ u

0

χη(r) dr =
u2

2η
χ{|u|<η} + (|u| − η

2
)χ{|u|≥η} ,

where χA denotes the characteristic function of the set A. Note that

lim
η→0

∫
Sη(u(t)) dx =

∫
|u(t)| dx .

So, (3.2) follows with q = 1. This completes the proof. �

The following results are similar to the ones in [12], so we omit their proofs.

Proposition 3.3. Let u be as in Proposition 3.2. Then, there is a constant C =
C(m0, u0) > 0 such that for every κ, ε, µ, ν > 0,

δ2‖L
s0
2
ε [Jκ(uε)]‖L2(QT ) ≤ C . (3.7)

Limit as ε→ 0.

Proposition 3.4. Let uε be the solution of problem (3.1). Then, there exists a
subsequence of {uε}ε>0 (still denoted as {uε}ε>0 ) such that for any R > 0,

uε → u, in L2(BR × (0, T )).

Moreover, u ∈ L∞(0, T ;L1(RN ))∩L∞(QT )∩L2(0, T ;H1(RN )) is a solution of the
problem

ut − δ1∆u− div(Hν(u)∇(−∆)−s[Gν(u)]) + δ2(−∆)s0Jκ(u) + |u|β−1uΦµ(u)

= 0, in QT .
(3.8)



8 N. A. DAO EJDE-2021/29

3.1. Limit as κ→ 0.

Proposition 3.5. Let uκ be the solution of problem (3.8). Then, for any R > 0 it
holds

uκ → u, in L2(BR × (0, T ))

up to a subsequence. Moreover, u ∈ L∞(0, T ;L1(RN ))∩L∞(QT )∩L2(0, T ;H1(RN ))
is a solution of the problem

ut− δ1∆u− div Θν(u) + δ2(−∆)s0(|u|m0−1u) + |u|β−1uΦµ(u) = 0, in QT , (3.9)

where we denote Θν(u) = Hν(u)∇(−∆)−s[Gν(u)].

3.2. Limit as ν → 0.

Proposition 3.6. Let uν be the solution, obtained in Proposition 3.5. Then, there
exists a subsequence of {uν}ν>0 converging to a function u in L2(BR × (0, T )) for
any R > 0. Moreover, u ∈ L∞(0, T ;L1(RN )) ∩ L∞(QT ) ∩ L2(0, T ;H1(RN )) is a
solution of the equation

ut− δ1∆u− div Θ(u) + δ2(−∆)s0(|u|m0−1u) + |u|β−1uΦµ(u) = 0, in QT . (3.10)

Recall that Θ(u) = H(u)∇(−∆)−s[G(u)], with H(u) = |u|m1 and G(u) = |u|m2−1u.

3.3. Limit as δ1, δ2 → 0.

Proposition 3.7. Let uδ2 be a solution of (3.10) above. Then, there exists a
subsequence of {uδ2}δ2>0, converging to a function u in L2(BR × (0, T )) for any
R > 0. Moreover, u ∈ L∞(0, T ;L1(RN )) ∩ L∞(QT ) ∩ L2(0, T ;H1(RN )) is a weak
solution of the problem

ut − δ1∆u− div Θ(u) + |u|β−1uΦµ(u) = 0, in QT . (3.11)

We emphasize that the estimates in the proof of Proposition 3.7 are also inde-
pendent of δ1.

Proposition 3.8. Let uδ1 be a solution of (3.11). Then there exists a subsequence
of {uδ1}δ1>0, converging to a function u in L2(BR×(0, T )) for any R > 0. Further-
more, u ∈ L∞(0, T ;L1(RN )) ∩ L∞(QT ), which is a weak solution of the equation

ut − div Θ(u) + |u|β−1uΦµ(u) = 0, in QT . (3.12)

In addition, div(Θ(u)) satisfies the following regularity:

• If s ∈ [1/2, 1), then

div(Θ(u)) ∈ L2(0, T ;H−1(BR)) . (3.13)

• If s ∈ (0, 1/2), then

div(Θ(u)) ∈ Lp(0, T ;W−2,p(RN )) , (3.14)

for p > 1 such that m2p
p−1 ≥ 1, and W−2,p(RN ) is the dual space of W 2,p(RN ).



EJDE-2021/29 FINITE TIME EXTINCTION OF SOLUTIONS 9

Limit µ→ 0.

