On Existence of Entropy Solution for a Doubly Nonlinear Differential Equation with L1 -Data ()
1. Introduction
This paper is devoted to the study of a class of doubly nonlinear history-dependent initial boundary value type probems of the form
(1)
Our aim is to prove existence of entropy solutions to the problem
. In our problem, the framework is the following: Ω is boundary domain of
, T is positive number,
is the space-time cylinder,
where
denotes the boundary of Ω,
stands for the
gradient of v with respect to the spatial,
is a real number,
,
is a singular kernel which is type
, i.e.
, nonnegative, non-increasing, such that there exists a function
satisfying
for every
and the expression
represents the convolution operation over the positive half-line in relation to the time variable
We further assume that the kernel k satisfies additional assumptions which are introduced in Section 2.3. Under these assumptions on k, our work cover the case of a time-fractional derivative of order
, i.e.
,
where Γ denotes the Gamma function.
Here, the partial derivative with respect to time of the product of the functions k and u, denoted as
, can be expressed as a distributed order derivative. This type of derivative is employed to characterize ultraslow diffusion scenarios, where the mean square displacement exhibits logarithmic growth. Such behaviour of the mean square displacement has been observed in various systems, including polymer physics and signal processing, as documented in references such as [1] and others cited therein. The diffusion term,
is a Leray-Lions operator which is coercive, monotone and which grows like
with respect to
. The
-valued function F, representing the convection flux term is assumed to be defined and locally Lipschitz continuous on the whole
. Let us stress that because the convection flux F is assumed merely Lipschitz continuous, existence techniques for
are those of entropy solutions. The lack of regularity of F and b are the only reasons why the doubling of variables in space can be needed for existence solution of type problems
. The
valued function b is assumed to be a
-function defined on the whole
, which is non-decreasing and satisfies the renormalization condition
. Finally, f represents a source term. The data
and f are such that
, and
is measurable function with
.
We should note that equations of the form
finds application in describing nonlinear heat flow in certain dielectric materials at extremely low temperatures. Experimental observations have revealed a finite speed of propagation for thermal disturbances in this situation. Various models have been proposed to explain this phenomenon, with [2] presenting a model in which the constitutive relations for internal energy and heat flux, unlike Fourier’s law, depend on the history of temperature and temperature gradient, respectively. As demonstrated in [3] , this formulation leads to a problem in the form of
under certain assumptions on the relaxation functions of internal energy and heat flux.
It is worth noting that the assumptions on k are driven by the need to ensure the positivity of solutions, which is a crucial physical requirement in several applications. When modeling nonlinear heat flow in materials with memory, the function
in problem
is considered to represent the absolute temperature at the location x in the domain Ω at time t. Such assumptions were initially introduced in [4] , giving rise to the concept of complete positivity, as discussed in [3] and [5] .
Under all of these assumptions and for
, the above problem
does not admit in general a weak solution for
-data, since that the fields
do not belong
in general, see e.g. ( [6] , Appendix I). As it has been in [7] and [8] . When the problem
, in the case
, it has been shown in ( [9] , section 3.1) that the problems of nonexistence and nonuniqueness of weak solutions to the elliptic problem carry over to the time-fractional case with
,
,
.
To address the challenges associated with the nonexistence and nonuniqueness of weak solutions, two novel solution concepts have been introduced. In [10] [11] , the existence and uniqueness of renormalized solutions are demonstrated for elliptic and parabolic problems, respectively. Additionally, [12] explores the uniqueness of renormalized solutions for elliptic-parabolic problems without history dependence.
The second concept, known as entropy solutions, is equivalent to the notion of renormalized solutions for problems without history dependence and was initially proposed in [6] for an elliptic problem. For the parabolic problem, refer to [13] .
These innovative concepts share the characteristic that a solution is not expected to be found as an element of a Sobolev space. Instead, the objective is to identify a measurable function, denoted as v, such that all truncations
of v belong to a specific Sobolev space. In the case problems of type
, this notion was introduced by Jakubowski and Wittbold in [14] .
