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Hazal Yüksekkaya, Erhan Pișkin, Salah Mahmoud Boulaaras, Bahri Belkacem Cherif, Sulima Ahmed Zubair, "Existence, Nonexistence, and Stability of Solutions for a Delayed Plate Equation with the Logarithmic Source", Advances in Mathematical Physics, vol. 2021, Article ID 8561626, 11 pages, 2021. https://doi.org/10.1155/2021/8561626
Existence, Nonexistence, and Stability of Solutions for a Delayed Plate Equation with the Logarithmic Source
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.
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.  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 . 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  studied the equation as follows: where the authors in  proved the existence and uniqueness of the solutions for equation (3). Gorka  obtained the global existence results of solutions for one-dimensional equation (3). Bartkowski and Gŏrka  considered the weak solutions and obtained the existence results.
In , Hiramatsu et al. studied the equation as follows:
In , 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 , 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 , 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  extended this result. In the presence of the strong damping term (), Pișkin and Polat  proved the global existence and decay of solutions for equation (7). For more results about plate problems, see [20–22].
In , 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 , Xu et al., by using the spectral analysis approach, established the same result similar to  for the one space dimension.
In , 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 , 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 , 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.
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.
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 ) (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.☐
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
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
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
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).☐
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.☐
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