1. Introduction
For many phenomena in engineering and science, fractional calculus offers an appropriate platform for their studies [
1,
2,
3,
4,
5,
6,
7]. In fact, for anomalous processes and disordered systems, where the mean square displacement MSD (measure of the deflection of the actual position of a particle with regards to a given reference position) grows in a non-linear fashion, one is forced to consider fractional derivatives. Complex diffusion phenomena are therefore better described by non-integer-order fractional derivatives.
The problem to be discussed here is the following non-linear fractional initial boundary value problem of order between 1 and 2
where
is the Caputo fractional derivative of order
defined below,
k is a non-negative function,
and
are the initial data of the state,
will be equal to 1 or 0 depending on whether they are effective or not in the equation,
is to be determined later and
is a bounded domain in
with smooth boundary
.
It is well known that the relationship between the heat flux vector and the temperature gradient is given by
where
k is the thermal conductivity. Then, having in mind the conservation of energy
where
is the heat capacity, the basic heat equation
is obtained. Taking into account the history dependence between the gradient and the flux
we are lead to the fractional heat conduction equation
for
and
(see [
5]).
A sub-diffusive problem has been derived in [
8] from the Rayleigh–Stokes problem
The authors proved that the solutions of Stoke’s first problem follow as a special case.
In case of a material with memory, the relationship between the heat flux vector and the temperature gradient will involve a term of the form
The fractional equation
is a fractional version interpolating the heat and wave equations. It describes anomalous processes and corresponds to the (fast) super-diffusion (particles move faster than in the normal case) (see [
9,
10]) when the mean square displacement grows non-linearly in time (
). The importance of super-diffusion is explained and highlighted in many works, see [
3,
6,
7,
11] to cite but a few.
A linear growth of the MSD (
) corresponds to the ordinary case (normal diffusion), when (
), we are in presence of sub-diffusion (particles move slower than in the ordinary case), whilst
corresponds to the ballistic diffusion. Apart from the normal diffusion and the ballistic diffusion, the other cases cannot be described adequately by the standard heat or wave equations. Therefore, accordingly, the equation
interpolates the heat conduction equation [
12]
and the viscoelastic equation [
13]
The stability of these problems has been investigated by many researchers: as parabolic problems with memory [
14,
15] and as viscoelastic problems [
16,
17,
18,
19,
20,
21]. For the viscoelastic problem, it has been proved, roughly, that the memory term produces a weak damping which is able by itself to drive the system to rest. It is the kernel which determines the rate of stability. Roughly, the stability rate is similar to the decay rate of the kernel. Various kinds of stability rates (exponential, polynomial and arbitrary) may be found in the literature [
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22]. It is clear now that these models are not only interesting from the mathematical point of view but also from the point of view of applications. They may describe the vibrations of viscoelastic structures in anomalous media.
Moreover, having in mind the telegraph equation [
1,
2,
22]
Orsingher and Beghin [
4] showed that the law of iterated Brownian motion and the telegraph process with Brownian time are expressed by the fractional telegraph equation
Therefore, our present model (1) is a kind of non-linear fractional telegraph problem with memory.
The telegraph equation in (6) is dissipative. This is due to the presence of a first-order derivative which is often called ’frictional’ damping. In fact, it is easy to see that the energy of the system tends to zero in an exponential manner as time tends to infinity. Hence, it is interesting to know whether the lower-order fractional derivative is a dissipative term for the fractional problem in (1). In the affirmative, what would be the rate of convergence?
While we can find a good number of results on the well-posedness of similar problems to ours for orders between 0 and 1 and for problems without the Laplacian memory term, there are very few papers dealing with the exact form in the problem in (1). We identified three papers: one by El-Sayed and Herzallah [
23], one by Ponce [
24] (for an infinite history) and one by Agarwal et al. [
25], dealing with abstract operators. We mention here the result in [
25].
Let
be a Banach space and
be closed linear operators defined on domains
and
dense in
X. We denote by
the graph norm in
and
the resolvent of
The problem
with initial data (
) in
(see [
24,
25]), admits a classical solution
such that
under the following assumptions
(A1) For some
and every
there is a constant
such that
and
with
(A2) For each is strongly measurable on There exists a locally integrable function with Laplace transform and In addition, (the space of bounded linear operators from into X) has an analytic extension to verifying and as
(A3) There exists a subspace S dense in and such that and
In the present situation our operator is the well-known Laplacian which fortunately satisfies the above conditions in an appropriate space. Therefore, we can profit from this result to ensure the well-posedness of our problem in (1).
In our case, because of the non-linearity present in the equation, we only obtain a local existence result (see [
26]). Assuming that the initial data are in
(
A is the operator in our problem in (1)), there exists a unique local solution in the space
for
where
For simplicity and convenience, we shall rather consider the problem
Because the fractional composition
is not correct in general, the two problems are not exactly the same. The case
corresponds to the parabolic case [
12,
15]. In the case where
, we recover the viscoelastic problem studied extensively by many authors [
7,
14,
18,
19,
20,
21,
22].