In particular, if
, then the problem
is a purely elliptic problem and the existence of entropy solution has been shown in this case in [15] . Note that for
, the problem
is a special case of the problems considered in [14] . The authors prove existence of entropy solution in the particular case
, but the uniqueness is only shown in the general case where b is increasing. When
is replaced by
, this problem has been studied by M. Scholtes and P. Wittbold in [16] and by N. Sapountzoglou in [17] . In [16] , the authors show the existence of entropy solutions for a strictly increasing function b and in [17] the author proves the existence for any non-decreasing function b.
The main novelty of the work that the present here comes from the fact that we make an extension of [17] , adding a convection term. We will combine the techniques of [17] and the approach developed in [16] .
In this article, we will show the existence of entropy solution to initial boundary value problem
. Our definition of entropy solution for above problem is similar to the definition in [16] .
The organization of the paper is the following. In the next section, we prepare assumptions, some tools, namely the adaptation of the regularization method of R-Landes (see [18] ) and some fundamental equality and inequality, which play a crucial role in our proofs. Finally, in the third section, we will state the Definition of our solution, main existence result and its proof. More precisely, we prove that under Leray Lions assumptions on the vector field a, F locally Lipschitz-continuous and b increasing that the generalized solution of an associated Volterra equation is an entropy solution.
Numerous references are provided at the conclusion of the paper. This list is by no means exhaustive, and additional pertinent references can be found in the cited works.
2. Preliminaries
2.1. Assumptions
Throughout the paper, we assume that the following assumptions hold true:
Ω is boundary domain of
with boundary
,
is given and we set
is the space-time cylinder,
,
is a real number,
and
.
is Carathéodory function, (H1)
i.e.
is measurable for all
, and
is continuous vector field a.e.
. Moreover, we assume that a satisfies the classical Leray-Lions conditions, i.e., for some real number
, we assume that a is monotone
(H2)
coercive
(H3)
and satisfies a growth condition
(H4)
Thus, assumptions on
are rather general.
Next, we assume that
The scalar kernel
is of type
, (H5)
i.e.
, nonnegative, nonincreasing and such that there exists a function
satisfying
for every
.
The function
is continuous, nondecreasing and
satifying the normalization condition
. (H6)
F is locally Lipschitz-continuous function defined on
with value in
, (H7)
i.e.
with
is continuous on
for
.
is a measurable function defined on Ω such that
belongs to
. (H8)
, is an element of
. (H9)
In the following subsection we give some of the notation, functions, definitions and the basic results which will be used later.
2.2. Notations and Functions
If
is a Lebesgue measurable set, we will denote its Lebesgue measure by
and by
its characteristic function.
For any real number
, we denote by
, the truncation function at the level K, defined by
More precisely, for
, the functional space
can be defined by
For more details about the class of functions
whose truncations
belong, for every
, to some Sobolev space see, e.g., [6] [19] and [20] . We will frequently use the notation
, for
. For
,
is the function defined by
if
and
if
.
Throughout the paper, for the sake of simplicity for any
and for K a positive real number, we write
for the set
. In addition, we set
: Absolutely continuous functions on
and
In the sequel, C denotes a constant that may change from line to line.
2.3. Approximating the Kernel k
In this subsection, we adapt the regularization method of R-Landes [18] to kernels of type
. This regularization will be a fundamental tool for the proof of our existence result. Note that those type of kernels have been introduced by [21] . The readers can also see [16] [22] [23] for the same expositions.
Definition 2.1 A kernel
is called to be of type
if it is nonnegative, nonincreasing and there exists a kernel
such that
In this case, we say that
is a
pair and write
.
From
it follows that l is completely positive (see Theorem 2.2 in [23] ). We next discuss an important method regularizing kernels of type
.
