EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 1, 2021, 204-233
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Radon measure-valued solutions for nonlinear strongly
degenerate parabolic equations with measure data

Quincy Stévène Nkombo1,∗, Fengquan Li1

1 School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning,
China

Abstract. In this paper, we prove the existence of Radon measure-valued solutions for nonlinear
strongly degenerate parabolic equations with nonnegative bounded Radon measure as initial data.
Moreover, we show the uniqueness of the Radon measure-valued solutions when the Radon measure
as a forcing term is diffuse with respect to the parabolic capacity and the Radon measure as a initial
value is diffuse with respect to the Newtonian capacity. We also deduce that the concentrated part
of the Radon measure-valued solution with respect to the Newtonian capacity depends on time.

2020 Mathematics Subject Classifications: 35K20, 35K65, 35K59, 35R06, 28A33

Key Words and Phrases: Radon measure-valued solutions, Nonlinear degenerate parabolic
equations, Capacity

1. Introduction

In this work we address the nonhomogeneous nonlinear strongly degenerate parabolic
equations having the nonnegative bounded Radon measure on the right-hand side with
the nonnegative bounded Radon measure as initial data. This problem is described as
follows 

ut −∆ψ(u) = µ in Q := Ω× (0, T ),

u = 0 on ∂Ω× (0, T ),

u(x, 0) = u0 in Ω,

(P)

where T > 0, Ω ⊂ RN (N ≥ 2) is an open bounded domain with smooth boundary ∂Ω, the
initial value data u0 is a nonnegative bounded Radon measure on Ω and µ is a nonnegative
bounded Radon measure on Q.
The nonlinear strongly degenerate parabolic equations (P ) is the special case derived

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v14i1.3877

Email addresses: quincysnk@yahoo.fr (Q. S. Nkombo), fqli@dlut.edu.cn (F. Li)

http://www.ejpam.com 204 c© 2021 EJPAM All rights reserved.



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 205

from the study of quasilinear parabolic equations with degenerate coercivity involving a
quadratic gradient term (see [4, 7]). The general model of the problem (P ) is given by

ut − div (α(u)∇u) = β(u) | ∇u |2 +f(x, t) in Q := Ω× (0, T ),

u = 0 on ∂Ω× (0, T ),

u(x, 0) = u0 in Ω,

(S)

where α and β are real continuous functions, moreover α is positive bounded and may
vanish at ±∞, u0 ∈ L∞(Ω) and f ∈ Lm(Ω)(m > 1 + N

2 ) (see [4]). For the problem (S),
the typical example of functions α and β are expressed as follows

α(s) =
1√

1 + s2
and β(s) =

1√
(1 + s2)3

.

In [7], the authors studied the problem (S) with more general assumptions in which (S)
is a nonlinear degenerate parabolic equation. Meanwhile, in [32] Bogelein, Duzaarr and
Gianazza dealt with nonhomogenous porous medium type equations related to Cauchy-
Dirichlet problem in a space-time cylinder Q := Ω × (0, T ) (see also [13]). Likewise,
Fiorenza, Mercaldo and Rakotoson [1] studied some regularity and uniqueness results of
the evolution N-Laplacian equation with right hand term µ ∈ L1((0, T ),M(Ω)). Further-
more Porzio, Smarrazzo and Tesei [23] introduced the definition of Radon measure-valued
solutions to quasilinear parabolic equations with initial value as measure data. More pre-
cisely, in [23] authors proved the existence, uniqueness and qualitative properties of Radon
measure-valued solutions to the following problem

ut = ∆ϕ(u) in Q,

u = 0 on ∂Ω× (0, T ),

u(x, 0) = u0 in Ω,

(F )

where u0 ∈M+(Ω) is a bounded Radon measure and

ϕ(s) = γ

[
1− 1

(1 + s)σ

]
(A.1)

with γ ∈ (0,+∞), σ > 0. Since ϕ increases monotonically to limiting value γ as s→ +∞.
Therefore, ϕ′(s) → 0, thus the problem (F ) is strongly degenerate parabolic equation at
infinity.
Another interesting problems similar to the problem (P ) has been investigated in [18, 22,
24, 28, 30, 31] in which authors showed the existence and uniqueness of Radon measure
valued solutions to nonlinear parabolic equations.
To obtain the problem (P ), we replace the function ϕ by ψ which is defined by

ψ(s) =

∫ s

0
e−|z|

m
dz (0 < m ≤ 1) (1.1)



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 206

The function ψ increases monotonically to limiting value γ as s → +∞. Therefore, the
problem (P ) is nonlinear strongly degenerate parabolic equation at infinity and the func-
tion ψ is given by Oleinik-Kruzhkov in [26].
The choice of the special function ψ in (1.1) is motivated by the connection with the
function ϕ in (A.1), such as ψ′ ≤ ϕ′ in R+. This comparison leads to the connection of
the problem (P ) with the previous study problem (F ).
In order to construct the problem (P ), we add a Radon measure as a forcing term
µ ∈M+(Q)( a nonnegative bounded Radon measure with respect to the parabolic capac-
ity) to the problem (F ).
The first difficulty when studying the problem (P ) is due to the presence of a forc-
ing term µ and the second difficulty is a lack of coercivity of the differential operator
u→ div(ψ′(u)∇u).
In the study of degenerate parabolic equations, a physical model may be imagined in
which the degenerate parabolic equations described arise in nonlinear fluid mechanics,
heat transfer or diffusion. Moreover the Radon measures involved as data describe the
distribution of mass in the length area, and volume.
The last decades some authors studied the parabolic and elliptic equations involving mea-
sure data, but the solutions of these equations are not measures (see [2, 17, 25]). Due to
this reason, the main purpose of this paper is to study the degenerate parabolic equations
with measure data which the solutions of such equations are measures as well. This result
is possible because of the definition of weak Radon measure-valued solutions introduced
in [23], hence the main motivation to study of the problem (P ).
The unique point of the novelty of this paper is the study of the uniqueness of the Radon
measure-valued solutions when the Radon measure as a forcing term is diffuse with respect
to the parabolic capacity and the Radon measure as initial data is diffuse with respect to
the Newtonian capacity.
To the best of our knowledge there is no existing results of the problem (P ) are known in
the literature. Hence, this interesting case will be discussed in this paper.
The plan of this paper is organized as follows. In the next section, we recall some prelimi-
naries about capacity and Radon measures. Then in Section 3, we state the main results,
while in Section 4-6, we prove the main results.
2. Preliminaries
2.1 About capacity and measures
For any Borel set E ⊂ Ω, the C2-capacity of E in Ω is defined as

C2(E) = inf

{∫
Ω
| ∇u |2dx/u ∈ ZEΩ

}
where ZEΩ denotes the set of u belongs to H1

0 (Ω) such that 0 ≤ u ≤ 1 almost everywhere
in Ω, and u = 1 almost everywhere in a neighborhood E (see [23]).
Let W =

{
u ∈ L2((0, T ), H1

0 (Ω)) and ut ∈ L2((0, T ), H−1(Ω))
}

endowed with its natural
norm ‖ u ‖W=‖ u ‖L2((0,T ),H1

0 (Ω)) + ‖ ut ‖L2((0,T ),H−1(Ω)) a Banach space. For any open
set U ⊂ Q, we define the parabolic capacity as

Cap(U) = inf
{
‖ u ‖W /u ∈ VUQ

}



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 207

where VUQ denotes the set of u belongs to W such that 0 ≤ u ≤ 1 almost everywhere in Q,
and u = 1 almost everywhere in a neighborhood U (see [16]). Let M(Ω) be the space of
bounded Radon measures on Ω, and
M+(Ω) ⊂ M(Ω) the cone of nonnegative bounded Radon measures on Ω. For any µ ∈
M(Ω) a bounded Radon measure on Ω, we set

‖ µ ‖M(Ω):=| µ | (Ω)

where | µ | stands for the total variation of µ.
The duality map 〈·, ·〉Ω between the space M(Ω) and Cc(Ω) is defined by

〈µ, ϕ〉Ω =

∫
Ω
ϕdµ.

For any µ ∈ M(Ω) and any Borel set B ⊆ Ω, the restriction µxB of µ to B is defined by
setting

(µxB)(A) := µ(B ∩A) for every Borel set A ⊆ Ω.

It is worth observing that (µxB)(∅) = 0.
M+

s (Ω) denotes the set of nonnegative measures singular with respect to the Lebesgue
measure, namely

M+
s (Ω) :=

{
µ ∈M+(Ω)/∃ a Borel set E ⊆ Ω ; | E |= 0 , µ = µxB

}
we will consider | · | the Lebesgue measure on RN . Similarly,M+

ac(Ω) the set of nonnega-
tive measures absolutely continuous with respect to the Lebesgue measure, namely

M+
ac(Ω) :=

{
µ ∈M+(Ω)/µ(E) = 0, for every Borel setE ⊆ Ω ; | E |= 0

}
.

