Existence, Nonexistence, and Stability of Solutions for a Delayed Plate Equation with the Logarithmic Source

Department of Mathematics, Dicle University, Diyarbakir, Turkey Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Oran, 31000 Oran, Algeria Preparatory Institute for Engineering Studies in Sfax, Tunisia Department of Mathematics, College of Education, Juba University, Sudan


Introduction
In this article, we consider a plate equation with frictional damping, delay, and logarithmic terms as follows: where Ω ⊂ R N , N ≥ 1, is a bounded domain with smooth boundary ∂Ω. τ > 0 denotes time delay, and α, β, and γ 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 (ujuj p−2 ln juj k ) 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][8][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 Eð0Þ < 0.
In [17], Messaoudi studied the equation as follows: and obtained the existence results and obtained that, if m ≥ p, the solution is global and blows up in finite time if m < p. Later, Chen and Zhou [18] extended this result. In the presence of the strong damping term (−Δu t ), Pișkin and Polat [19] proved the global existence and decay of solutions for equation (7). For more results about plate problems, see [20][21][22]. In [7], Nicaise and Pignotti studied the equation as follows: where a 0 , a > 0. They proved that, under the condition 0 ≤ a ≤ a 0 , the system is exponentially stable. The authors obtained a sequence of delays that shows the solution is unstable in the case a ≥ a 0 . 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 d 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][29][30][31][32][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 ðβu t ðx, t − τÞÞ term and logarithmic (u ln juj γ ) 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.

Preliminaries
In this part, we show the norm of X by k·k X for a Banach space X. We give the scalar product in L 2 ðΩÞ by ð·, · Þ. We show k·k 2 by k·k, for brevity. Let B 1 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 γ in (1) satisfies To get the main result, we have the lemmas as follows.

Corollary 2.
For any u ∈ H 2 0 ðΩÞ, where k is a positive real number.
Remark 3. Assume that inequality (17) holds for all k > 0, and we choose the constant k that satisfies where μ is any real number with Lemma 4 (see [12]) (Logarithmic Gronwall inequality). Suppose that c > 0 and l ∈ L 1 ð0, We define for v ∈ H 2 0 ðΩÞ; then, Suppose that Then, it satisfies (see, e.g., [36][37][38]) where N is the well-known Nehari manifold, denoted by Lemma 5. I and J are the functions that satisfy Proof. We obtain, for λ ≥ 0, and therefore, we obtain the desired result. ☐ Remark 6. JðλvÞ has the absolute maximum value at λ * , such that for v ∈ H 2 0 ðΩÞ.

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

Advances in Mathematical Physics
Proof. By Corollary 2, (13), and (18), we have Taking the limit k ⟶ ffiffiffiffiffiffiffi π/γ p , we obtain Taking into consideration this and (28), we get and therefore, Hence, we have by (24) and (31) From the definition of d given in (25), we obtain the result. ☐

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 (βu t ðx, t − τÞ) term and logarithmic (u ln juj γ ) source term; hence, our work is the generalization of the above studies. Firstly, we introduce the new function Hence, problem (1) takes the form 3.1. Local Existence. In this part, we establish the local existence results similar to [8,39].
Proof. Let fv i g i∈N be the orthogonal basis of H 2 0 ðΩÞ that is orthonormal in L 2 ðΩÞ. Define φ i ðx, 0Þ = v i ðxÞ, and we extend φ i ðx, 0Þ by φ i ðx, ηÞ over L 2 ðΩ × ð0, 1ÞÞ. We denote V n = spanfv 1 , v 2 ,⋯,v n g and W n = spanfφ 1 , φ 2 ,⋯,φ n g for n ≥ 1. We consider the Faedo-Galerkin approximation solution ðu n , y n Þ ∈ V n × W n of the form solving the approximate system Advances in Mathematical Physics where Since problem (42)-(44) is a normal system of ordinary differential equations, there exists a solution ðu n , y n Þ on the interval ½0, t n Þ, t n ∈ ð0, T. The extension of that solution to the ½0, TÞ is a consequence of the estimate below.
By replacing v by u n t ðtÞ in (42) and by using the relation ð Ω u n ln u n j j γ u n t dx = d dt we have d dt By replacing φ by ωy n ðη, tÞ in (43), we see that Summing (47) and (48), we obtain where where Utilizing Young's inequality and the fact that y n ðx, 0, tÞ = u n t ðx, tÞ, we obtain where Taking into consideration this and Corollary 2, we have By using (18), we obtain and therefore, where the sequel c j , j = 1, 2, ⋯, shows a positive constant. Also, we know that Utilizing Cauchy-Schwarz's inequality and (57), we obtain 5 Advances in Mathematical Physics From Lemma 4, we arrive at f ðsÞ = s ln s is the function which is continuous on ð0, ∞Þ , lim s⟶0 + f ðsÞ = 0, lim s⟶+∞ f ðsÞ = +∞, and f decreases on ð0, e −1 Þ and increases on ðe −1 ,+∞Þ; hence, we get by (57) and (60) Hence, there exists a subsequence of fðu n , y n Þg, which we still denote fðu n , y n Þg, such that Utilizing the Aubin-Lions compactness theorem, we conclude that The function s ⟶ s ln jsj γ is continuous on R; hence, u n ln u n j j γ ⟶ u ln u j j γ a:e:in Ω × 0, T ð Þ: ð64Þ Let Thus, we obtain where we used s ln s j j≤ 1 e for 0 < s < 1, By (57) and (66), we conclude that where B 2 is the Sobolev imbedding constant of Therefore, we get from (68) From the Lebesgue bounded convergence theorem, (64), and (70), we arrive at u n ln u n j j γ ⟶ u ln u j j γ strongly in L 2 0, T ; L 2 Ω ð Þ À Á : We pass the limit m ⟶ ∞ in (42) and (43). The remainder of the proof is standard and similar to [39,40]. ☐

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 ω is the positive constant given in (51). We see that Advances in Mathematical Physics By the same arguments similar to (52), we infer that where C 1 and C 2 , given in (54), are positive constants.

Lemma 10.
Suppose that (H1) and (H2) are satisfied. If Eð0 Þ < d and Iðu 0 Þ > 0, then the solution u of problem (1) satisfies where T is the maximal existence time of the solutions.
Proof. We know that Iðu 0 Þ > 0 and u is continuous on ½0, TÞ; hence, we have Let t 0 be the maximum of t 1 satisfying (76). Assume that t 0 < T; then, Iðuðt 0 ÞÞ = 0, 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 EðtÞ is a nonincreasing function. Proof. It suffices to show that ku t k 2 + kΔuk 2 is bounded independent of t. 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 k ⟶ ρ + 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). ☐

Nonexistence
In this part, similar to [41][42][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 Eð0Þ < E 1 and Iðu 0 Þ < 0, then the solution u of problem (1) satisfies where T is the maximal existence time of the solutions.

Lemma 14.
Under the conditions of Lemma 10, for C 3 , C 4 > 0, we obtain

Conclusions
Recently, there have been many published works related to wave equations with time delay. There were no local existence, global existence, nonexistence, and stability results of the plate equation with delay and logarithmic source terms, to the best of our knowledge. Firstly, we have obtained the local and global existence results. Then, we have obtained the nonexistence of solutions. Finally, we have proved stability results under sufficient conditions.

Data Availability
No data were used to support the study.

Conflicts of Interest
The authors declare that they have no competing interests.