We shall shed some light on the effects of the non-linear term, the fractional lower-order term and the memory term on the stability of the system. The lower-order term and the memory term will be considered separately. It is well known, in the integer-order case, that having both at the same time enhances the dissipation. Thus, the challenging question is to consider only one at a time. We have purposefully written the lower-order term as a half-order fractional derivative of the leading fractional derivative as it corresponds to the one in (7). It would be interesting, at least from a mathematical point of view, to treat other orders as well. We will show below that it is a dissipative term. More precisely, we will prove that this lower-order term is by itself (without the memory term) sufficient to guarantee the stability of the system in the Mittag–Leffler manner (solutions of the system will decay as the Mittag–Leffler functions under certain norms). However, without the lower-order fractional term, the Mittag–Leffler stability is only shown for ’small’ kernels. We are unable to decide on the dissipativity of the system in the early stages. This is justified by the fact that the memory term alone is not capable of producing a considerable amount of damping. The product rule (Proposition 5 below) gives rise to a complicated term. It is defined by three singular integrals involving functions of undefined signs. Moreover, the stability is only of local character due to the difficulty to control the non-linear terms. In fact, this local character is not suitable to fraction, it is also the case in integer-order problems. The objective of this paper is to suggest a way to remedy to these difficulties. Unfortunately, we could not compare our results with existing ones as, to the best of our knowledge, this is the first paper treating this as a non-linear issue.
In case both the lower-order fractional term and memory term are present, the system remains Mittag–Leffler stable for reasonable classes of kernels. In fact, the memory term in this situation does not contribute significantly. On the contrary, it only adds some complexity to the estimations. To keep the size of the paper reasonable, we refrain from treating this case. Instead a simple combination of the first two cases is pursued (see Remark 1 below).
The next section contains some preliminary definitions and results. It is followed by
Section 3 on the existence and uniqueness issue. In
Section 4 we treat the case where only the lower-order term is present. In
Section 5 we determine an appropriate function to work with.
Section 6 contains the Mittag–Leffler stability in the presence of the memory term. We end the paper by
Section 7.
6. The Case and
Here, in the absence of the fractional damping, there remains only the viscoelastic term. Without loss of generality, we pick . Our assumptions on the kernel k are:
(
K)
is a non-negative continuous function satisfying
and
and
for some positive constant
To deal with the non-linearity we need to assume that
(Γ) if and if
This assumption will allow us to use the embedding when if and if
It is easy to see from Proposition 1 that
for some
and from Propositions 2 and 3 that such kernels are summable and
In fact, this family of functions satisfies (K). Moreover, our proofs below are valid for initial data in
Lemma 2. The functional satisfies along solutions of (8), for Proof. Clearly, a direct fractional differentiation of
, using Proposition 5, yields for
Along solutions of (8) (see (17)) and in view of Proposition 4, we find
We conclude by using Proposition 6 and the evaluation
This finishes the proof. □
The last term in the estimation of needs to be controlled somehow. This term does not appear in the integer-order case (second-order) and, whatever evaluation we derive, its weight is going to be considerable. We suggest a new functional in the next lemma.
Lemma 3. The fractional derivative of the functionalsatisfiesfor Proof. With the help of Proposition 5, we can write
and the relation (9) gives
Next, the summability of
k and Proposition 4 imply
or, for
Observing that (see Proposition 3)
and
it appears that
for
□
The second functional we introduce is in the next lemma.
Lemma 4. For the functionalit holds thatforwhereis the Poincaré constant. Proof. By Proposition 5 we have
Therefore, for
and
The proof is complete. □
Proof. This follows immediately from Proposition 4 and (K). □
As we have removed the lower-order fractional term, which represented the damping and was responsible for the appearance of the nice term we need to find a way to come up with this missing and essential term. The next suggested functional will take care of this matter.
Lemma 6. If the exponentsatisfies (
Γ),
then for the functionalwe have, along solutions of problem (8)for where for some
.
Proof. By Proposition 5, we see that
and along solutions of (8), we find for
Next, we apply Proposition 4
or, for
Another application of Proposition 5 gives
Therefore, for
We can apply the Cauchy-Schwarz inequality to the last term in (19) to get
In view of assumption (
K), Proposition 3 and the fact that
(for some positive constant
) away from zero
for
where
Therefore
for
where
Moreover, it is clear that
and the embedding mentioned above (
when
if
and
if
) gives
where
is the embedding constant. Observe also that
implies
Finally, for
Taking into account the relations (20)–(24) in (18), we obtain
The proof is complete. □
One more term needs to be controlled. To this end we introduce the functional
Lemma 7. The above functional fulfills Proof. Therefore,
or, using (14), we find
that is
Furthermore, we may evaluate
and as
we deduce that
Therefore,
for
Further, using a similar argument to the one used to derive (20), for
and
we obtain
This ends the proof. □
Let
for some positive constants
to be determined later.
Theorem 3. Assume that k satisfies (K) and γ satisfies (Γ). Then, the solution to the problem in (8) (with ) is locally Mittag–Leffler stable, that is there exist positive constants and such thatfor small K (or, alternatively, a large Proof. The evaluations obtained in the previous lemmas, with
imply
The strategy consists of choosing the parameters so as to give rise to a term of the form
, in addition to
, on the right-hand side of (25). Observe that, as
decreases towards zero, choosing
large enough allows us to ignore this term. Incidentally, we may also ignore
and
. Let
and recall that
Therefore, for small values of
K (or large values of
) and consequently small
,
and
, it suffices that
To this end, we may pick
and
. Finally, we go back and select the remaining parameters to ensure the fractional inequality
for some
and by the equivalence of
and
Let
be real numbers (we can assume that
) such that
and
so that
Consequently,
Assume that there exists
such that
then as
we have
This is a contradiction. Starting from we can continue the process forever; thus, the proof is complete. □
Remark 1. The case and , results from the above arguments. The functionals in the first case are good enough to obtain an explicit upper bound for the kernels ensuring Mittag–Leffler stability.