Recall that M+
s (Ω) ∩ M+

ac(Ω) = {0}. Moreover, by the Lebesgue decomposition and
Radon-Nikodym theorem (see [9]), for any µ ∈M+(Ω):

(i) there exists a unique couple µac ∈M+
ac(Ω), µs ∈M+

s (Ω) such that

µ = µac + µs (2.1)

(ii) there exist a unique nonnegative function ur ∈ L1(Ω) called the density of the measure
µac such that

µac(E) =

∫
E
urdx, for every Borel set E ⊆ Ω. (2.2)

LetM+
c,2(Ω) be the set of nonnegative measures on Ω which are concentrated with respect

to the Newtonian capacity

M+
c,2(Ω) :=

{
µ ∈M+(Ω)/∃ a Borel set E ⊆ Ω ; µ = µxE and C2(E) = 0

}
.



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 208

Notice thatM+
c,2(Ω) can be also defined as the set of all measures µ inM+(Ω) which are

singular with respect to the Newtonian capacity, i.e.

M+
c,2(Ω) :=

{
µ ∈M+

s (Ω)/∃ a Borel set E ⊆ Ω ; C2(E) = 0
}
.

It is clear to observe that M+
c,2(Ω) ⊆M+

s (Ω) (see [12]).

M+
d,2(Ω) denotes the set of nonnegative measures on Ω which are diffuse with respect to

the Newtonian capacity

M+
d,2(Ω) :=

{
µ ∈M+(Ω)/µ(E) = 0, for every Borel setE ⊆ Ω ;C2(E) = 0

}
.

Due to C2(E) = 0 implies that | E |= 0 (see [9]), we observe that M+
ac(Ω) ⊆M+

d (Ω).

It is known that a measure µd,2 ∈ M+
d,2(Ω) if there exist f0 ∈ L1(Ω) and G0 ∈

[
L2(Ω)

]N
such that

µd,2 = f0 − divG0 in D′(Ω). (2.3)

For any µ ∈ M+(Ω), if there exists a unique couple µd,2 ∈ M+
d,2(Ω), µc,2 ∈ M+

c,2(Ω) such
that

µ = µd,2 + µc,2. (2.4)

Notice that µc,2 = [µ]c,2 and µd,2 = [µ]d,2.
For the above assertions we can also refer to ([18, 23, 30] and references therein).
Let M(Q) be the space of bounded Radon measures on Q, and
M+(Q) ⊂M(Q) the cone of nonnegative bounded Radon measures on Q.
For any µ ∈M(Q), we set

‖ µ ‖M(Q):=| µ | (Q)

where | µ | denotes the total variation of µ.
For any diffuse measure µ0 ∈ M+

d,2(Q), there exist f ∈ L1(Q), g ∈ L2((0, T ), H1
0 (Ω)) and

G ∈
[
L2(Q)

]N
µ0 = f − divG+ gt in D′(Q) (2.5)

(see [10, 11, 16]). The rest of statements of M(Q) can be deduce from the properties of
M(Ω).
Let E be a Borel subset of Ω, for t0 ∈ (0, T ) fixed, one has Cap(E × {t0}) = 0 if and only
if | E |= 0 and for any 0 ≤ t0 < t1 ≤ T , there holds Cap(E × (t0, t1)) = 0 if and only if
C2(E) = 0 (see [16]).
The relationship between parabolic capacity and Newtonian capacity is given in [27] such
that :
(i) There exist positive constants 0 < k1 < k2 such that

k1C2(E) ≤ Cap(E × {t0}) ≤ k2C2(E).

(ii) For any 0 < t0 < t1, there exist positive constants 0 < l1 < l2 such that

l1C2(E) ≤ Cap(E × (t0, t1)) ≤ l2C2(E).



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 209

Let U ⊂ Q an open set and K ⊂ Q a compact set with Cap(K) = 0, then there exists
ϕn ∈ C∞c (U) such that
(iii) 0 ≤ ϕn ≤ 1 a.e in Q, (iv) ϕn = 1 a.e in K, (v) ϕn → 0 in W , (vi) ϕn converges to
zero Cap-quasi continuous (see [27, Proposition 2.2]).
On the other hand, assume that V ⊂ Ω an open set and K ⊂ Ω a compact set with
Cap(K) = 0, then there exists φn ∈ C∞c (V ) such that (vii) 0 ≤ φn ≤ 1 a.e in Ω, (viii)
φn = 1 a.e in K, (ivx) φn → 0 in H1

0 (Ω), (x) ϕn converges to zero Cap-quasi continuous
(see [15, Lemma 4.E.1].
By L∞ ((0, T ),M+(Ω)), the set of nonnegative Radon measures u ∈M+(Q) which satisfy
the following property: For almost every t ∈ (0, T ), there exists a measure u(·, t) ∈M+(Ω)
such that

(a) for every ξ ∈ C(Q), the map t 7→ 〈u(·, t), ξ(·, t)〉Ω is Lebesgue measurable and there
holds

〈u, ξ〉Q =

∫ T

0
〈u(·, t), ξ(·, t)〉Ωdt (2.6)

(b) for every Borel set E ⊆ Ω, the map t 7→ u(·, t)(Et) is Lebesgue measurable and there
holds

u(E) =

∫ T

0
u(·, t)(Et)dt

where Et = {x ∈ Ω/(x, t) ∈ E}

(c) there exists a constant C > 0 such that

ess sup
t∈(0,T )

‖ u(·, t) ‖M(Ω)≤ C.

In the following, we will use the notation

‖ u ‖L∞((0,T ),M(Ω))= ess sup
t∈(0,T )

‖ u(·, t) ‖M(Ω) .

If u ∈ L∞((0, T ),M(Ω)), it is easily seen that uac, us ∈ L∞((0, T ),M(Ω)) as well
and that ur ∈ L∞((0, T ), L1(Ω)).
Moreover, the inequality (2.6) implies that for every ξ ∈ C(Q)

〈uac, ξ〉Q =

∫
Q
urξdxdt

and

〈us, ξ〉Q =

∫ T

0
〈us(·, t), ξ(·, t)〉Ωdt



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 210

Notice that uac(·, t) = [u(·, t)]ac, ur(·, t) = [u(·, t)]r and us(·, t) = [u(·, t)]s (see [18, 23, 30]).
Assume that the function ψ satisfies the following conditions:

(I)


(i) ψ ∈ L∞(R+) ∩ C2(R+), ψ(0) = 0, ψ′ > 0 in R+,

(ii) ψ(j) ∈ L∞(R∗+), for any j = 1, 2, . . . , n if 0 < m ≤ 1,

(iii) ψ(s)→ γ as s→ +∞,

where R+ ≡ [0,+∞) and γ ∈ R∗+ ≡ (0,+∞) . By ψ′ and ψ(j) we denote the first and
j th derivative of the function ψ. The assumption (I)-(iii) stems from (I)-(i), hence we
extend the function ψ in [0,+∞] defining ψ(+∞) = γ.
To prove the well-posedness of (P) (if N ≥ 2) we will need further assumption

(J)


There exist γ > 0, s < s and l1, l2 > 0, l1 < l2 such that

(i) ψ′(s) ≥ l1e−|s|
m
,

(ii) ψ′(s) ≤ l2e−|s|
m
,

for any s < s < s.

where l1, l2 can be expressed as follows

l1 = min
s∈[s,s]

ψ′(s)e|s|
m

and l2 = max
s∈[s,s]

ψ′(s)e|s|
m

(0 < m ≤ 1).

3. Statement of main results
Definition 3.1. For any u0 ∈ M+(Ω) and µ ∈ M+(Q), a measure u is called a weak
solution of the problem (P ), if u ∈M+(Q) such that
(i) u ∈ L∞((0, T ),M+(Ω))
(ii) ψ(ur) ∈ L1((0, T ),W 1,1

0 (Ω))
(iii) for every ξ ∈ C1([0, T ], C1

0 (Ω)), ξ(·, T ) = 0 in Ω, u satisfies the identity∫ T

0
〈u(·, t), ξt(·, t)〉Ωdt =

∫
Q
∇ψ(ur)∇ξdxdt−

∫
Q
ξdµ− 〈u0, ξ(·, 0)〉Ω (3.1)

where ur is the density of the absolutely continuous part of the Radon-measure with re-
spect to the Lebesgue measure such that 0 ≤ ur ∈ L∞((0, T ), L1(Ω)).
Remark 3.1 In (3.1), we can choose test functions ξ in C1(Q) which vanish on ∂Ω× [0, T ]
and t = T .
The following theorem gives necessary conditions on the measures µ and u0 for the ex-
istence of weak solutions to the problem (P ) with respect to the parabolic capacity and
Newtonian capacity respectively.

Theorem 3.1. Assume that (I), (J), µ ∈ M+(Q) and u0 ∈ M+(Ω) hold. If u is a
weak solution to the problem (P ). Then µ and u0 ⊗ δ{t=0} are absolutely continuous
measures with respect to the parabolic capacity.



