Global Existence of Solutions for the Viscoelastic Kirchhoff Equation with Logarithmic Source Terms

In this paper, a nonlinear viscoelastic Kirchhoff equation in a bounded domain with a time-varying delay term and logarithmic nonlinearity in the weakly nonlinear internal feedback is considered, where the global and local existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo-Galerkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given. /ese results extend and improve many results in the literature.

where Ω is a bounded domain in R n , n ∈ N * , with a smooth boundary zΩ, l > 0, ], μ 1 , and μ 2 are positive real numbers, h is a positive function which decays exponentially, τ(t) > 0 is a In the absence of delay term (i.e., μ 2 � 0), Han and Wang in [1] considered the following nonlinear viscoelastic equation with damping: Time delay is often present in applications and practical problems. In recent years, the control of PDEs with time delay effects has become an active area of research (see, for example, [2][3][4]). For example, in [5], it has been proven that a small delay in a boundary control could turn a well-behaved hyperbolic system into a wild one, thus showing that delay can be a source of instability.
Wu [6] treated problem (1) for a constant time delay τ and g 1 (x) � g 2 (x) � x. He proved the local existence result using the Faedo-Galerkin method and established the decay result employing suitable Lyapunov functionals under appropriate conditions on μ 1 and μ 2 and on the kernel h.
Benaissa et al. [7] considered the case of constant time delay τ, with l � 0 and M(r) � 1. ey proved the global existence and uniform decay for the following problem: e same problem (3) was also treated by Kirane and Said-Houari [8] for g 1 (x) � g 2 (x) � x and a homogeneous right hand side with τ, a constant time delay. Daewook [9] considered a viscoelastic Kirchhoff equation, with a timevarying delay and a nonlinear source term, given as u tt − M x, t, ‖∇u‖ 2 Δu + t 0 h(t − s)div(a(x)∇u(s))ds +|u| m u + μ 1 u t (x, t) + μ 2 u t (x, t − τ(t)) � 0, inΩ ×]0, +∞[, (4) is equation describes axially moving viscoelastic materials. Using the smallness condition with respect to Kirchhoff coefficient and the relaxation function and by assuming 0 ≤ m ≤ (2/(n − 2)) if n > 2 or 0 ≤ m if n ≤ 2, he obtained the uniform decay rate of the Kirchhoff-type energy.
In [10], the authors studied homogeneous problem (1) without the viscoelastic term, with l � 0 and M(r) � 1. In addition, μ 1 g 1 and μ 2 g 2 are multiplied by a positive nonincreasing function σ of C 1 (R + ) satisfying +∞ 0 σ(s)ds � +∞ and |σ ′ (t)| ≤ cσ(t). ey proved the global existence, and using a multiplier method with some properties of convex functions to get decay rate of the energy (when t goes to infinity) depends on the function σ and on the function H which represents the growth at the origin of g 1 .
Apart from the aforesaid attention given to polynomial nonlinear terms, logarithmic nonlinearity has also received a great deal of interest from both physicists and mathematicians.
is type of nonlinearity was introduced in the nonrelativistic wave equations describing spinning particles moving in an external electromagnetic field and also in the relativistic wave equation for spinless particles [11]. Moreover, the logarithmic nonlinearity appears in several branches of physics such as inflationary cosmology [12], nuclear physics [13], optics [14], and geophysics [15]. With all this specific underlying meaning in physics, the global-intime well-posedness of solution to the problem of evolution equation with such logarithmic-type nonlinearity captures lots of attention. Birula and Mycielski [16,17] studied the following problem: which is a relativistic version of logarithmic quantum mechanics and can also be obtained by taking the limit p goes to 1 for the p-adic string equation [18,19]. In [20], Cazenave and Haraux considered and they established the existence and uniqueness of the solution for the Cauchy problem. Gorka [21] used some compactness arguments and obtained the global existence of weak solutions, for all to initial boundary value problem (5) in the one-dimensional case. Bartkowski and Górka [22] proved the existence of classical solutions and investigated the weak solutions for the corresponding one-dimensional Cauchy problem for equation (6). Hiramatsu et al. [23] introduced the following equation: to study the dynamics of Q-ball in theoretical physics and presented a numerical study. However, there was no theoretical analysis for the problem. In [24], Han proved the global existence of weak solutions, for all to initial boundary value problem (8) in R 3 .
In the present paper, we investigate the stabilization of a dynamic model describing a string with a rigid surface and an interior somehow permissive to slight deformations. is leads to a varying material density |u t | l and a Kirchhoff term M(‖∇u‖ 2 ) that depends on ‖∇u‖ 2 . We prove the existence of global solutions in suitable Sobolev spaces by combining the energy method with the Fadeo-Galerkin procedure. We also establish an explicit and general decay result using a perturbed energy method with some techniques due to Mustafa and Messaoudi [25], as well as some properties of convex functions. ese convexity arguments were introduced and developed by Lasiecka et al. [26][27][28] and used, with appropriate modifications, by Liu and Zuazua [29], Alabau-Boussouira [30], and others. e paper is organized as follows: In Section 2, we give some hypotheses and state our main result. en, in Section 3, 2 Complexity we prove the global existence of weak solutions. Furthermore, in Section 4, the uniform decay of the energy is derived.
(A1) Assume that l satisfies (A2) e relaxation function h: and suppose that there exists a positive constant ζ satisfying (A3) g 1 : R ⟶ R is a nondecreasing function of class C 1 and H: R + ⟶ R + is convex, increasing and of class where ε, c 1 , and c 2 are positive constants. g 2 : R ⟶ R is an odd nondecreasing function of class C 1 (R) such that there exist c 3 , α 1 , and α 2 > 0, where τ 0 and τ 1 are positive numbers.
(A5) We also assume that (A6) We define the energy associated to the solution of system (12) by
e following lemma states an important property of the convolution operator. □ Lemma 5 (see [34] Remark 1. Let us denote by Φ * the conjugate function of the differentiable convex function Φ, i.e., en, Φ * is the Legendre transform of Φ, which is given by (see Arnold [35], p. 61-62) and Φ * satisfies the generalized Young inequality: Lemma 6. Let (u, z) be a solution of problem (12). en, the energy functional defined by (22) satisfies