To do this, for
,
and from ( [24] ; p37, p44), we shall denote by
(resp
) the unique solution in
(resp in
) of the scalar Volterra equations:
(2)
(3)
Note that the function
is called the resolvent of
. Recall that, according to ( [24] , part 1, chapter 2, Theorem 3.5) the solution of the equation (2) is given by the variation of constants formula:
Next, for
and a real Banach space X, we consider the operator L defined by:
(4)
and for
(2.4)
with
. According to [( [5] , Theorem 3.1) and [25] , it is know that this operator is m-accretive in
. Its resolvent
,
, which is of the form
where
is the solution the Equation (3) (see Theorem 2.1 in [23] ). Therefore,
,
, the Yosida approximation of L can be written in the form
where
. By the Equation (3), we get
(5)
Note that, since l is completely positive on
, then according to ( [23] , Proposition 2.1),
is nonnegative and nonincreasing on
. Since
belongs to
, then by the equality (5), we have
For
, we obtain that
Then,
which entails that
for all
. Thus, it follows that
for any
as
, which proof that
in
. In particular,
(6)
as
(see [16] ). We can now to introduce a modification of the regularization in time by R-Landes (see e.g. Definition 2.2 in [16] ).
Definition 2.2. Let X be a real Banach space,
its dual.
For
we define
by
In the sequel the letter
is used in this meaning only. Note that
, where
is the adjoint operator of L. Consequently, we have for any
:
To be able to proof to the existence of an entropy solution to
, let us make same further assumptions on k and
:
(K1)
(K2)
and
(K3)
The most important example of a kernel which satisfies conditions (K1)-(K3) is the kernel corresponding to the case of fractional derivatives, i.e.;
,
. Moreover, also the kernel corresponding to exponential weighted fractional derivatives, i.e.,
,
,
, satisfies these
assumptions. For more details and another example on these kernel types see [16] .
We next recall a fundamental identity for integro-differential operators of the form
which will be needed for the energy estimate.
Lemma 2.3. ( [16] , Lemma 2.4). Let
and U be an open subset of
. Let further
,
and
with
for almost every
. Suppose that
. Then, we have for almost every
,
An integrated version of (2.3) can fund in ( [24] , p574, Lemma 18.4.1). Equality (2.3) is highly important for deriving a priori estimates for problems of the form
.
2.4. Approach to the Abstract Volterra Equations
The proof of our an entropy solution existence result presented in this article will be based on the theory of G. Gripenberg (see [26] ) for abstract nonlinear Volterra integro-differentials equations of the form
(7)
in a real Banach space X. Here
is a nonnegative constant, k is a scalar kernel that is assumed to be locally integrable, nonnegative and nonincreasing function on
, A is an m-acrretive, possibly multivalued operator in X,
and
.
In this subsection, we recall the definitions and the main results of the abstract theory. We limit ourselves to the case which will be treated in our purpose, i.e.
and k of type
. The abstract problem (7) then takes form
(8)
The theory of G. Gripenberg is to consider for
the following approximating problem
(9)
Here,
are the kernels associated to the Yosida approximations of the operator given by (4).
Definition 2.4. A measurable function
is called strong solution to the approximating Equation (9), if
and there exists
such that
and
(10)
for almost every
.
The abstract approximating problem (9) admits a unique strong solution
, for every
in the sense of Definition 2.4 (see [26] ; Theorem 1]).
The generalized solution to (8) is defined as follows (see [26] ):
Definition 2.5. Let
be the strong solutions to the approximating problem (9).
If there exists a functions
such that
in
as
tends to 0, then u is called the generalized solution to (8).
By definition, the generalized solution is unique.
The following theorem is the main existence result of the abstract theory (for the proof, see ( [26] ; Theorem 1]).
Theorem 2.6. Let X be a real Banach space. Assume that A is an m-acrretive operator in
and
, then there exists a generalized solution to (8).