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 211

Since Newtonian capacity and parabolic capacity are equivalent, then µ and u0 ⊗ δ{t=0}
are absolutely continuous measures with respect to the C2-capacity as well.

Theorem 3.2. Assume that the hypothesis (I) holds. Let u be a weak solution to
the problem (P ). Then there exist a set F ⊂ (0, T ) with zero Lebesgue measure and
νt ∈M+(Ω) such that

[u(·, t)− u0]c,2 =
[
νt
]
c,2

(3.2)

for every t ∈ (0, T ) \ F .
Remark 3.2. Theorem 3.2 improves Theorem 2.4 in [23].

To prove the existence of solutions to the problem (P ), we will consider the approximating
problems 

unt = ∆ψn(un) + µn in Q := Ω× (0, T ),

un = 0 on ∂Ω× (0, T ),

u(x, 0) = u0n in Ω,

(Pn)

where {u0n} ⊆ C∞0 (Ω) and {µn} ⊆ C∞c (Q) satisfy
u0n

∗
⇀ u0 in M+(Ω),

u0n → u0r a.e in Ω,

‖ u0n ‖L1(Ω)≤‖ u0 ‖M+(Ω) .

(3.3)

And {
µn

∗
⇀ µ in M+(Q),

‖ µn ‖L1(Q)≤‖ µ ‖M+(Q) .
(3.4)

The approximating function ψn is such that

ψn(u) = ψ(u) +
1

n
(3.5)

for every n ∈ N.
By [3, 20], the approximating problem (Pn) has a solution un in C((0, T ), L1(Ω))∩L∞(Q).
Theorem 3.3. Assume that (I), µ ∈ M+(Q) and u0 ∈ M+(Ω) hold. Then there exists
a weak solution u to the problem (P ) obtained as a limiting point of the sequence {un}
of solutions to the problem (Pn) such that for every t ∈ (0, T ) \H∗, there holds

‖ u(·, t) ‖M+(Ω)≤ C
(
‖ µ ‖M+(Q) + ‖ u0 ‖M+(Ω)

)
. (3.6)

Moreover, there exists a Radon measure νt ∈M+(Ω) such that

[us(·, t)]± ≤ [u0s]
± + [νts]

± in M+(Ω) (3.7)

where C is positive constant and H∗ a zero Lebesgue measure set.
To get the uniqueness of the solution to the problem (P ), we define the notion of very



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 212

weak solutions as follows.
Definition 3.2. For any µ ∈ M+

d,2(Q) and u0 ∈ M+
d,2(Ω), a measure u is called a very

weak solution to the problem (P ) if u ∈ L∞((0, T ),M+(Ω)) such that∫ T

0
〈u(·, t), ξt(·, t)〉Ωdt = −

∫
Q
ψ(ur)∆ξdxdt−

∫
Q
ξdµ− 〈u0, ξ(0)〉Ω (3.8)

for every ξ ∈ C2,1(Q), which vanishes on ∂Ω× [0, T ], for t = T .
The notion of very weak solutions adapted to our study can be found in [18, 33].
Definition 3.3. Let u0 ∈M+

d,2(Ω) and µ ∈M+
d,2(Q) such that

u0 = f0 − divG0 , f0 ∈ L1(Ω) and G0 ∈
[
L2(Ω)

]N
.

µ = f − divG+ gt , f ∈ L1(Q) , G ∈
[
L2(Q)

]N
and g ∈ L2((0, T ), H1

0 (Ω)).

A measure u is called very weak solutions obtained as limit of approximation, if

un
∗
⇀ u in M+(Q) (3.9)

where {un} ⊆ L∞(Q) ∩ L2((0, T ), H1
0 (Ω)) is a sequence of weak solutions to the problem

(Pn) and satisfy 

µn = fn − Fn + gnt ∈ C∞0 (Q),

u0n = f0n − F0n ∈ C∞0 (Ω),

fn → f in L1(Q),

Fn → divG in L2((0, T ), H−1(Ω)),

gn → g in L2((0, T ), H1
0 (Ω)),

F0n → divG0 in H−1(Ω),

f0n → f0 in L1(Ω).

(3.10)

Notice that
µn

∗
⇀ µ in M+(Q) and u0n

∗
⇀ u0 in M+(Ω).

Theorem 3.4. Under assumptions of (I) and (J), then for every µ ∈ M+
d,2(Q) and

u0 ∈ M+
d,2(Ω) , there exists a unique very weak solution obtained as limit of approxima-

tion u of the problem (P ).
Notice that a very weak solution is also weak solution to the problem (P ), therefore the
problem (P ) possesses a unique weak solution obtained as limit of approximation.

4. Approximating problems and the persistence
Now we establish some technical statements which will be used in the proof of the exis-
tence solution.
Lemma 4.1. Assume that (I) and (J) are satisfied and un is the solution of the approx-
imation problem (Pn). Then there exists a zero Lebesgue measure set F ∗ ⊂ (0, T ) such
that

‖ un(·, t) ‖L1(Ω)≤‖ u0 ‖M+(Ω)) + ‖ µ ‖M+(Q) (4.1)



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 213

for every t ∈ (0, T ) \ F ∗ and n ∈ N.
Proof. Assuming that any sequence {Ωj} of smooth open sets such that

Ωj ⊂ Ωj+1 ⊂ Ωj+1 ⊂ Ω , Ω =
∞⋃
j=1

Ωj , dist(Ωj , ∂Ω) ≤ 1

j
.

Let {ρj} ⊆ C∞c (Ω) be any function such that

0 ≤ ρj ≤ 1 in Ω , ρj = 1 in Ωj , | ∇ρj |≤ j in Ω \ Ωj .

Then for any

| ∇ρj |≤ j ≤
1

d(x)

where d(x) := dist(x, ∂Ω) ≤ dist(Ωj , ∂Ω) ( see [24]).
Let us consider the truncated function η such that for any 0 ≤ t1 < t2 ≤ T

η(s) =


0 if 0 ≤ s ≤ t1,
1 if t1 < s < t2,

0 if s ≥ t2.

For any fixed j ∈ N, we choose ξj(x, s) = η(s)ρj(x) as a test function in the problems
(Pn) gives∫

Ω
un(x, t2)ρj(x)dx−

∫
Ω
un(x, t1)ρj(x)dx = −

∫ t2

t1

∫
Ω
η(s)∇ψ(un)∇ρj(x)dxds+

+

∫ t2

t1

∫
Ω
η(s)ρj(x)µn(x)dx. (4.2)

It is worth observing that∣∣∣∣∫
Ω
∇ψ(un)∇ρj(x)dx

∣∣∣∣ ≤| Ω \ Ωj |‖ ∇ψ(un) ‖L2(Ω) .

By letting j to infinity, we deduce that

lim
j→∞

∫
Ω
∇ψ(un)∇ρj(x)dx = 0. (4.3)

By the properties of the sequence functions {ρj}, we set t2 = t , t1 = 0 and then combining
together (4.2) with (4.3), there holds∫

Ω
un(x, t)dx ≤

∫
Ω
u0n(x)dx+

∫ t

0

∫
Ω
dµn. (4.4)

Hence the estimate (4.1) follows. �
To show the existence of the solutions to the problems (P ) we need a priori estimates of



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 214

sequences {ψ(un)}.

Proposition 4.1. Under the assumptions of (I) − (J) and un be the solution of the
approximation problem (Pn). Then we obtain

‖ ∇ψ(un) ‖L2(Q)≤ C. (4.5)

‖ ψ(un) ‖L∞((0,T ),H1
0 (Ω))≤ C. (4.6)

Proof. Since ψ(un) ≥ 0 in Q and ψ(un) = 0 on ∂Ω× (0, T ) for every
t ∈ (0, T ). The fact that un = ψ(ψ−1(un)) ∈ C1([0, T ], H1

0 (Ω)). Take ψ(un) as a test
function in (Pn), we get∫

Q
| ∇ψ(un) |2 dxdt =

∫
Ω

(∫ u0n(x)

0
ψ(s)ds

)
dx−

∫
Ω

(∫ un(x,T )

0
ψ(s)ds

)
dx

+

∫
Q
µnψ(un)dxdt.

It follows that∫
Q
| ∇ψ(un) |2 dxdt ≤

∫
Ω

(∫ u0n(x)

0
ψ(s)ds

)
dx+

∫
Q
µnψ(un)dxdt.

By (I)-(i) and the assumption (3.3), there exists a positive constant C such that (4.5)
holds.
Assume that {ηj} a sequence such that ‖ ηj ‖L1(Ω)≤ C and ηj

∗
⇀ δt0(t) in M+(0, T ).

