General Decay Rate of Solution for Love-Equation with Past History and Absorption

: In the present paper, we consider an important problem from the point of view of application in sciences and engineering, namely, a new class of nonlinear Love-equation with inﬁnite memory in the presence of source term that takes general nonlinearity form. New minimal conditions on the relaxation function and the relationship between the weights of source term are used to show a very general decay rate for solution by certain properties of convex functions combined with some estimates. Investigations on the propagation of surface waves of Love-type have been made by many authors in different models and many attempts to solve Love’s equation have been performed, in view of its wide applicability. To our knowledge, there are no decay results for damped equations of Love waves or Love type waves. However, the existence of solution or blow up results, with different boundary conditions, have been extensively studied by many authors. Our interest in this paper arose in the ﬁrst place in consequence of a query for a new decay rate, which is related to those for inﬁnite memory (cid:118) in the third section. We found that the system energy decreased according to a very general rate that includes all previous results.


Introduction
The Love equation is a one-dimensional mathematical model that is used to determine a many physical phenomenon. This theory is a continuation of the Euler-Bernoulli beam theory and it was developed in 1888 by Love. This kind of systems appears in the models of nonlinear Love waves or Love type waves. It is a generalization of a model introduced by [1][2][3].
The original Love's equation is derived in [4,5] by the energy method. Under the assumptions that the Kinetic energy per unit of length is and the potential energy per unite of length is where F is an area of cross-section and w is a cross-section radius of gyration about the central line.
Using, in (2), the corrected form of tension, we have Subsequently, the variational equation of motion is given by and we then obtain the equation of extensional vibrations of rods as The parameters in (5) have the following meaning: u is the displacement, µ is a coefficient, E is the Young modulus of the material, and ρ is the mass density.
This type of problem describes the vertical oscillations of a rod, and it was established from Euler's variational equation of an energy functional associated with (5). A classical solution of problem (5), with null boundary conditions and asymptotic behavior are obtained using the Fourier method and method of small parameter. In this article, we consider a nonlinear Love-equation in the form where Ω = [0, L], L > 0 and F[y] = F x, t, y, y x , ∂ t y, ∂ t y x ∈ C 1 Ω × R + × R 4 .
The given functions , f will be specified later. With F = F(x, t, y 1 , . . . , y 4 ), we put D 1 F = ∂F ∂x , D 2 F = ∂F ∂t , D i+2 F = ∂F ∂y i , with i = 1, . . . , 4. Equation (6) satisfies the homogeneous Dirichlet boundary conditions y(0, t) = y(L, t) = 0, t > 0, (8) and the following initial conditions We call the Sobolev space of order 1 on Ω the space The space H 1 is endowed with the norm that is associated to the inner product and we note the corresponding norm We have the generalization of such spaces. Let m ∈ N. A function v ∈ L 2 (Ω) belongs to the Sobolev space of order m, denoted H m (Ω), if all of the derivatives of v up to order m, in the distributional sense, belong to L 2 (Ω). By convention, we note H 0 (Ω) = L 2 (Ω).
In order to deal with a wave equation with infinite memory, we assume that the kernel function satisfies the following hypothesis: and there exists an increasing strictly convex function H ∈ C 1 (R + , R + ), satisfying where H −1 is the inverse of H function.
We need the following assumptions on source forces. .

Hypothesis 5.
There exists a constant m 0 ≥ 0, such that Our results are very interesting from an application point of view and, as for as, we know that there is no results for equations of Love waves or Love type waves with the presence of finite/infinite memory term ([1-3,6-12]).
Without infinite memory term, Triet et al. in [12] considered an initial boundary value problem for a nonlinear Love equation where λ, λ 1 > 0 are constants andũ 0 ,ũ 1 ∈ H 1 0 (Ω) ∩ H 2 (Ω); f , F and G are given functions. First, under suitable conditions, the existence of a unique local weak solution has been proved and a blow-up result for solutions with negative initial energy is also established. A sufficient condition ensuring the global existence and exponential decay of weak solutions is given in the last section. These results will be improved in [11,12] to the Kirchhoff type.
Concerning problems with infinite history, we mention the work [15], in which the authors onsidered the following semi-linear hyperbolic equation, in a bounded domain of R 3 , with K(0), K(∞) > 0, ∂ t K ≤ 0 and they proved the existence of global attractors for the problem. Next, in [22], the authors considered a fourth-order suspension bridge equation with nonlinear damping term |∂ t u| m−2 ∂ t u and source term |u| p−2 u. The authors found necessary and sufficient conditions for global existence and energy decay results without considering the relation between m and p. Moreover, when p > m, they gave a sufficient condition for finite time blow-up of solutions. The lower bound of the blow-up time is also established.
Recently, in [23], the authors studied a three-dimensional (3D) visco-elastic wave equation with nonlinear weak damping, supercritical sources, and prescribed past history where the relaxation function k is monotone decreasing with k(+∞) = 1, m ≥ 1, 1 ≤ p < 6. When the source is stronger than the dissipation, i.e., p > max{m, k(0)}, they obtained some finite time blow-up results with positive initial energy. In particular, they obtained the existence of certain solutions, which blow-up in finite time for initial data at an arbitrary energy level (see [24]).
The outline of our work: in the next section, the existence results in Theorems 1 and 3 are obtained by using a new combined methods. A decay rate of energy, which is very general, is derived in the last section: Theorem 4 extends the results that were obtained in [20], where the authors established a general decay rate for relaxation functions satisfying for a positive function H ∈ C 1 (R + ) and H is linear or strictly increasing and strictly convex C 2 function on (0, r], 1 > r. This improves the conditions that were introduced by [13] on the relaxation functions: where χ is a non-negative function, strictly increasing and strictly convex on (0, k 0 ], k 0 > 0. Furthermore, the authors required that and proved a decay result for the energy in a bounded domain. In addition to these assumptions, if then, in this case, an explicit decay rate is given.

