Polynomial Decay Rate for Kirchhoff Type in Viscoelasticity with Logarithmic Nonlinearity and Not Necessarily Decreasing Kernel

This paper describes a polynomial decay rate of the solution of the Kirchhoff type in viscoelasticity with logarithmic nonlinearity, where an asymptotically-stable result of the global solution is obtained taking into account that the kernel is not necessarily decreasing.


Introduction
Consider the quasilinear viscoelastic problem: u (x, 0) = u 0 , u t (x, 0) = u 1 for x ∈ Ω with: where Ω ⊂ R N is a bounded domain with a sufficiently smooth boundary ∂Ω, γ ≥ 2, k, ξ 0 and ξ 1 are positive constants, ρ ≥ 0 for N = 1, 2 and 0 ≤ ρ ≤ 4 N−2 for N ≥ 3, and h is a relaxation function, which will be specified later.From the physical point of view, the problem (1)-( 3) is related to the spillover problem with memory and the panel flutter equation.Viscoelasticity has an important role in the study of biological phenomena, and it can have a strong effect on some factors that have a correlation with biological phenomenain general.A viscoelastic material will return back to its original size after an influential force has been detached, even though it will remain for a specific time in its original shape; see [1][2][3][4][5][6][7][8][9][10][11].
Recently, viscoelasticity problems have been handled carefully in several papers, and other results relating the global existence and decay of the global solution have been found (see [1][2][3][4][5][6][7][10][11][12][13][14][15][16][17][18][19][20][21][22][23]), where the function of the relaxation was supposed to be either exponential decay or polynomial decay.In [7], the mathematicians searched for a viscoelastic equation for more general decaying kernels and obtained some general decay results, from which the common exponential or polynomial rates were more specific samples of them.Later, many papers, using almost identical techniques, achieved analogous general decay outcomes.See [11,24].Our decay rate obtained in the third section is less general than that obtained in [7,8], where a common decay rate outcome was established in order to let the functions of the relaxation satisfy: g (t) ≤ −H(g(t)), t ≥ 0, H(0) = 0 (5) a positive function H ∈ C 1 (R + ), where H is either linear or a strictly-increasing and strictly-convex C 2 function on (0, r], 1 > r.In [7], the exponential decay for the viscoelastic wave equation of the solution with an unnecessarily decreasing kernel was handled.More specifically, the mathematicians had an exponential decay outcome as time tended to infinity h (t) + γh(t) ≥ 0 for all t ≥ 0 provided that (h (t) + γh(t)) e αt ∈ L 1 (0, ∞) for some α > 0. Their proof depended on another technique, the so-called "Lyapunov functional".In this present work, we follow up with the same steps of the previous result in [7,10] for a new class of Kirchhoff hyperbolic equations on bounded domain polynomial decay of the Kirchhoff type in viscoelasticity combined with the right-hand side defined as a logarithmic nonlinearity, and the kernel is not necessarily decreasing taking into account that the similar conditions in the last ones in ( [7,10]) are considered.The logarithmic nonlinearity is of much use in physics, since it naturally appears in inflation cosmology and supersymmetric filed theories, quantum mechanics, and nuclear physics (see [5,14]).This kind of problem can be applied in many different areas of physics such as nuclear physics, optics, and geophysics (see [3,15]).The outline of the paper is as follows.In the second section, we give some basic concepts related to our problem given by ( 1)- (3).In Section 3, we prove our principle result.
As in previous work in ( [7,10]), we impose the following conditions on the relaxation function h.Namely, we suppose that the kernel h (t (A3) There exists a positive differentiable function ξ (t) such that: where 1 ≤ r < 3 2 and for α > 0, and ξ (t) satisfies some positive constant L, Furthermore, where 1 < r < 3 2 for any t ≥ 0, there exists a positive constant C r depending only on r, such that: t Notation: We denote by l, l, l α , l α and h the following expressions: and we recall the binary notation: Lemma 1.The classical energy associated with (1)-( 3) is defined by: and its derivative is: Proof.Multiplying (1) by u t and by integration over (Ω) , we have: By using: we get by direct calculation: then: By using integration by parts, we have: Integrating by parts, we have: Also integrating by parts, we get: By using: we get: By replacing (12)-( 16) in (11) , we get: then (17) is equivalent to: and by using (9) in (18) , we get (10) .
The proof is complete.
Lemma 2. The modified energy for (1)-( 3) is defined by: e (t) : = 1 ρ+2 u t (t) and its derivative satisfies the following: Proof.Now, for the estimation terms, we have: and using: ds.

General Decay and Polynomial Decay
In the previous work, it was supposed that h (t) ≤ 0. Therefore, from (20), we see that e (t) ≤ 0. This implies that e (t) ≤ e (0) , for all t ≥ 0. In our case, we are not assuming that h (t) ≤ 0. In fact, we are allowing the function h (t) to oscillate.
To prove our result, we need to introduce the following auxiliary functional: and: where: and: l (t) is defined in (8).Further, we consider the functional: where 1 ≤ r < 3 2 and for some positive constants ε, η, k 1 , k 2 , k 3 , k 4 , and k 5 to be determined later.
Proof.It suffices to note that: and applying Holder's inequality for The proof of the lemma is completed.
and h be a continuous function on [0, T] and suppose that 0 < θ < 1 and ρ > 1.Then, there exists a constant C > 0 such that: Proof.By using Lemma 3 with r := (ρ , we obtain: It is easy to see that: By combining (45) and (46) , the proof of the lemma is completed.Lemma 5. Let v ∈ L ∞ (0, T) ; L 2 (Ω) be such that v x ∈ L ∞ (0, T) ; L 2 (Ω) and h be a continuous function on [0, T] and suppose that ρ > 1.Then, there exists a constant C > 0 such that: Proof.We use (45) for θ = 1 to arrive at: .
It suffices to note that: to obtain (47).This completes the proof. Let: and: Now, we are in a position to state and prove our first result.
Then, the classical energy E (t) of (1)-( 3) decays to zero exponentially and polynomially.That is, there exist positive constants C 1 , C 2 , and C 3 such that: Proof.Now, a differentiation of V (t) definite in (31) with respect to time gives: By a differentiation of (27), we have: By using u as a solution in (1)-(3) , we get: Integrating by parts, we get: Also integrating by parts, we get: By using integration by parts, we get: By direct calculation, we get: By replacing (53)-( 56) in (52) , we get: By replacing (57) in (51) , we get: and: On the other hand, we have: By using: we get: By direct calculation, we get: Replacing (62) and (63) in (61) , we get: By replacing (64) in (60) , we get: (65) By using direct calculation, we get: By using direct calculation and Using (66) and (67) in (65) , we see that: Taking into account (20) , (59), and (68) in (50), we obtain: Next, we use the estimate (69) .