Global Existence and Decay of Solutions for a Class of a Pseudo-Parabolic Equation with Singular Potential and Logarithmic Nonlocal Source ()
1. Introduction
In this paper, we focus on the Neumann initial boundary problem:
(1)
where
is a bounded domain with smooth boundaries
, n is the outer normal vector of
while
.
with
. As is well known, according to the law of conservation, many diffusion processes with reactions can be described by the following equation (see [1] ):
(2)
Among them,
represents the mass concentration in the chemical reaction process or the temperature in thermal conduction. At position x and time t in the diffusion medium, the function D is called the diffusion coefficient or thermal diffusion rate, the term
represents the rate of change caused by diffusion, and
is the rate of change caused by the reaction.
In the past few years, many researchers have paid attention to Equation (2). For source
and
, a lot of work has been obtained. Many scholars have studied the global existence [2] [3] , blow-up conditions, blow-up time estimates, and asymptotic behavior of solutions to such problems. Interested individuals can read reference materials [4] [5] [6] .
Yan et al. [7] considered the following parabolic equation:
(3)
According to the logarithmic Sobolev inequality and energy estimation method, the results of blow-up and non-extinction of solutions under appropriate conditions are given, which generalizes some recent results.
Taking inspiration from these studies, we will consider the problem with logarithmic nonlocal sources in this paper. As far as we know, this is the first work to consider the singular parabolic Laplace equation with strong damping and logarithmic nonlocal sources. This work has great significance and can fill the research gap in this area.
The organizational structure of this article is as follows. In Section 2, we will introduce some symbols, definitions and basic lemmas that will be used in this paper. In Section 3, we present the main results of the paper, which are the local existence of weak solutions and the global existence and decay estimation of weak solutions under certain conditions.
2. Preliminaries
In this section, we will introduce some symbols and lemmas that will run through this paper. In the following text, we denote by
the norm in
and by
the
inner product. First, for Problem (1), we introduce the potential energy functional:
(4)
and the Nehari functional:
(5)
by a direct computation:
(6)
By
and
, we define the potential well:
and the Nehari manifold:
The depth of potential well is defined as:
Lemma 1. [8] [9] Let
be a positive number. Then we have the following inequalities:
and
Lemma 2. [8] Let
is a bounded smooth region in
, then for any
and
, we have:
Lemma 3. [7] [9] [10] Let
. Then,
and there exists a constant
such that:
(7)
Lemma 4. [11] For any
, we have the following inequality:
(8)
where
,
.
Lemma 5. [8] [11] Let
be a nonincreasing function and
be a positive constant such that:
Then, we have:
1)
, for all
, whenever
.
2)
, for all
, whenever
.
Lemma 6. Assume that
, then:
1)
,
.
2) There exists a unique
such that
.
3)
is increasing on
, decreasing on
, and attains the maximum at
.
4)
for
,
for
, and
.
The following is the definition of weak solution for Problem (1). To avoid confusion, we also write
as
.
Definition 7. [7] [12] (Weak solution) A function
with:
is called a weak solution of Problem (1) on
if
in
and:
for any
.
3. Main Results
In this section, we present two theorems. Firstly, we present the local existence and uniqueness theorems for weak solutions to Problem (1). Next, we present the existence theorem for the global weak solution of Problem (1), and also provide an estimate of the exponential decay of the solution in the theorem.
Theorem 8. Let
. Then, there exist a
and a unique weak solution
of (1) with:
satisfying
. Moreover,
satisfies the energy equality:
Proof. We divide the proof of Theorem 8 into 3 steps.
Step 1. Local existence
To deal with the influence of singular potentials, we introduce a cut-off function:
We denote the solutions corresponding to
of Problem (1) as
. We can know that:
where
as
. Let
be a linear independent basis in
and construct the approximate solution:
solving the problem
(9)
and
(10)
as
. Noticing that
, it is not hard to verify for any fixed j:
From above equality, we know that
is determined by the following Cauchy problem:
where
The standard theory of ODE states that there exists a
such that
.