Suppose that ξ(x, t) = ψ(un)(T − t)α
∫ T
t ηj(s)ds (1 < T − t < τ , α > 1) as a test

function in the approximating problem (Pn), there holds

−
∫

Ω

(∫ u0n(x)

0
ψ(s)ds

)
Tα
∫ T

0
ηj(s)ds+

+

∫
Ω

(∫ un(x,t)

0
ψ(s)ds

){
(T − t)α

∫ T

0
ηj(s)ds+

∫ T

0
ηj(s)(T − t)αdt

}
=

=
1

1 + α

∫
Ω
| ∇ψ(un) |2 dx

(∫ T

0
ηj(s)χ(0,T )(s)ds

)
(T−t)α−

∫
Q
µnψ(un)(T−t)α

∫ t

T
ηj(s)ds.

(4.7)
This leads to the following result(∫ T

0
ηj(s)χ(0,T )(s)ds

)∫
Ω
| ∇ψ(un) |2 dx ≤ C

(
‖ u0 ‖M+(Ω) + ‖ µ ‖M+(Q)

)



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 215

Letting j → +∞ the assertion (4.6) holds true. �
Proposition 4.2. Suppose that (I) − (J) and (1.1) hold. Let un be the solution of the
problem (Pn) and φ ∈ C1(R+) be the function defined by

φ(s) =

∫ s

0
ψ(z)dz. (4.8)

Then the sequence [φ(Tk(ψ(un))]t is bounded in L2((0, T ), H−1(Ω)) + L1(Q). Where
Tk(s) = min{s, k}.
Proof. We choose ψ(un)ϕ as a test function in (Pn), with ϕ ∈ C2,1

c (Q), there holds

[φ(Tk(ψ(un))]t−div [ψ(Tk(ψ(un)))∇ψ(Tk(ψ(un)))] + | ∇ψ(Tk(ψ(un))) |2= ψ(Tk(ψ(un)))µn.

It follows that
‖ [φ(Tk(ψ(un)))]t ‖L2((0,T ),H−1(Ω))+L1(Q)

≤‖ ψ(Tk(ψ(un)))∇ψ(Tk(ψ(un))) ‖L2(Q) + ‖ ∇ψ(Tk(ψ(un))) ‖2L1(Q) + ‖ ψ(Tk(ψ(un)))µn ‖L1(Q) .

By the condition (I), we obtain the sequence {[φ(Tk(ψ(un))]t} is bounded in L2((0, T ), H−1(Ω))+
L1(Q). �

Proof of Theorem 3.1. This proof is similar to ([21, Theorem 1.1]). As in ([27, Propo-
sition 3.1]), it is enough to show that for any compact K ⊂ Q such that µ−(K) = 0 ,(
u−0 ⊗ δ{t=0}

)
(K) = 0 and Cap(K) = 0, then µ+(K) = 0 and

(
u+

0 ⊗ δ{t=0}
)

(K) = 0.
By the equivalence of the capacity, we have Cap(E × {t = 0}) = 0, where E a com-
pact set of Ω with u−0 (E) = 0. Let ε > 0 and we choose an open set U such that(
| µ | + | u0 | ⊗δ{t=0}

)
(U \ K) < ε and K ⊂ U ⊂ Q. Then there exists a sequence

{ϕn} ⊆ C∞0 (Q) such that

(i) 0 ≤ ϕn ≤ 1 in Q, ϕn ≡ 1 in K.

(ii) ‖ ∆ϕn ‖L1(Q)→ 0 as n→∞.

In particular, ϕn → 0 in W , indeed∫
Q
| ∇ϕn |2 dxdt = −

∫
Q
ϕn∆ϕndxdt ≤

∫
Q
| ∆ϕn | dxdt.

Let us consider ϕn as a test function in (P ), there holds∫
Q
ϕndµ+

∫
Ω
ϕn(0)du0 = −

∫
Q
ψ(ur)∆ϕndxdt. (4.9)

On the other hand, we get∫
Q
ϕndµ+

∫
Ω
ϕn(0)du0 ≥ µ+(K) +

(
u+

0 ⊗ δ{t=0}
)

(K)



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 216

−
(
| µ | + | u0 | ⊗δ{t=0}

)
(U \K).

It follows that ∫
Q
ϕndµ+

∫
Ω
ϕn(0)du0 ≥ µ+(K) +

(
u+

0 ⊗ δ{t=0}
)

(K)− ε. (4.10)

Combining (4.11) with (4.12), we obtain that

µ+(K) +
(
u+

0 ⊗ δ{t=0}
)

(K) ≤‖ ψ ‖L∞(Q)‖ ∆ϕn ‖L1(Q) +ε.

Letting n to infinity, we infer that

µ+(K) =
(
u+

0 ⊗ δ{t=0}
)

(K) = 0. �

Proof of Theorem 3.2. Let K ⊆ Ω be any compact set such that C2(K) = 0, there exists
a sequence {φn} ⊆ C∞c (Ω) satisfying (iv) and (viii) as stated in preliminaries, Section 2.
Furthermore, ρV ∈ C∞c (V ) be any smooth function such that

(iii) 0 ≤ ρV ≤ 1 in Ω , ρV ≡ 1 in K.

By standard regularization argument, we consider φτ (x, s) = ρ(x)ητ (s) as a test function
in (3.8), where

ητ (s) =


1 if 0 ≤ s ≤ t,
1
τ (t+ τ − s) if t ≤ s ≤ t+ τ,

0 if s ≥ t+ τ,

for any ρ ∈ C2
0 (Ω) and τ > 0. There holds

1

τ

∫ t+τ

t
〈u(s), ρ〉Ω ds− 〈u0, ρ〉Ω =

∫ T

0
ητ (s)ds

∫
Ω
ψ(ur)∆ρdx+

∫ T

0
ητ (s)

∫
Ω
ρdµ.

Since ητ (s) → χ(0,t] for every s ∈ (0, T ) as τ → 0 and we replace the test function ρ by
φn(x)ρV (x).
Then we infer that

〈u(·, t), φnρV 〉Ω − 〈u0, φnρV 〉Ω =

∫ t

0

∫
Ω
ψ(ur)∆(φnρV )dxds+

∫ t

0

∫
Ω

(φnρV )dµ.

By ([14, Theorem 8, p.85]), the measure µ ∈M+(Q) can be decomposed as λ ∈M+(0, T )
and νt ∈M+(Ω) such that for φnρV ∈ C(Ω), there holds

〈µ, φnρV 〉Q =

∫
(0,T )

dλ(s)

∫
Ω
φnρV dν

t

with λ(s) := δ(0,T )(s), where δ(0,T ) a Dirac measure on (0,T). Therefore,

〈[u(·, t)]c,2, φnρV 〉Ω + 〈[u(·, t)]d,2, φnρV 〉Ω =



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 217

=

∫ t

0

∫
Ω
ψ(ur)∆(φnρV )dxds+

〈
[νt]c,2, φnρV

〉
Ω

+

+
〈
[νt]d,2, φnρV

〉
Ω

+ 〈[u0]c,2, φnρV 〉Ω + 〈[u0]d,2, φnρV 〉Ω . (4.11)

By the assumptions stated above, we infer that

lim
n→∞

∫ t

0

∫
Ω
ψ(ur)∆(φnρV )dxds = 0.

Moreover, since [u(·, t)]d,2, [νt]d,2 , [u0]d,2 belong to L1(Ω)+H−1(Ω) and φn
∗
⇀ 0 in L∞(Ω)

, φn → 0 in H1
0 (Ω) so that

lim
n→∞

〈[u(·, t)]d,2, φnρV 〉Ω = lim
n→∞

〈
[νt]d,2, φnρV

〉
Ω

= lim
n→∞

〈[u0]d,2, φnρV 〉Ω = 0.

It follows that (4.11) can be rewritten as

〈[u(·, t)]c,2, φnρV 〉Ω =
〈
[νt]c,2, φnρV

〉
Ω

+ 〈[u0]c,2, φnρV 〉Ω . (4.12)

Since K is a subset compact of Ω, then

[u(·, t)− u0]c,2 (K) ≤ lim sup
n→∞

〈[u(·, t)− u0]c,2, φnρV 〉Ω = lim sup
n→∞

〈
[νt]c,2, φnρV

〉
Ω
≤
[
νt
]
c,2

(K).

On the other hand, we get[
νt
]
c,2

(K) ≤ lim sup
n→∞

〈
[νt]c,2, φnρV

〉
Ω

= lim sup
n→∞

〈[u(., t)− u0]c,2, φnρV 〉Ω ≤ [u(., t)− u0]c,2 (K).

The above inequality implies that

[u(·, t)− u0]c,2 (K) ≤ inf
{[
νt
]
c,2

(V ) | K ⊂ V, open
}

=
[
νt
]
c,2

(K).

Similarly, we have[
νt
]
c,2

(K) ≤ inf
{

[u(·, t)− u0]c,2 (V ) | K ⊂ V, open
}

= [u(., t)− u0]c,2 (K).

