#### Abstract

In this work, we study a plate equation with time delay in the velocity, frictional damping, and logarithmic source term. Firstly, we obtain the local and global existence of solutions by the logarithmic Sobolev inequality and the Faedo-Galerkin method. Moreover, we prove the stability and nonexistence results by the perturbed energy and potential well methods.

#### 1. Introduction

In this article, we consider a plate equation with frictional damping, delay, and logarithmic terms as follows: where , , is a bounded domain with smooth boundary . denotes time delay, and , , and are real numbers that will be specified later. Generally, logarithmic nonlinearity seems to be in supersymmetric field theories and in cosmological inflation. From quantum field theory, that kind of () logarithmic source term seems to be in nuclear physics, inflation cosmology, geophysics, and optics (see [1, 2]). Time delays often appear in various problems, such as thermal, economic, biological, chemical, and physical phenomena. Recently, partial differential equations have become an active area with time delay (see [3, 4]). In 1986, Datko et al. [5] indicated that, in boundary control, a small delay effect is a source of instability. Generally, a small delay can destabilize a system which is uniformly stable [6]. To stabilize hyperbolic systems with time delay, some control terms will be needed (see [7–9] and references therein).

For the literature review, firstly, we begin with the studies of Bialynicki-Birula and Mycielski [10, 11]. The authors investigated the equation with the logarithmic term as follows: where the authors proved that, in any number of dimensions, wave equations including the logarithmic term have localized, stable, soliton-like solutions.

In 1980, Cazenave and Haraux [12] studied the equation as follows: where the authors in [12] proved the existence and uniqueness of the solutions for equation (3). Gorka [2] obtained the global existence results of solutions for one-dimensional equation (3). Bartkowski and Gŏrka [1] considered the weak solutions and obtained the existence results.

In [13], Hiramatsu et al. studied the equation as follows:

In [14], Han established the global existence of solutions for equation (4).

In [15], Al-Gharabli and Messaoudi were concerned with the plate equation with the logarithmic term as follows:

They established the existence results by the Galerkin method and obtained the explicit and decay of solutions utilizing the multiplier method for equation (5).

In [16], Liu introduced the plate equation with the logarithmic term as follows:

The author proved the local existence by the contraction mapping principle. Also, he studied the global existence and decay results. Moreover, under suitable conditions, the author proved the blow-up results with .

In [17], Messaoudi studied the equation as follows: and obtained the existence results and obtained that, if , the solution is global and blows up in finite time if . Later, Chen and Zhou [18] extended this result. In the presence of the strong damping term (), Pișkin and Polat [19] proved the global existence and decay of solutions for equation (7). For more results about plate problems, see [20–22].

In [7], Nicaise and Pignotti studied the equation as follows: where . They proved that, under the condition , the system is exponentially stable. The authors obtained a sequence of delays that shows the solution is unstable in the case . In the absence of delay, some other authors [23, 24] looked into exponential stability for equation (8). In [9], Xu et al., by using the spectral analysis approach, established the same result similar to [7] for the one space dimension.

In [25], Nicaise et al. studied the wave equation in one space dimension in the presence of time-varying delay. In this article, the authors showed the exponential stability results with the condition where is a constant and

In [26], Kafini and Messaoudi studied wave equations with delay and logarithmic terms as follows:

The authors proved the local existence and blow-up results for equation (11).

In [27], Park considered the equation with delay and logarithmic terms as follows:

The author showed the local and global existence results for equation (12). Also, the author investigated the decay and nonexistence results for equation (12). In recent years, some other authors investigate hyperbolic-type equations with delay terms (see [28–33]).

In this work, we studied the local existence, global existence, nonexistence, and stability results of plate equation (1) with delay and logarithmic terms, motivated by the above works. There is no research, to our best knowledge, related to plate equation (1) with the delay term and logarithmic () source term; hence, our work is the generalization of the above studies.

This work consists of five sections in addition to the introduction. Firstly, in Section 2, we recall some assumptions and lemmas. Then, in Section 3, we obtain the local and global existence of solutions. Moreover, in Section 4, we establish the nonexistence results. Finally, in Section 5, we get the stability of solutions.

#### 2. Preliminaries

In this part, we show the norm of by for a Banach space . We give the scalar product in by . We show by , for brevity. Let be the constant of the embedding inequality

We have the following assumptions related to problem (1):

*(H1)*. The weights of delay and dissipation satisfy

*(H2)*. The constant in (1) satisfies

To get the main result, we have the lemmas as follows.

Lemma 1 (see [34, 35]) (Logarithmic Sobolev inequality). *For any ,
where is a positive real number.*

Corollary 2. *For any ,
where is a positive real number.*

*Remark 3. *Assume that inequality (17) holds for all , and we choose the constant that satisfies
where is any real number with

Lemma 4 (see [12]) (Logarithmic Gronwall inequality). *Suppose that and . If a function satisfies
then
**We define
for ; then,
**Suppose that
**Then, it satisfies (see, e.g., [36–38])
where is the well-known Nehari manifold, denoted by
*

Lemma 5. * and are the functions that satisfy
for any with , where
*

*Proof. *We obtain, for ,
and therefore, we obtain the desired result.☐

*Remark 6. * has the absolute maximum value at , such that
for .