3. Definition of Entropy Solution and Main Result
The definition of a entropy solutions for problem
can be stated as follows:
Definition 3.1. A measurable function
is called an entropy solution of (1) if
(P1)
and
for any
, (3.1) and for any functions
,
with
,
, for all, we have
(P2)
(11)
where
and
.
The following remarks are concerned with a few comments on Definition 3.1.
Remark 3.2. Note that in Definition 3.1,
and
does note general make sense in the first equation of problem (1), but that to (3.1) each term in inequality (11) has meaning in
.
We can now formulate our main existence result of a entropy solution of
which is given by the following Theorem:
Theorem 3.3. Assume that b satisfies (H6), the vector fields a and F satisfy (H1)-(H4) and (H7) and that the scalar kernel k satisfy (K1)-(K3). Let
and
a measurable function. Then there exists at least one entropy solution v of problem
.
Proof of Theorem 3.3
To prove Theorem 3.3, we will use several techniques and approximation procedures. First, we will construct the abstract problem corresponding to our problem
.
3.1. Abstract Problem Corresponding to
Since our objective is to apply the abstract theory of G.Gripenberg, let then the graph of the possibly multivalued operator
be defined by
(12)
This characterization of the operator
is based on the results that are shown in [15] .
Thus, using this characterization of the operator
and by the same arguments as in [ [27] , Lemma 3.3.4], we can establish the following Lemma:
Lemma 3.4. Let
the operator defined (12). Then
implies that
(13)
for all
and all
.
Proof: Let
. From characterizations of the operator
in (12), we have
. Since
, it follows by Lemma 2.3 of [28] that
. We may therefore take
,
as an admissible test function in (12) which yields
(14)
for all
. By Lebesgue’s theorem, we get
where we used that
almost everywhere on Ω. Since
, we obtain that
for any K sufficiently large which shows that
Next, we have
The coercivity condition (H3) entails that
for all
. On the over hand, we may conclude by Lebesgue’s theorem that
In order to pass to the limit in the left-hand side of (14), observe that
To the first integral on the right-hand side, since
a.e. on Ω, then we have
for any L sufficiently large which shows that
For the second integral, Lebesgue’s theorem yields
Thus, we get that
which show the equality (13).
Using the result of Lemma 3.4 we can prove the following result:
Corollary 3.5 Let
the operator defined (12). Then
implies that
(15)
for all
and all
.
We state some properties of the operator
in the following Proposition (for the proof see [29] pp44 and Proposition 4.1.1).
Proposition 3.6 The operator
satisfies the following properties:
1)
is m-accretive in
,
2)
.
Now, taking the Banach space
and using the operator L defined in (4), since the operator
is m-accretive, then Theorem 2.6 entails that the abstract Volterra equation
(16)
admit for almost all
, all
and
a unique generalized solution
. But, it is a priori not clear in which sense the generalized solution satisfies
.
In order to show that
where v satisfies
, we will define approximate and perturbed problems associated to
in the next subsection.
3.2. Regularised and Perturbed Problem Corresponding to
Note that, for general b, we can not expect to find a strong solution which solves the inclusion problem (16).
In order to overcome this difficulty, let us introduce the following regularizations:
1)
, for
.
2)
, for
.
3)
, for
.
4)
, for all
.
5)
the perturbed operator defined as:
(17)
Furthermore, we set
.
By [30] (see also [29] [31] ), we affirm that operator
is m-accretive in
and we have
Note that, the function
is a strictly increasing approximation of b,
for each
,
a.e. in
, hence
in
and almost everywhere in
. Similarly, we have
,
a.e. in Ω,
in
and a.e. in Ω as m and n tend to infinity and l tends to infinity.
Let us now consider the following regularized problem:
We cannot expect to have a strong solution to the abstract problem corresponding to
(18)
for almost everywhere
. However, we know by [ [26] , Theorem 1] that the corresponding approximating abstract problem with respect to the Yosida approximating of L defined by:
(19)
for almost everywhere
, admits a unique strong solution
in
in the sense of Definition 2.4 and through the Theorem 2.6, there exists a measurable function
in
such that
in
as
tends to 0, where
is the generalized solution to (18) in the sense of Definition 2.5.