Whence, the following statement[
νt
]
c,2

(K) = [u(·, t)− u0]c,2 (K) (4.13)

holds true. According to the arbitrariness of K, (4.13) is satisfied for every Borel set
E ⊆ Ω with C2(E) = 0. By the definition of concentrated measure with respect to the
Newtonian capacity, we have for any t ∈ (0, T ) \ F ,

[u(·, t)]c,2 = [u(·, t)]c,2 xB1(t) ,
[
νt
]
c,2

=
[
νt
]
xB2(t) and [u0]c,2 = [u0]c,2 xA

for some Borel sets B1(t), B2(t), and A is a zero Newtonian capacity, then (4.13) yields

[u(·, t)]c,2 ((B1(t) ∪B2(t)) \A) =
[
νt
]
c,2

((B1(t) ∪B2(t)) \A) =



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 218

= [u0]c,2 ((B1(t) ∪B2(t)) \A) = 0.

Therefore for every t ∈ (0, T ) \F , [u(·, t)]c,2 ,
[
νt
]
c,2

, [u0]c,2 are concentrated measures on

the set B∗(t) such that B∗(t) = (B1(t)∩A)∪ (B2(t)∩A). Therefore, for every set E ⊆ Ω
and t ∈ (0, T ) \ F , there holds

[u(·, t)− u0]c,2 (E) =
(

[u(·, t)− u0]c,2 xB
∗(t)
)

(E) = [u(·, t)− u0]c,2 x(B
∗(t) ∩ E) =

=
[
νt
]
c
x(B∗(t) ∩ E) =

([
νt
]
c,2
xB∗(t)

)
(E).

Hence, the proof is achieved. �

5. Existence results
We prove the existence result of the problem (P ).
Proposition 5.1. Assume that (I) and (J) hold. Let un be the solution to the approxima-
tion problem (Pn), then there exist a subsequence

{
unj

}
⊆ {un} and v ∈ L2((0, T ), H1

0 (Ω))∩
L∞((0, T ), H1

0 (Ω)) ∩ L∞(Q) with 0 ≤ v ≤ γ in Q such that

ψ(unj )
∗
⇀ v in L∞(Q). (5.1)

∇ψ(unj ) ⇀ ∇v in
[
L2(Q)

]N
. (5.2)

ψ(unj )→ v a.e in Q. (5.3)

Proof. By the assumption (I)-(ii), the sequence {ψ(un)} is uniformly bounded in L∞(Q),
then from [5] there exists a function v ∈ L∞(Q) such that the convergence in (5.1) holds
true. Furthermore, the convergence (5.2) stems from estimate (4.5).
By (4.6), we have

| ∇φ(Tk(ψ(un))) |=| ∇Tk(ψ(un)) || ψ(Tk(ψ(un))) |

≤ γ | ∇Tk(ψ(un)) | . (5.4)

It follows that,∫
Q
| ∇φ(Tk(ψ(un))) | dxdt ≤ γ | Q |

[∫
Q
| ∇Tk(ψ(un)) |2 dxdt

] 1
2

.

Since Tk(ψ(un)) ∈ L2((0, T ), H1
0 (Ω)) then there exists a positive constant C such that∫
Q
| ∇φ(Tk(ψ(un)) | dxdt ≤ C. (5.5)

By Proposition 4.2, the sequence [φ(Tk(ψ(un)))]t is bounded in
L2((0, T ), H−1(Ω)) + L1(Q). According to the compactness theorem in [29], then there



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 219

exists a subsequence denoted again
{
ψ(unj )

}
(possibly for k > 0, Tk(ψ(un)) = ψ(unj ) and

| ψ(unj ) |≤ k ) and a function v ∈ L1((0, T ),W 1,1
0 (Ω)) ∩ L1(Q) such that

φ(ψ(unj ))→ v a.e in Q. (5.6)

Therefore, we get
ψ(unj )→ φ−1(v) a.e in Q. (5.7)

Combining (5.6) with (5.1) gives φ−1(v) = v, this proves (5.3). �

We recall the following sclicing property of the bounded Radon measure
u ∈ M(Q). The proof is omitted since it follows from the more general result in ([14,
Theorem 8, p.35]).

Proposition 5.2. Assume that µ ∈M+(Q). Then there exists a measure λ ∈M+(0, T )
and for λ almost everywhere t ∈ (0, T ), there exists a probability νt ∈ M+(Ω) with the
following properties

(i) for any Borel set E ⊆ Q
µ(E) =

∫
(0,T )

νt(Et)dλ(t) (5.8)

where Et = {x ∈ Ω/(x, t) ∈ E}

(ii) for every ξ ∈ C(Q)

〈µ, ξ〉Q =

∫
(0,T )

dλ(t)

∫
Ω
ξ(x, t)dνt(x). (5.9)

Proposition 5.3. Let
{
unj

}
and v as in Proposition 5.1. Then the following assertions

hold
(i) ψ−1(unj ) ∈ L1(Q) and we have

unj (x, t)→ [ψ−1(v)](x, t) a.e (x, t) ∈ Q. (5.10)

(ii) There exist λ1, λ2 ∈ L∞((0, T ),M+(Ω))) and we can extract a subsequence still de-
noted

{
unj

}
such that

u+
nj

∗
⇀ [ψ−1(v)]+ + λ1 in M+(Q), (5.11)

u−nj

∗
⇀ [ψ−1(v)]− + λ2 in M+(Q), (5.12)

unj

∗
⇀ [ψ−1(v)] + λ in M+(Q), (5.13)

where λ := λ1 − λ2 in L∞((0, T ),M+(Ω)).
Proof. From (5.3), (4.1) and ψ−1(unj ) ∈ L1(Q), then by Fatou’s Lemma, we get∫

Q
[ψ−1(v)](x, t)dxdt ≤ lim inf

j→∞

∫
Q
unj (x, t)dxdt. (5.14)



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 220

By (5.3) the convergence (5.11) is satisfied.
Since the sequence

{
unj

}
is uniformly bounded in L1(Q) and by (4.6), there exist a

subsequence
{
unj

}
which still denote

{
unj

}
and Radon-measures

u , ũ ∈M+(Q) such that

u+
nj

∗
⇀ u in M+(Q). (5.15)

u−nj

∗
⇀ ũ in M+(Q). (5.16)

Let us prove that u , ũ ∈ L∞((0, T ),M+(Ω)). To prove this, we consider λi ∈ M+(0, T )
and λi-a.e t ∈ (0, T ). Let νti ∈ M+(Ω) be the measure given by Proposition 5.2 in
correspondence with each u , ũ. Let us show that the measures λi ∈ M+(0, T ) are
absolutely continuous with respect to the Lebesgue measure over (0, T ). In this direction,
fix arbitrarily t ∈ (0, T ) and choose r, s > 0 such that Jr,s ≡ (t−r−2s, t+r+2s) ⊆ (0, T ).
Then for every function ηr,s ∈ C1

c (0, T ) such that

ηr,s ≡ 1 in [t− r − 2s, t+ r + 2s] , 0 ≤ ηr,s ≤ 1 , suppηr,s ⊆ Jr,s.

By the estimate (4.1), we have∫
Q
u±nj

ηr,s(t)dxdt ≤ 2(r + 2s) ‖ µ ‖M+(Q) +2(r + 2s) ‖ u0 ‖M+(Ω) . (5.17)

By (5.15), (5.16) and (5.17), there holds∫
[t−r,t+r]

dλi(t) ≤
∫

(t−r−2s,t+r+2s)
νti (Ω)dλi(t) ≤ lim inf

k→∞

∫
Q
u±nj

(x, t)ηr,s(t)dxdt.

Thus ∫
[t−r,t+r]

dλi(t) ≤ 2(r + 2s) ‖ µ ‖M+(Q) +2(r + 2s) ‖ u0 ‖M+(Ω) .

Noting s is arbitrary, thus we divide both sides of the above inequality by 2r, we obtain

1

2r

∫
[t−r,t+r]

dλi(t) ≤‖ µ ‖M+(Q) + ‖ u0 ‖M+(Ω) .

Therefore there exists hi ∈ L1(0, T ), hi ≥ 0 such that dλi(t) = hi(t)dt, this means that
the Radon-measure M+(0, T ) is regular (e.g, [9]).
Since u , ũ ∈ M+(Q) are nonnegative Radon-measures, letting r → 0 in the previous
inequality yields

0 ≤ hi(t) ≤ C
(
‖ µ ‖M+(Q) + ‖ u0 ‖M+(Ω)

)
for almost every t ∈ (0, T ). Finally, defining

u(t) = h1(t)νt1 and ũ(t) = h2(t)νt2 for almost everywhere t ∈ (0, T ).

From (5.7) and (5.8) we obtain that u, ũ ∈ L∞((0, T ),M+(Ω)).
Since unj → ψ−1(v) almost everywhere in Q, then u±nj

→ [ψ−1(v)]± almost everywhere in



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 221

Q.
By (5.14) and (5.16), then we infer from Fatou’s Lemma∫

Q
[ψ−1(v)]+ξ(x, t)dxdt ≤ lim inf

j→∞

∫
Q
u+
nj
ξ(x, t)dxdt ≤ 〈u, ξ〉Q .