Lemma 7. *The potential depth in (25) satisfies
*

*Proof. *By Corollary 2, (13), and (18), we have
Taking the limit , we obtain
Taking into consideration this and (28), we get
and therefore,
Hence, we have by (24) and (31)
From the definition of given in (25), we obtain the result.☐

#### 3. Existence

In this part, we have studied the local existence, global existence, nonexistence, and stability results of plate equation (1) with delay and logarithmic terms, motivated by the above works. There is no research, to our best knowledge, related to plate equation (1) with the delay () term and logarithmic () source term; hence, our work is the generalization of the above studies. Firstly, we introduce the new function

Hence, problem (1) takes the form

*Definition 8. *Assume that . is a local solution of problem (39) if it satisfies

##### 3.1. Local Existence

In this part, we establish the local existence results similar to [8, 39].

Theorem 9. *Suppose that (H1) and (H2) are satisfied. Then, for the initial data , , and , there exists a local solution for problem (39).*

*Proof. *Let be the orthogonal basis of that is orthonormal in . Define , and we extend by over . We denote and for . We consider the Faedo-Galerkin approximation solution of the form
solving the approximate system
where
Since problem (42)–(44) is a normal system of ordinary differential equations, there exists a solution on the interval , . The extension of that solution to the is a consequence of the estimate below.

By replacing by in (42) and by using the relation
we have
By replacing by in (43), we see that
Summing (47) and (48), we obtain
where
where
Utilizing Young’s inequality and the fact that , we obtain
where
Taking into consideration this and Corollary 2, we have
By using (18), we obtain
and therefore,
where the sequel , , shows a positive constant. Also, we know that
Utilizing Cauchy-Schwarz’s inequality and (57), we obtain
From Lemma 4, we arrive at
is the function which is continuous on , , , and decreases on and increases on ; hence, we get by (57) and (60)
Hence, there exists a subsequence of , which we still denote , such that
Utilizing the Aubin-Lions compactness theorem, we conclude that
The function is continuous on ; hence,
Let
Thus, we obtain
where we used
By (57) and (66), we conclude that
where is the Sobolev imbedding constant of
Therefore, we get from (68)
From the Lebesgue bounded convergence theorem, (64), and (70), we arrive at
We pass the limit in (42) and (43). The remainder of the proof is standard and similar to [39, 40].☐

##### 3.2. Global Existence

In this part, we obtain the global existence results for problem (39). For this goal, we define the energy functional of problem (39): where is the positive constant given in (51). We see that

By the same arguments similar to (52), we infer that where and , given in (54), are positive constants.

Lemma 10. *Suppose that (H1) and (H2) are satisfied. If and , then the solution of problem (1) satisfies
where is the maximal existence time of the solutions.*

*Proof. *We know that and is continuous on ; hence, we have
Let be the maximum of satisfying (76). Assume that ; then, , that is,
Therefore, we obtain by (26)
We see that this is in contradiction to the relation as follows:
By (74) and Lemma 10, we see that is a nonincreasing function.☐

Theorem 11. *The solution is global, under the conditions of Lemma 10.*

*Proof. *It suffices to show that is bounded independent of . By Lemma 10, (73), and (74), we get
In a similar way, we get
By Corollary 2 and (23), we conclude that
By taking the limit in this inequality and from (81), we obtain
By Lemma 7 and (18), we get
Therefore, we see by (81) and (83) that
Hence, we conclude that
Therefore, we complete the proof by (80) and (86).☐

#### 4. Nonexistence

In this part, similar to [41–43], we get the nonexistence results for problem (1). Firstly, we need the lemma as follows.

Lemma 12. *Assume that (H1) and (H2) are satisfied. If and , then the solution of problem (1) satisfies
where is the maximal existence time of the solutions.*

*Proof. *We know that and is continuous on ; hence, we have
Let be the maximal time satisfying (89) and assume that ; then, , such that
Therefore, we obtain
This is in contradiction to Lemma 7. Thus, (87) is proved. By Lemma 7, (31), and (87), we conclude that
Therefore, the proof is completed.☐

Theorem 13. *Suppose that (H1) and (H2) are satisfied. Let , where , and . Then, the solution of problem (1) blows up at infinity.*

*Proof. *Firstly, we set
By (74), we obtain
Utilizing (72), (88), and (94), we see that
We define
By (39) and (72), we get
Utilizing Young’s inequality and (94), we obtain
By adapting this to (97) and from (88) and (93), we have
Firstly, fix such that and then choose small enough so that . Then, by (94), we get
Also, we conclude that
Taking small enough again, we obtain
By (100) and (102), we get
Utilizing (100) and (101), we see that
and therefore,
Therefore, blows up at infinity. Consequently, the proof is completed.☐

#### 5. Stability

In this part, we obtain the stability of global solutions. Firstly, we define the perturbed energy by where , , and .

Lemma 14. *Under the conditions of Lemma 10, for , we obtain
*

*Proof. *Utilizing Lemma 10 and Young’s inequality, we have
Taking small enough, we complete the proof.☐

Theorem 15. *Assume that (H1) and (H2) are satisfied. Suppose that and . Hence, for , we obtain
*

*Proof. *From (39) and Young’s inequality, we get
By using the second equation of (39) and the integration by parts, we obtain
Summing these and (74), we obtain
Adding and subtracting with , we get
Utilizing the logarithmic Sobolev inequality, we have