Proposition 3.9. Let uµ be a solution of (3.12). Then, there exists a subsequence
of {uµ}µ>0, converging to a function u in L2(BR× (0, T )) for any R > 0. Further-
more, u ∈ L∞(0, T ;L1(RN ))∩L∞(QT ), which is a weak solution of equation (1.1).
Also div(Θ(u)) satisfies either (3.13) if s ∈ [1/2, 1), or (3.14) if s ∈ (0, 1/2).

Then, it is clear that solution u, obtained from Proposition 3.9 is a weak solution
of (1.1). Moreover, u also satisfies the energy inequality

1

p

d

dt

∫
|u(x, t)|p dx

+ (p− 1)

∫ ∫
(G(u(x))−G(u(y)))(|u|m1+p−2u(x)− |u|m1+p−2u(y))

|x− y|N+2(1−s) dx dy ≤ 0 ,

see (4.1) below. Thus, we can mimic the proof of [12, Theorem 2] to obtain decay
estimate (1.5). This completes the proof of Theorem 1.4.

4. Finite time extinction of solutions

Proof of Theorem 1.5. For every p > 1, it follows from (3.3) that

1

p

d

dt

∫
|u(x, t)|p dx+

∫
|u(x, t)|p−1+β dx

+ (p− 1)

∫ ∫
(G(u(x))−G(u(y)))(|u|m1+p−2u(x)− |u|m1+p−2u(y))

|x− y|N+2(1−s) dx dy ≤ 0 .

(4.1)
Thanks to Lemma 2.3, we obtain

1

p

d

dt

∫
|u(x, t)|p dx+

∫
|u(x, t)|p−1+β dx

+ C(p− 1)

∫ ∫ ∣∣|u|θ0−1u(x)− |u|θ0−1u(y)
∣∣2

|x− y|N+2(1−s) dx dy dt ≤ 0 ,

with θ0 = (m1 +m2 + p− 1)/2. Next, applying the Sobolev embedding yields

‖|u(t)|θ0‖L2? ≤ C‖|u(t)|θ0‖Ḣ1−s ,

with C = C(N, s), and 2? = 2N
N−2(1−s) = 2

α0
. Combining these inequalities yields

1

p

d

dt
‖u(t)‖pLp + C(‖u(t)‖p−1+β

Lp−1+β + ‖u(t)‖2θ0
L2?θ0

) ≤ 0 . (4.2)

Thanks to the interpolation inequality, we obtain

‖u(t)‖Lp ≤ ‖u(t)‖γ
Lp−1+β‖u(t)‖1−γ

L2?θ0

= (‖u(t)‖p−1+β
Lp−1+β )

γ
p−1+β (‖u(t)‖2θ0

L2?θ0
)

1−γ
2θ0

≤ (‖u(t)‖p−1+β
Lp−1+β + ‖u(t)‖2θ0

L2?θ0
)

γ
p−1+β+ 1−γ

2θ0 ,

(4.3)

where 1
p = γ

p−1+β + 1−γ
2?θ0

. Note that 2?θ0 > p since m1 + m2 ≥ α0. By (4.2) and

(4.3), we obtain

d

dt
‖u(t)‖pLp + C‖u(t)‖pλ0

Lp ≤ 0 ,



10 N. A. DAO EJDE-2021/29

with

λ0 =
1

1 + p(1−γ)
θ0

( 1
2 −

1
2? )
∈ (0, 1) .

Thus, y(t) = ‖u(t)‖pLp satisfies

y′(t) + Cyλ0(t) ≤ 0 . (4.4)

This implies that there exists a finite time T0 > 0 such that y(t) = 0 for t > T0.
Thus, we obtain (1.7).

Finally, to estimate T0, we solve directly (4.4) and obtain

y1−λ0(t) + Ct ≤ y1−λ0(0) = ‖u0‖(1−λ0)p
Lp .

Thus, (1.8) follows. This completes the proof. �

Acknowledgements. This research was funded by the University of Economics,
Ho Chi Minh City, Vietnam.

References

[1] P. Biler, C. Imbert, G. Karch; The nonlocal porous medium equation: Barenblatt profiles and

other weak solutions. Arch. Ration. Mech. Anal., 215 (2015), 497–529.