Similarly, we have∫
Q

[ψ−1(v)]−ξ(x, t)dxdt ≤ lim inf
j→∞

∫
Q
u−nj

ξ(x, t)dxdt ≤ 〈ũ, ξ〉Q

for every ξ ∈ Cc(Q), ξ ≥ 0, thus defining

λ1 = u− [ψ−1(v)]+ and λ2 = ũ− [ψ−1(v)]−.

Hence, λ1 , λ2 ∈ L∞((0, T ),M+(Ω)) hods true. �

Proposition 5.4. Let u and v be in Proposition 5.3 and Proposition 5.1. Then for
almost every t ∈ (0, T ), we have

unj (·, t)→ [ψ−1(v)](·, t) a.e in Ω. (5.18)

u+
nj

(t)
∗
⇀ [ψ−1(v)]+(·, t) + λ1(·, t) in M+(Ω). (5.19)

u−nj
(·, t) ∗⇀ [ψ−1(v)]−(·, t) + λ2(·, t) in M+(Ω). (5.20)

unj (·, t)
∗
⇀ [ψ−1(v)](·, t) + λ(·, t) in M+(Ω). (5.21)

Proof. This proof is similar to that given in [18, 24]. Let us recall the statement of the
function F which belongs to C2(R+) (see [18, Proposition 4.3]. Let un be the solution of
the problem (Pn), and F ∈ C2(R+), then for any ρ ∈ C1

c (Ω), ρ(x) ≥ 0 and there exists a
zero Lebesgue measure set H such that (0, T ) \H, the following identity is satisfied∫

Ω
F(un)(x, t)ρ(x)dx−

∫
Ω
F(un)(x, 0)ρ(x)dx =

=

∫ T

0

∫
Ω

{
−F ′(un)∇ψ(un)∇ρdx− F

′′(un)

ψ′(un)
| ∇ψ(un) |2 ρ

}
dxdt+

+

∫ T

0

∫
Ω
µnF ′(un)ρdxdt. (5.22)

The convergence (5.18) immediately follows from (5.3). Next let us fix J > 1 and we
consider the functions FJ , RJ ∈ C2(R+) defined as follows

FJ(s) =


0 if 0 ≤ s ≤ J,
s− J if J ≤ s ≤ J + 1,

s− J if s ≥ J + 1,



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 222

and RJ(s) = s−FJ(s) (s ∈ R+) and RJ(s)χ{s≥J+1} = J .
Let us consider the function Hn belongs to C1(R+) by setting

Hn,ρ(t) =

∫
Ω
FJ(un(x, t))ρ(x)dx.

By (4.1), there exists a positive constant C such that∫ T

0
| Hn,ρ(t) | dt ≤‖ ρ ‖L∞(Ω)

∫ T

0

∫
Ω
u+
n (x, t)dxdt ≤ C

where C = C
[
T, ‖ ρ ‖L∞(Ω), ‖ u0 ‖M+(Ω), ‖ µ ‖M+(Q)

]
> 0.

Thus Hn,ρ ∈ L1(0, T ) for every ρ ∈ C1
c (Ω). Furthermore by (5.22) yields∫ T

0

∣∣∣∣dHn,ρ(t)dt

∣∣∣∣ dt ≤ ∫ T

0

∫
Ω
F ′J(un) | ∇ψ(un) |2| ∇ρ | dxdt+

+

∫ T

0

∫
Ω

F ′′(un)

ψ′(un)
| ∇ψ(un) |2 ρdxdt+

∫ T

0

∫
Ω
µnF ′(un)ρdxdt. (5.23)

By properties of sequence {FJ(un)}J>1 mentioned above and ρ ∈ C1
c (Ω), there exists a

positive constant
C = C

[
‖ ρ ‖L∞(Ω), ‖ u0 ‖M+(Ω), ‖ µ ‖M+(Q)

]
> 0 such that∫ T

0

∣∣∣∣dHn,ρ(t)dt

∣∣∣∣ dt ≤ C.
Thus the family Hn,ρ is uniformly bounded in W 1,1(0, T ).
Hence there exist a subsequence

{
Hnj ,ρ

}
⊆ {Hn,ρ} and a function

Hρ ∈ L1(0, T ) such that
Hnj ,ρ → Hρ in L1(0, T ). (5.24)

By the properties of the function FJ , the function RJ is continuous and bounded in R+,
then the convergence (5.10) and the dominated convergence theorem imply that

RJ(unj )→ RJ
(
ψ−1(v)

)
in L1(Q). (5.25)

By (5.10), (5.11) and the definition of RJ , we have

FJ(unj ) = u+
nj
−RJ(unj )

∗
⇀
[
ψ−1(v)

]+
+λ1−RJ

(
ψ−1(v)

)
= FJ

(
ψ−1(v)

)
+λ1 in M+(Q).

(5.26)
In view of (5.24) and (5.26), for any h ∈ Cc(0, T ) and ρ ∈ C1

c (Ω) we get∫ T

0
Hρ(t)h(t)dt = lim

j→∞

∫ T

0
Hnj ,ρ(t)h(t)dt = lim

j→∞

∫
Q
FJ(unj )ρ(x)h(t)dxdt =

=

∫ T

0
h(t)

〈
FJ
(
ψ−1(v)(·, t)

)
+ λ1(·, t), ρ

〉
Ω
dt.



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 223

Then by the above equality, we deduce that

Hρ(t) =
〈
FJ
(
ψ−1(v)(·, t)

)
+ λ1(·, t), ρ

〉
Ω

for almost every t ∈ (0, T ) and

Hj,ρ →
〈
FJ
(
ψ−1(v)(·, t)

)
+ λ1(·, t), ρ

〉
Ω

in L1(0, T )

for any ρ ∈ C1
c (Ω). �

Proof of Theorem 3.3. Let us show that for every ρ ∈ C1
c (Ω), ρ ≥ 0 and for almost

every τ ∈ (0, T ), there exists a Radon measure ντ ∈M+(Ω) such that

〈λ1(τ), ρ〉Ω ≤
〈
[u0s]

+ + [ντs ]+, ρ
〉

Ω
, (5.27)

〈λ2(τ), ρ〉Ω ≤
〈
[u0s]

− + [ντs ]−, ρ
〉

Ω
. (5.28)

We prove the first inequality (5.27) and the second one follows by similar argument. Fix
any ρ ∈ C1

c (Ω), ρ ≥ 0 and we consider the sequence {FJ(un)} as mentioned above and we
use it in (5.22), then we obtain for every τ ∈ (0, T )∫

Ω
FJ(un)(x, τ)ρ(x)dx−

∫
Ω
FJ(u0n)(x)ρ(x)dx

≤ −
∫ τ

0

∫
Ω
F ′J(un)∇ψ(un)∇ρdxdt+

∫ τ

0

∫
Ω
µnF ′J(un)ρdxdt. (5.29)

Let us consider
{
unj

}
the sequence given in Proposition 5.1 and Proposition 5.2 and let

us take the limit as j tends to infinity in (5.29) (with n = nj). By (5.2), (5.3) and the fact
that

{
F ′J(unj )

}
is bounded in L∞(Q), there holds

lim
j→∞

∫ τ

0

∫
Ω
F ′J(unj )∇ψ(unj )∇ρdxdt =

∫ τ

0

∫
Ω
F ′J(ψ−1(v))∇v∇ρdxdt.

In view of the definition of the sequence
{
F ′J(unj )

}
, yields

0 ≤ F ′J(unj ) ≤ 1 , F ′J(unj )→ 0 as J →∞ and ψ−1(v) ∈ L1(Q).

It follows that

lim
J→∞

lim
j→∞

∫ τ

0

∫
Ω
F ′J(ψ−1(v))∇v∇ρdxdt = 0. (5.30)

On the other hand, by (5.26) one has

lim
j→∞

∫
Ω
FJ(unj (x, τ))ρ(x)dx =

∫
Ω
FJ(ψ−1(v))(x, τ)dx+ 〈λ1(τ), ρ〉Ω .

Referring to the definition of the sequence {FJ(un)}J>1, we infer that

0 ≤ FJ(unj ) ≤ 1 , FJ(unj )→ 0 as J →∞ and ψ−1(v) ∈ L1(Q).



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 224

Then we obtain

lim
J→∞

lim
j→∞

∫
Ω
FJ(un(τ))ρ(x)dx = 〈λ1(τ), ρ〉Ω . (5.31)

Let us consider the sequence {u0n(x)} satisfies (3.3), then

FJ(u0nj ) = [u0nj ]
+ −RJ(u0nj ) ≤ [u0rnj ]

+ + [u0sn]+ −RJ(u0nj ).