Multiply (9) by
, sum for
and recall
to find:
(11)
Integrating both sides of (11) in
, we get:
(12)
where
(13)
From Lemma 1, we get:
(14)
Let
, then from (14), Lemma 2 and Young’s inequality, we obtain:
(15)
where
, and
,
is the best embedding constant. We note that since
,
holds. Let
then
since
.
From (12), (13) and (15), we get:
that is
(16)
where
,
,
.
From Gronwall inequality, we obtain:
(17)
where
is a constant which dependent on T.
Multiplying (11) by
, summing on
and then integrating on
, we know that, for all
, we have:
(18)
By the continuity of the functional J and (10), there exists a constant
satisfying:
, for any positive integer n and k. (19)
Applying (4), (13), (15), (17), (18) and (19), we obtain:
(20)
where
. From (18) and (20), for any
, it follows that:
(21)
(22)
(23)
(24)
By (22), (23) and Aubin-Lions-Simon Lemma, we get:
(25)
Combining (10) with that
in
, we observe that
in
.
By (25), we have
, a.e.
. That means:
That is to say, there is:
From Lemma 1 and Lemma 2, we get:
where
is the best constant of the Sobolev embedding
↪
. Here, we choose
, we know that:
, for any positive integer n and k. (26)
According to the Holder inequality, we obtain:
(27)
where
.
By (21)-(24), (26), and (27), there exist functions u and a subsequence of
, which we still denote it by
such that:
weakly star in
(28)
weakly in
(29)
weakly in
(30)
weakly star in
(31)
By (28)-(31), passing to the limit in (9), (10) as
, it follows that u satisfies the initial condition
:
(32)
for all
.
Step 2. Energy equality
Multiplying
at both ends of Problem (1), integrating from 0 to t and combining (4), we have:
Step 3. Uniqueness
Assuming
and
are two solutions to Problem (1), we have:
(33)
and
(34)
Let
and
, then by subtracting (33) and (34), we can derive:
Let
, we obtain:
Integrating it on
, we obtain:
(35)
where
and
satisfy locally Lipschitz continuity. That means:
By Gronwall’s inequality, we have
.
The proof of Theorem 8 is complete.¨
Theorem 9. Assume that
and
, then Problem (1) admits a global solution
,
with
, and
for
. Moreover, if
, then
where
,
.
Proof. Now, we prove Theorem 9. In order to prove the existence of global weak solutions, we consider two following cases.
1) Global existence
Case 1. The initial data
and
.
Taking a weak solution
, which satisfies:
(36)
Among them,
is the maximum existence time of the solution
. We need to prove that
. Thanks to
and (36), we obtain:
(37)
We will assert that:
(38)
In fact, using the method of proof to the contrary, assuming that (38) does not hold, let
is the minimum time for
. So, considering the continuity of
, it can be inferred that there is
. The following conclusion can be drawn:
(39)
and
(40)
It is evident that (39) could not occur by (37) while if (40) holds then, by the definition of d, we have:
which also contradicts with (37). As a consequence, it follows from this fact and the definition of functional J that:
(41)
namely,
(42)
From Lemma 2, we have:
(43)
Combining above inequality, (36) and (42), we obtain:
(44)
This estimate allows us to take
. Hence, we can conclude that there exists a unique global weak solution
of Problem (1), which satisfies that:
Case 2. The initial data
and
.
Firstly, we choose a sequence
such that
. Then, we consider the following problem:
(45)
where
.
Due to
, it can be inferred from Lemma 3 that
. Therefore, we obtain
and
. Use arguments similar to Case 1. We found that Problem (45) allows for global weak solutions u.
The remainder of the proof can be processed similarly as Case 1.
2) Decay estimates
We are now in a position to prove the algebraic decay results. Thanks to
, and Lemma 3, we get
. Fro (5), (38) and (40), we have:
(46)
Combining with the first equality of Problems (1), (5) and Lemma 3, we obtain:
(47)
where
.
By (46) and (47), we get:
Let
in above inequality, by Lemma 5, it follows that:
The proof of Theorem 9 is complete.¨
Acknowledgements
Sincere thanks to the professional performance of JAMP members, and special thanks to the editors for their dedication to this article.