As the function
is bijective, then there exists a unique measurable function
such that
.
3.3. Entropy Solution to Approximate and Perturbed Problem with
-Data
In this subsection, our plan is to show existence of entropy solution to
. This is done via the study to approximate problem
. The next proposition will give us existence of entropy solutions
of
for each
.
Proposition 3.7 For
, there exists a function
which is a entropy solution of
in the sense of Definition 2.5.
Proof of Proposition 3.7. The proof will be divided into five steps.
Step 1: In this step, we will show existence of entropy solution to
.
Corollary 3.8. The generalized solution
of (18) is of the form
where
is an entropy solution to
.
Proof of Corollary 3.8. We use the following result which gives a few basic a priori estimates of the sequence
(for
and
fixed) which are going through standard method of
-theory.
Lemma 3.9. Let
be the sequence of strong solutions of (19). Then, the sequence
satisfies,
1)
for every
, where
is a constant independant of
and n.
2)
.
Proof: The proof is classical (see [16] and [17] ). For the sake of completness, let us recall the arguments.
(i): For K fixed, we choose
as a test function into (19). Using the characterization (17) of operator
and the representation (4) of the Yosida approximation, we find
(20)
for almost everywhere
.
Note that applying Gauss-Green Theorem and boundary condition
for almost all
. Moreover, by the monotonicity of
, we have
for almost all
. Thus, applying Lemma 2.5 of [16] to the first term in (20), integrate in time over
, taking into account the assumption of coercivity (H3) of vector field a, we have
(21)
for all
, all
and all
where
is a positive constant independent of
and n. Then, defining a function H as
for every
and every
, we deduce that
for almost all
.
By the convexity of the function H, we obtain that each term on the left-hand side in (21) is nonnegative. So, we have
for some constant C independent of
and n. By applying Poincaré’s inequality, this involves (i). The proof of estimate (ii) follows the same lines as the proof of ( [17] , Lemma 3.3). Indeed, taking
,
in (19) as test function where
, taking into account the boundary condition and the Lipschitz character of F, then by divergence theorem, we obtain
for almost all
.
The following result states useful convergences result (see [16] [17] ).
Lemma 3.10. As
, we have (up to subsequences):
1)
almost everywhere in
.
2)
almost everywhere in
, where
is a measurable function satisfying
a.e. in
.
3)
weakly in
and almost everywhere in
.
4)
, weakly in
, for every
.
Remark 3.11. Note that, by (ii) of Lemma 3.10, a immediate consequence of Lemma 3.9 is the following result:
(22)
Hence, by the coercivity condition (H3) and Lemma 3.9, we find following result:
(23)
where
is a constant independent of l.
Now, thanks to Lemma 3.9 and 3.10, we will show that
satisfies the inequality of type (11).
Let
,
and
,
. Let R be a positive real number such that
and define
.
Pointwise multiplication of the approximate abstract problem (19) by
and integration over
leads to
(24)
Here, we used Corollary 3.5. Next, for
, we define
and
. It is obvious to see that
and since
in
then
and
in
where
are as in the definition of the entropy solution. Applying Lemma 2.6 of the reference [16] to
, we obtain
(25)
In whats follows we pass to the limit-inf as
tends to 0 in (25).
• Limit of
.
We know that the convergence
a.e. in
entails
Since
is bounded and continuous function, then we have
As
in
, there exists a function in
independent of
which dominates
. Thus, by Lebesgue’s convergence theorem, it follows
and
As
in
, it follows by Young’s inequality that
and for a subsequence if necessary a.e. in
. Hence,
in
.