Since u0rnj → u0r in L1(Ω) and the sequence
{
RJ(u0rnj )

}
is bounded in L∞(Ω), we obtain

[u0rnj ]
+ −RJ(u0nj )→ [u0r]

+ −RJ(u0r) = FJ(u0r) in L1(Ω)

which leads to

lim
J→∞

lim sup
j→∞

∫
Ω
FJ(u0nj )ρ(x)dx ≤

〈
[u0s]

+, ρ
〉

Ω
. (5.32)

Let us now consider the function ηr,s constructs from the function ηr,s given in Proposition
5.2 as follows

ηr,s(t) =

∫ t

t+r+2s
ηr,s(θ)dθ for every θ ∈ (0, T )

we deduce that∫ τ

0

∫
Ω
µnjF ′J(unj )ρdxdt =

∫ τ

0

∫
Ω
µnj

(
1− ηr,s(t)

)
F ′J(unj )ρdxdt+

+

∫ τ

0

∫
Ω
µnjηr,s(t)F ′J(unj )ρdxdt. (5.33)

Since
{
µnj

}
is a nonnegative bounded Radon-measure, and the function

1− ηr,s(t) is bounded in R+, there holds

lim sup
j→∞

∫ τ

0

∫
Ω
µnj

(
1− ηr,s(t)

)
F ′J(unj )ρdxdt ≤

∫ τ

0

〈
µ, ρFJ(ψ−1(v))

(
1− ηr,s(t)

)〉
dt.

Letting J to infinity, we obtain

lim
J→∞

lim sup
j→∞

∫ τ

0

∫
Ω
µnj

(
1− ηr,s(t)

)
F ′J(unj )ρdxdt = 0. (5.34)

By [11, Theorem 8, p.85], there exist νtnj
∈M+(Ω) and δ0 ∈M+(0, T ) for µnj ∈M+(Q)

such that (5.33), becomes∫ τ

0

∫
Ω
µnjηr,s(t)F ′J(unk

)ρdxdt ≤ ηr,s(0)

∫
Ω
ντnj
FJ(unj )ρdx ≤ (4r + 2s)

∫
Ω
ντnj
FJ(unj )ρdx.

Setting r = 1
8 and s = 1

4 , then∫ τ

0

∫
Ω
µnjηr,s(t)F ′J(unj )ρdxdt ≤

∫
Ω

[ντs ]+nj
ρdx+

∫
Ω

[ντr ]+nj
FJ(unj )ρdx.



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 225

Therefore,

lim
J→∞

lim sup
j→∞

∫ τ

0

∫
Ω
µnjF ′J(unj )ρdxdt ≤

〈
[ντs ]+, ρ

〉
Ω
. (5.35)

Combining (5.30), (5.31), (5.32), (5.34) and (5.35) together. Hence (5.27) holds true. �
Remark 5.1. By the assumptions (I) and (J), it has been proved that
(i) the set

S̃ = {(x, t) ∈ Ω/ψ(ur)(x, t) = γ}
has zero Lebesgue measure (see [23, Proposition 5.2]).
(ii) There hold

supp(u(x, t)) ⊆ S̃ and ur = ψ−1(v) a.e in Q \ S̃
(see [30, Proposition 4.1]).
6. Monotonicity and Uniqueness Results
Lemma 6.1. Under assumption (I). If u is a weak solution of the problem (P ). Then
(i) there exist a zero Lebesgue measure set D ⊆ (0, T ) and a positive constant c such that

ess lim
t→0+

∫
Ω
u(·, t)dx = c (6.1)

(ii) for any ρ ∈ C2
0 (Ω), ρ ≥ 0, there holds

ess lim
t→0+

〈u(·, t), ρ〉Ω = 〈u0, ρ〉Ω (6.2)

for almost every t ∈ (0, T ) \D.
Proof. Let us consider for every τ > 0, the smooth function ητ ∈ C1

0 (0, T ), 0 ≤ ητ ≤ 1
such that

ητ (t) =



0 if 0 ≤ t ≤ t1 − τ,
1
τ (t+ τ − t1) if t1 − τ ≤ t ≤ t1,
1 if t1 ≤ t ≤ t2,
1
τ (−t+ τ + t2) if t2 ≤ t ≤ t2 + τ,

0 if t2 + τ ≤ t ≤ T.

Let us choose ρj(x)ητ (t) as a test function in (P ), there holds∫ T

0

∫
Ω

{
−uρj(x)η′τ (t)− ψ(ur)ητ (t)∆ρj(x)

}
dxdt =

∫ T

0

∫
Ω
µρj(x)ητ (t)dxdt.

It is worth observing that the first term of the left hand side of the above equality becomes∫ T

0

∫
Ω
−uρj(x)η′τ (t)dxdt = −1

τ

∫ t1

t1−τ

∫
Ω
u(x, t)ρj(x)dxdt+

1

τ

∫ t2+τ

t2

∫
Ω
u(x, t)ρj(x)dxdt.

Let us consider a zero Lebesgue measure set Dj in (0, T ) such that for any t1, t2 ∈
(0, T ) \Dj , one has

lim
τ→0

∫ T

0

∫
Ω
−uρj(x)η′τ (x, t)dxdt = −

∫
Ω
u(x, t1)ρj(x)dx+

∫
Ω
u(x, t2)ρj(x)dx.



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 226

We use a sequence {ρj(x)}j∈N of test functions in Ω such that

ρj(x) ∈ C2
0 (Ω), 0 ≤ ρj(x) ≤ 1, ρj(x) → 1 in Ω and −∆ρj(x) ≥ 0 (for instance, ρj(x) =

1−(1− φ)j , where φ is the first eigenfunction of −∆ in H1
0 (Ω), with normalization maxφ =

1)(see [6] reference therein). For every s ∈ (0, T ) \Dj , there holds∫
Ω
u(x, t)ρj(x)dx−

∫
Qt

ψ(ur)∆ρj(x)dxds =

∫
Qt

ρj(x)dµ+

∫
Ω
ρj(x)du0.

Let j goes to infinity, then we get that

D ≡
⋃
j∈N

Dj

which leads to ∫
Ω
u(x, t)dx ≤

∫
Qt

dµ+

∫
Ω
du0.

Now let us consider {φk} be a sequence of C0(Ω) functions such that
0 ≤ φj ≤ 1, φk → 1 as j →∞.
By [12, Lemma 5.1], the following statement hold∫

Ω
φjdu0 ≤

1

j
and

∫
Qt

φjdµ ≤
1

j

then∫
Qt

dµ+

∫
Ω
du0−

∫
Ω
u(x, t)dx =

∫
Qt

(1− φj) dµ+

∫
Qt

φjdµ+

∫
Ω

(1− φj) du0 +

∫
Ω
φjdu0−

−
∫

Ω
u(x, t)φjdx+

∫
Ω
u(x, t) (φj − 1) dx.

Since φj ≤ 1 yields∫
Qt

dµ+

∫
Ω
du0 −

∫
Ω
u(x, t)dx ≤

∫
Qt

(1− φj) dµ+

∫
Ω

(1− φj) du0 −
∫

Ω
u(x, t)φjdx+

2

j
.

Since u(x, t) converges to δx, we get

lim sup
t→0+

∣∣∣∣∫
Ω
du0 −

∫
Ω
u(x, t)dx

∣∣∣∣ ≤ ∫
Ω

(1− φj) du0 +
2

j
.

Let j to infinity, there exists a positive constant c such that (6.1) holds. Using the same
method as the previous, it is obvious that for every ρ ∈ C2

0 (Ω)

ess lim
t→0+

〈u(x, t), ρ〉Ω = 〈u0, ρ〉Ω .



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 227

Hence (6.2) is satisfied. �
For every g ∈ C1(R)

G(s) =

∫ s

0
g(ψ(z))dz. (6.3)

Assuming (I) holds. Let us state the following definition.
Definition 6.1. For any µ ∈ M+

d,2(Q) and u0 ∈ M+
d,2(Ω), a measure u is called a weak

entropy solution, if u is a weak solution of (P ) such that for every g ∈ C1(R), g′ ≥ 0,
g(γ) = 0, the inequality holds∫

Q

{
g′(ψ(ur)) | ∇ψ(ur) |2 φ+ g(ψ(ur))∇ψ(ur)∇φ−G(ur)φt

}
dxdt

≤
∫
Q
g(ψ(ur))φdµ+

∫
Ω
G(u0r)φ(0)dx (6.4)

for every φ ∈ C1([0, T ], C1
0 (Ω)), φ(., T ) = 0 in Ω and φ ≥ 0.

By the Definition 6.1, the existence of weak entropy solutions of problem (P ) is the same
as stated in [23, Theorem 2.8]. For that we use entropy inequality to prove the mono-
tonicity of solutions given by the following proposition.