e First Estimate.
Since the sequences u k 0 , u k 1 , and z k 0 converge and from Lemma 6, we can find a positive constant As h is a positive nonincreasing function, we get where By applying the Logarithmic Sobolev inequality, (58) yields 6 Complexity where C 2 is a positive constant depending only on we obtain k − (]σ 2 /2π) > 0 and (]/2) + (](1 + lnσ)) > 0. is selection is possible thanks to (A6). So we get Let us note that en, by using Cauchy Schwarz's inequality, we get Hence, (62) gives where C � max 2Tc, 2‖u k (0)‖ 2 . Applying the Logarithmic Gronwall inequality to (65), we obtain Hence, from (58), we obtain the first estimate: e estimate implies that the solution (u k , z k ) exists in [0, T) and it yields Noting that M(‖∇u k ‖ 2 ) ≥ a and by using Lemma 5, we obtain By using the Green formula, we have Consequently, equation (74) yields Complexity 7 To estimate the term on the right-hand side of (76), we apply Lemma 4 with ϵ 0 � (1/2) and use repeatedly Young's, Cauchy-Schwartz's, and the embedding inequalities as follows: Combining (76) and (77) to have Replacing ϕ j by − Δϕ j in (54), multiplying by d jk , and summing over j from 1 to k, it follows that en, we get We integrate over (0, 1), and we find Combining (78) and (81) and using (A2), we get 8 Complexity From the first estimate (67) and Young's inequality, we get Using (17) and Chaucy-Schwarz's inequality, we obtain Taking into account (83) and (84) into (82) yields Multiplying (51) by c jk tt and summing over j from 1 to k, it follows that Differentiating (54) with respect to t, we get Multiplying by d jk t and summing over j from 1 to k, it follows that en, we have 1 2 Integrating over (0, 1) with respect to ρ, we obtain Complexity 9 Summing (87) and (91), we get By Cauchy-Schwarz's, Sobolev's, and Young's inequalities, the right hand side of (92) can be estimated as follows: and from (16), 10 Complexity Using Lemma 6, Jensen's inequality, and the concavity of From (17) Similar to (77), we get Combining (85) and (99), we get Complexity en, from (67) and by integration over (0, t), (100) yields For a suitable η, we get Using Gronwall lemma, we obtain We observe from the estimates (67) and (103) that there exists a subsequence u m { } of u k and functions u, z, χ, and ψ such that u m tt ⇀u tt weakly star in L 2 0, T, z m ⇀z weakly star in L ∞ 0, T, H 1 0 Ω, L 2 (0, 1) , (108) z m t ⇀z t weakly star in L ∞ 0, T, L 2 (Ω ×(0, 1)) , Now, we will prove that u is the solution of (1). First, we will treat the nonlinear terms.
(1) Term |u k t | l u k t : from the first estimate (67) and Lemma 1, we deduce On the other hand, from Aubin-Lions theorem (see Lions [36]), we deduce that there exists a subsequence of which implies that Hence, where A � Ω × (0, T). us, using (117), (114), and Lions Lemma, we derive 12 Complexity which implies z m ⟶ z almost everywhere in A.
(2) Term u k ln|u k |: using (103), we have u k being bounded in L ∞ (0, T, H 2 0 (Ω)) which implies the boundedness of u k in L 2 (A). Similarly, u k t is bounded in L 2 (A). en, from Aubin-Lions theorem, we find a subsequence such that which implies Since the map s ⟶ ]s ln|s| is continuous, we have the following convergence: (0, T)). Next, taking into account the Lebesgue bounded convergence theorem (Ω is bounded), we get ]u m ln u m ⟶ ]u ln|u| strongly in L 2 0, T; L 2 (Ω) .
is completes the proof of eorem 1.

Uniform Decay of the Energy Proof of Theorem 2
In this section, we study the solution's asymptotic behavior of system (1).
To prove our main result, we construct a Lyapunov functional F equivalent to E. For this, we define some functionals which allow us to obtain the desired estimate. Lemma 11. Let (u, z) be a solution of problem (12). en, the functional satisfies the estimates Proof.

Complexity
where η > 0 and c s is the Sobolev embedding constant. Proof.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this manuscript.