• Limit of
By the triangle ineguality, we have the following estimate:
Since
verifies
for any
and
. So, from Young’s inequality, we have
for any
and
. As
is non-negative and non-increasing, then
for any
and
, where
denotes the variation on the interval
. Thus, we have just shown that
as
.
Moreover, we may conclude by [ [26] , Lemma 3.4] that
as
. Its follows that
in
as
. Since, S is bounded and continuous, the convergence
a.e. in
implies that
converges to
a.e. in
and
weak-
. Hence,
• Limit of
.
Here, we will try to make an estimate of
. Recall that it has been assumed that
and
where
. Furthermore,
is identified with the term
Thus, from the monotonicity assumption (H2), weak convergences in Lemma 3.10 and the same argument as in ( [16] , p. 494-495), we obtain
for any
,
and
,
.
• Limit of
.
We have
Due to
, the weak convergence
in
and a.e. in
, we deduce that
as
tends to 0, while
is uniformly bounded with respect to
and converges a.e. in
to
. As a consequence, it follows that for
strongly in
and
weakly in
. So,
• Limit of
.
Recalling that
belongs to
,
a.e. in
,
where is a positive constant independent of
(see Lemma 3.9 and 3.10) and that
is bounded and convergences to
a.e.
. as
tends to 0. Then, it possible to obtain
as
tends 0. Moreover, by definition of the function
, we also have
is uniformly bounded with respect to
and converges a.e. in
to
as
tends to 0. So, we also have
as
tends 0.
As a consequence of the above convergence results, we can say
satisfies
(26)
Moreover, we know that
and
Hence,
is entropy solution to
. This ends the proof of Corollary 3.8.
Using the existence result with
-data of the Corollary 3.8, we get a strong convergence in the following step.
Step 2: In this step, we will show that a subsequence of
(still denoted by
, for simplicity) converges in
as
.
In order to pass at the limit-inf with
in inequality (26), we need some type of strong convergence of an subsequence of entropy solution
to
in
. It is the perturbation term that allows us to prove this result by comparing two different entropy solutions
and
. To this end, we use Kruzhkov’s method of doubling variables to show the strong convergence of
in the following Lemma. Before, stating the lemma, we should note that by ( [26] , Theorem 5) we have
in
where
is the generalized solution to abstract problem
for almost all
. Then, by Remark 3.11, we can deduce that
where
is a constant which depends on
and b.
Lemma 3.12. Let
be the sequence of entropy solution
. Then, there exist a measurable function
such that (up to subsequences):
in
and a.e. in
as
.
Proof: The proof follows the same lines as the proof of Lemma 3.5 in [17] . First of all let us note that by an approximation argument an entropy solution of
additionally satisfies inequality (26) for all
with
.
We apply Kruzskov’s method of doubling variables in time. Let
two variables in
and consider
the entropy solution to
as a function of
and
the entropy solution to
as a function of
. Moreover, let
with
, we take
as a test function in the of entropy solution for
and
as a test function in the of entropy solution for
with for
. Adding the two variational inequalities, we obtain,
(27)
Here, for simplicity, we set
and use the abbreviations
,
,
,
and furthermore
Dividing inequality (27) by L, we get
(28)
Now, we will pass to the limit with
in (28).
Note that the term
is nonnegative by the monotonicity assumption (H2). Moreover, as F is locally Lipschitz continuous, let
be the Lipschitz constant of F. Then, we find follows
Since,
almost everywhere in
as
, it follows that
Therefore, using the same arguments as in [ [17] ; p. 9-12], we get
(29)
for almost every
. So,
is a Cauchy sequence in
. Since,
is uniformly bounded and by (23), we have (up to subsequence)
then from (29), we deduce that
for a subsequence.