Proposition 6.1. Suppose that the assumption (I) holds. Let u be a weak entropy
solution to the problem (P ).
For any ρ ∈ H1

0 (Ω), ρ ≥ 0, then

〈us(·, t2), ρ〉Ω ≤ 〈us(·, t1), ρ〉Ω ≤ 〈u0s, ρ〉Ω (6.5)

hols, for almost every t1, t2 ∈ (0, T ); t1 < t2.
Proof. Let Gj be the function given in (6.3) and we take g = gj for any j ∈ N. By the
Definition 6.1, we obtain∫

Q

{
g′j(ψ(ur)) | ∇ψ(ur) |2 φ+ gj(ψ(ur))∇ψ(ur)∇φ−Gj(ur)φt

}
dxdt

≤
∫
Q
gj(ψ(ur))φdµ+

∫
Ω
Gj(u0r)φ(0)dx (6.6)

for every φ ∈ C1([0, T ], C1
0 (Ω)), φ(·, T ) = 0 in Ω and φ ≥ 0, where

gj(s) =


−1 if s ≤ γ − 1

j ,

j(s− γ) if γ − 1
j ≤ s ≤ γ,

0 if s ≥ γ.

To avoid repeating the same calculation we refer to the proof of [23, Theorem 2.9].
Then by letting j to infinity, we get∫

Q
{urφt −∇ψ(ur)∇φ} dxdt ≤ −

∫
Q
φdµ−

∫
Ω
u0rφ(0)dx. (6.7)



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 228

Combining (6.7) with (3.1), we have

−
∫ T

0
〈us(·, t), φt〉Ω dt ≤ 〈u0s, φ(0)〉Ω . (6.8)

For any fix 0 ≤ t1 < t2 ≤ T . We consider

χr(t) =


1
r

(
t− t1 + r

2

)
if t1 − r

2 < t < t1 + r
2 ,

1 if t1 + r
2 < t < t2 − r

2 ,

−1
r

(
t− t2 − r

2

)
if t2 − r

2 < t < t2 + r
2 ,

0 otherwise,

where 0 < r < t2 − t1, such that [t1 − r
2 , t2 + r

2 ] ⊂ (0, T ) (see [30, Theorem 2.5]. For any
φ ∈ C1

0 (Ω), ρ ≥ 0 we choose φ(x, t) = ρ(x)χr(t) as a test function in (6.8), one has

−1

r

∫ t1+ r
2

t1− r
1

〈us(t), ρ〉Ω dt+
1

r

∫ t2+ r
2

t2− r
2

〈us(t), ρ〉Ω dt ≤ 0

for almost every 0 < t1 < t2 < T and letting r → 0 in the above inequality, there holds

〈us(·, t2), ρ〉Ω ≤ 〈us(·, t1), ρ〉Ω .

Similarly, let us consider for every fixed t1 ∈ (0, T )

χr(t) =


1 if 0 ≤ t ≤ t1,
−1
r (t− t1 − r) if t1 ≤ t ≤ t1 + r,

0 if t ≥ t1 + r.

Therefore, we can deduce that

1

r

∫ t1+r

t1

〈us(·, t), ρ〉Ω dt ≤ 〈u0s, ρ〉Ω .

Hence the estimate (6.5) holds true. �

Proof of Theorem 3.4. Let u1 , u2 be two very weak solutions obtained as limit of ap-
proximation of (P ) with initial data u01n and u02n respectively . Let {u1n}, {u2n} ⊆
L∞(Q) ∩ L2((0, T ), H1

0 (Ω)) be two approximating sequences of solutions to the approxi-
mation problem (Pn) and satisfying the assumption (3.9).
For every ξ ∈ C2,1(Q) vanishing on ∂Ω× (0, T ) and ξ(·, T ) = 0 in Ω, there holds∫

Q
(u1n − u2n) ξtdxdt = −

∫
Q

(ψ(u1n)− ψ(u2n)) ∆ξdxdt−

−
∫
Q

(µ1n − µ2n) ξdxdt−
∫

Ω
(u01n − u02n) ξ(x, 0)dx, (6.9)



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 229

where {µ1n}, {µ2n}, {u01n}, and {u02n} are approximating Radon measures satisfying
(3.10).
For almost every (x, t) ∈ Q, we consider the function an(x, t) defined by

an(x, t) =

{
ψ(u1n(x,t))−ψ(u2n(x,t))

u1n(x,t)−u2n(x,t) if u1n(x, t) 6= u2n(x, t),

ψ′(u1n(x, t)) if u1n(x, t) = u2n(x, t).
(6.10)

Obviously an ∈ L∞(Q) and for every n ∈ N there exists a positive constant Cn such that

ess inf
(x,t)∈Q

an(x, t) ≥ Cn > 0.

This ensures that for every z ∈ C2
c (Q), the problem

ξnt + an∆ξn + z = 0 in Q

ξn = 0 on ∂Ω× (0, T )

ξn(·, T ) = 0 in Ω

(6.11)

has a unique solution ξn ∈ L∞((0, T ), H2(Ω)) ∩ L2((0, T ), H1
0 (Ω)) with ξnt ∈ L2(Q) (see

[8, 19]).
Moreover, it can be seen that

| ξn(x, t) |≤ (T − t) ‖ z ‖L∞(Q) . (6.12)

Let us consider the function η such that for any t1 + 1 < t2 and t1 , t2 ∈ (0, T )

η(t) =


0 if 0 ≤ t ≤ t1,
t− t1 if t1 < t < t2,

t2 − t1 if t ≥ t2.

Choosing η∆ξn as a test function in (6.11), then we obtain∫
Q
ξntη(t)∆ξndxdt+

∫
Q
η(t)an(x, t)[∆ξn]2dxdt+

∫
Q
zη(t)∆ξndxdt = 0. (6.13)

It follows that

1

2

∫
Q
| ∇ξn |2 dxdt+

∫
Q
an(x, t)[∆ξn]2dxdt ≤ C0(T, z) (6.14)

holds, for some constant C0(T, z) independent on n.
From (6.12) and (6.14), there exists a constant C1(T, z) such that

‖ ξn ‖L2((0,T ),H1
0 (Ω)) + ‖

√
an∆ξn ‖L2(Q)≤ C1(T, z). (6.15)

On the other hand, multiplying (6.11) by ∆ξn, we obtain

−
∫
Q
∇ξn∇ξnt +

∫
Q
an[∆ξn]2dxdt = −

∫
Q
ξn∆zdxdt



Quincy S. Nkombo, Fengquan Li / Eur. J. Pure Appl. Math, 14 (1) (2021), 204-233 230

which leads to

1

2

∫
Ω
| ∇ξn |2 (x, 0)dx+

∫
Q
an[∆ξn]2dxdt ≤ C2(T, z), (6.16)

where C2(T, z) =‖ ξn ‖L∞(Q)‖ z ‖C2(Q). Therefore, we get

‖ ξn(., 0) ‖H1
0 (Ω) + ‖

√
an∆ξn ‖L2(Q)≤ C2(T, z). (6.17)

By standard density argument and for ξ = ξn a test function in (6.9). Moreover, by
recalling (6.10) and (6.9), there holds∫

Q
(u1n − u2n) zdxdt =

∫
Q

(µ1n − µ2n) ξ(x, t)dxdt+

∫
Ω

(u01n − u02n) ξ(x, 0)dx. (6.18)

Letting n to infinity in (6.18). Then it is enough to observe from (6.15), there exists ξn ∈
L∞((0, T ), H2(Ω))∩L2((0, T ), H1

0 (Ω)) which is obtained by extracting the subsequence of
the sequence {ξn}, such that

ξn(x, t)
∗
⇀ ξ(x, t) in L∞(Q). (6.19)

∇ξn(x, t) ⇀ ∇ξ(x, t) in [L2(Q)]N . (6.20)

Since ξnt ∈ L2(Q), as stated in [19], we deduce that

ξnt(x, t)→ ξt(x, t) in L2(Q), (6.21)

ξn(x, t)→ ξ(x, t) a.e in Q. (6.22)

On one hand, it is enough to observe that from (6.17), there exists
ξ(·, 0) ∈ L∞(Ω) ∩H1

0 (Ω) such that the following statements

ξn(x, 0)
∗
⇀ ξ(x, 0) in L∞(Ω), (6.23)

ξn(x, 0) ⇀ ξ(x, 0) in H1
0 (Ω), (6.24)

holds true.
Combining (6.18)-(6.24) and (3.10), there holds

lim
n→∞

∫
Q

(u1n − u2n) zdxdt = lim
n→∞

∫
Q

(f1n − f2n) ξ(x, t)dxdt+

+ lim
n→∞

∫
Q

(F1n − F2n) ξ(x, t)dxdt− lim
n→∞

∫
Q

(g1n − g2n) ξt(x, t)dxdt+

+ lim
n→∞

∫
Ω

(g01n − g02n) ξ(x, 0)dxdt+ lim
n→∞

∫
Ω

(F01n − F02n) ξ(x, 0)dx = 0.

Therefore the following equality holds

〈u1 − u2, z〉Q = 0.

As we stated above in the previous proof

u1n
∗
⇀ u1 in M+(Q) and u2n

∗
⇀ u2 in M+(Q).

Thus we can deduce u1 = u1 holds. �



REFERENCES 231

Acknowledgements

This work was partially supported by National Natural Sciences Foundation of China,
No: 11571057.

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