[2] M. Bonforte, A. Figalli, X. Ros-Otón; Infinite speed of propagation and regularity of solutions
to the fractional porous medium equation in general domains, Comm. Pure Appl. Math., 70

(2017), no. 8, 1472–1508.

[3] M. Bonforte, A. Figalli, J. L. Vázquez; Sharp global estimates for local and nonlocal porous
medium-type equations in bounded domains. Analysis & PDE, 11 (2018), 945–982.

[4] M. Bonforte, Y. Sire, J. L. Vázquez; Existence, Uniqueness and Asymptotic behaviour for
fractional porous medium equations on bounded domains. Discrete Contin. Dyn. Syst., 35

(2015), no. 12, 5725–5767.

[5] M. Bonforte, J. L. Vázquez; Quantitative Local and Global A Priori Estimates for Fractional
Nonlinear Diffusion Equations. Advances in Math., 250 (2014), 242–284.

[6] M. Bonforte, J. L. Vázquez; A Priori Estimates for Fractional Nonlinear Degenerate Diffu-

sion Equations on bounded domains. Arch. Ration. Mech. Anal., 218 (2015), 317–362.
[7] L. Caffarelli, L. Silvestre; An extension problem related to the fractional Laplacian, Comm.

PDEs, 32 (2007), 1245–1260.

[8] L. A. Caffarelli, F. Soria, J. L. Vázquez; Regularity of solutions of the fractional porous
medium flow. J. Eur. Math. Soc., 15 (2013), 1701–1746.

[9] L. A. Caffarelli, J. L. Vázquez; Nonlinear porous medium flow with fractional potential pres-

sure. Arch. Rational Mech. Anal., 202 (2011), 537–565.
[10] L. A. Caffarelli, J. L. Vázquez; Asymptotic behaviour of a porous medium equation with

fractional diffusion, Discrete Cont. Dyn. Systems A, 29 (2011), 1393–1404.
[11] N. A. Dao; Instantaneous shrinking of the support of solutions to parabolic equations with a

singular absorption, RACSAM (2020), 114–165
[12] N. A. Dao, J. I. Dı́az; Energy and large time estimates for nonlinear porous medium flow

with nonlocal pressure in RN , Arch. Rational Mech. Anal., 238 (2020), 299–345.
[13] N. A. Dao, J. I. Dı́az, Q. B. H. Nguyen; Pointwise gradient estimates in multi-dimensional

slow diffusion equations with a singular quenching term, Adv. Nonlinear Stud., 20 (2020),
477–502.

[14] N. A. Dao, J. I. Dı́az, H. V. Kha; Complete quenching phenomenon and instantaneous
shrinking of support of solutions of degenerate parabolic equations with nonlinear singular
absorption, Proceedings of the Royal Society of Edinburgh, 149 (2019), 1323–1346.

[15] A. De Pablo, F. Quirós, A. Rodŕıguez, J. L. Vázquez; A fractional porous medium equation,

Advances in Mathematics. 226 (2011), 1378–1409.
[16] A. De Pablo, F. Quirós, A. Rodŕıguez, J. L. Vázquez; A general fractional porous medium

equation. Comm. Pure Appl. Math., 65 (2012), 1242–1284.

[17] D. Stan, F. del Teso, J. L. Vázquez; Finite and infinite speed of propagation for porous
medium equations with fractional pressure. Journal Diff. Eqns., 260 (2016), 1154–1199.



EJDE-2021/29 FINITE TIME EXTINCTION OF SOLUTIONS 11

[18] D. Stan, F. del Teso, J. L. Vázquez; Existence of weak solutions for porous medium equations

with nonlocal pressure. Arch. Ration. Mech. Anal., 233 (2019), no. 1, 451–496.

[19] J. L. Vázquez; The Porous Medium Equation. Mathematical Theory, vol. Oxford Mathemat-
ical Monographs, Oxford University Press, Oxford, 2007.

Nguyen Anh Dao

Institute of Applied Mathematics, University of Economics Ho Chi Minh City, Viet Nam

Email address: anhdn@ueh.edu.vn


	1. Introduction
	2. Functional setting
	Approximation of the fractional Laplacian

	3. Existence of solutions
	A regularized problem
	Limit as 0
	3.1. Limit as 0
	3.2. Limit as 0
	3.3. Limit as 1, 2 0
	Limit 0

	4. Finite time extinction of solutions
	Acknowledgements

	References