Remark 3.13. First, as a consequence of this Lemma, we have
(30)
Indeed, since
a.e. in
, then by (22), we deduce (30). Moreover, recall that for all
the function
is the unique strong solution to (19) and if we set
, then we have
. From the diagonal principle, there exists a subsequence
with
as
such that setting
, we have the following convergence results for
(31)
(32)
(33)
(34)
In addition, note that, by Lemma 3.9 and the growth condition (H4), the sequence
is bounded in
, So, there exist
and subsequence, still denotes by
, such that
(35)
In the next steps we will show that
in
for all
.
Step 3: In this step, we prove the following lemma, which is the main estimate in the arguments that will be developed in step 4. The idea of the proof is the same as in ( [17] , p. 13-19). Let
be defined by
for each
.
Lemma 3.14. For any
,
, the subsequence
defined above in Step 2 satisfies
(36)
for almost any
with
.
Here,
is the time regularization of
introduced in Definition 2.2.
Proof: We know that
converges a.e. on
towards
as
and from Lemma 2.3 in [16] , it is uniformly bounded by 2K. Since convergences (31), (32) and (34) holds true, then using the properties of
,
and Lebesgue’s theorem, we get that
for any
. So, it remains to show that
for almost any
.
If we apply Lemma 2.3 on
, we obtain
For,
, we define the following functions
for every
. As, we observe that
, then it follows that
and
Now, our objective is to show that
To do this end, we take
as a test function in (19).
We know that taking into account of
on
, the divergence theorem gives
Then, using coercivity condition (H3) and Lemma 2.5 in [16] , we get that
(37)
for any
where
is a constant. As
a.e. in
and by Lemma 3.9
a.e. in
, then we get that
Since all terms on the left-hand side in (37) are nonnegative, then it follows that
(38)
(39)
(40)
for any
. Noting that the above results hold true in case
replaced by
or
, we deduce that
for almost every
. Thus, it remains to show that
for almost every
. Applying Lemma 2.5 in [16] and integrating over
for some
, we get
(41)
Now, our objective is to estimate limit-inf of (41) with
. Since,
a.e. in
, b is continuous on
and
, then it comes that
Moreover, as
then there exist at least a subsequence of
, still denoted same way that
, which is dominated by an
-function. So, from Lebesgue’s theorem, we get
and
Since
in
, then using the Young inequality for convolution, it follows that (up to subsequence)
(42)
in
and a.e. in
. Next, note that the nonnnegativity of all terms in (21) implies that
for any
and some constant
independent of
and n. Since the kernel verifies assumptions (K1) and (K2), then the Lemma of Fatou gives
(43)
and
(44)
for any
. By convergences (42), (43) and (44), we can estimate lim-inf of (41) with
. Thus, passing to the limit in (41) with
, we obtain
(45)
for almost any
. Next, we want to pass to the limit with
as
. As
is bounded and compact support, then we observe that
and
So, taking into account that
, we obtain that
for all
where
denotes the Yosida approximation of operator L as defined in subsection 2.3. Here, we used that
involves
for
. In particular,
for any
and any
. Since,
is compactly supported, then we get that la sequence
is uniformly bounded. As,
then, from Lebesgue’s theorem, it comes that
as
.
being a bounded operator verifying
for any
as
, then by an application of the triangle inequality, it follows that
as
. Hence, the continuity of the Yosida approximation
gives
for any
. Now, using Lemma A5 in ( [17] , Appendix), we get that
(46)
for almost every
. As
in
, then by Young’s inequality, it follows that
(47)
in
and passing to a subsequence if necessary, a.e. in
. In addition, by the properties of
, we have
a.e. in
as
. The function b being monotone, it follows that
a.e. in
. As, the kernel k is nonnegative, then we have
Hence, Lebesgue’s theorem involves
(48)
in
, and, extracting subsequence if necessary, a.e. in
. Next, note that the monotonicity of b, the condition
and
involves that
Moreover, the same arguments entails that
From, (43) and (44), we see that
and
for any
with
. As,
is non-negative, non-increasing and verifies assumptions (K1)-(K3), then by application of Lebesgue’s theorem, we get that
(49)
for any
and using the same arguments as above we find