The Existence of Solution
We define the weak solution to of (6)-(9), as follows.

Definition 1.
A function y is said to be a weak solution of (6)- such that y satisfies the variational equation for all test function w ∈ H 1 0 (Ω), for almost all t ∈ (0, T).
The following famous and widely used technical Lemma will play an important role in the sequel.
Now, we state the existence of a local solution for (6)-(9).
Let f ≡ 0. Here, and in the sequel, we consider problem (6) with the boundary conditions (8) and the initial conditions (9) in the following form We introduce the energy functional E(t) that is associated to our problem (23), as follows where Now, we introduce the stable set, as follows (see [26,27]) where We notice that the mountain passes level d, given in (26), is defined by Additionally, by introducing the so called "Nehari manifold" The potential depth d is also characterized by This characterization of d shows that It is not hard to see this Lemma.

Lemma 2.
Suppose that (Hypothesis 1) holds. Let y be solution of our equation. Subsequently, the energy functional (24) is a non-increasing function, i.e., for all t ≥ 0, ν > 0, Proof. Multiplying (23), with p ≥ 2, by ∂ t y, integrating over Ω to obtain then, using Lemma 1, we obtain which completes the proof.
As in [25], we will prove the invariance of the set W, that is if for some t 0 > 0 we have y(t 0 ) ∈ W, then y(t) ∈ W, ∀t ≥ t 0 . The next Lemma ensures the existence of the potential depth. Now, we will show that our local solution y is global in time. For this purpose it suffices to prove that the norm of the solution is bounded, independently of t. This is equivalent to proving the following Theorem.
The next Theorem shows that our local solution is global in time.
Theorem 3. Suppose that (Hypothesis 1), (Hypothesis 5), p ≥ 2 and (33) hold. If y 0 (0) ∈ W, y 1 ∈ H 1 0 (Ω), then the local solution y is global in time and such that y ∈ G T , where Proof. It suffices to show that the following norm is bounded independently of t.
The last term can be treated, as follows By using Cauchy-Schwarz and Young's inequalities, we obtain, for any ν > 0, Subsequently, By the continuous embedding for p ≥ 2, we have Using (33) and choosing ν = l 1−l , we obtain Lemma 6. Assume that (Hypothesis 1), (Hypothesis 5), and p ≥ 2 hold. Subsequently, for ν < (1 − l), the functional introduced in (40) satisfies, along the solution, the estimate and for all For any ν > 0, we have and Furthermore, by using (13), we have the following estimate Now, because p ≥ 2, we have, by using (42) and the previous estimate, In the same way, we have The last term can be estimated, as follows A combination of all estimates gives where by (33), we have and for all ν > 0, we have Let us define the Lyapunov functional We need the next Lemma, which means that there is an equivalence between the Lyapunov and energy functions Lemma 7. There exist positive real numbers c 1 and c 2 , such that Proof. By (43), we have Thanks to Hölder and Young's inequalities, we have and The two last terms can be estimated, as follows Hence, there exists a constant C > 0, such that Therefore, we can choose ε 1 , so that Lemma 8. Assume that (Hypothesis 1) hold. Susequently, there exist strictly positive constants λ and c, such that Proof. By Lemmas 2, 5, and 6, we have where, by (33), we have and for all Now, we choose ν and, once this constant is fixed, we can select ε 1 , ε 2 small enough that give, for p ≥ 2, the existence of constants c, λ > 0, such that (46) holds true.

Conclusions
By imposing a new appropriate conditions (Hypothesis 1)-(Hypothesis 5), which seems not be used in the literature, with the help of some special results, we obtained an unusual a decay rate result while using properties of convex functions combined with some estimates, extending some earlier results known in the existing literature. The main results in this manuscript are the following. Theorem 3 for the global existence of solutions and Theorem 4 for the general decay rate.
A class of symmetric regularized long wave equations, which is known in abbreviation as (SRLWEs), is given by ∂ tt y − y xx − ∂ tt y xx = −y∂ t y x − u x ∂ t u. (50) Equation (50) was proposed as a model for propagation of weakly nonlinear ion acoustic and space charge waves; it is explicitly symmetric in the x and t derivatives and is very similar to the regularized long wave equation, which describes shallow water waves and plasma drift waves. The SRLWE and its symmetric version also arises in many other areas of mathematical physics. We remark that Equation (50) is special form of the equation that is discussed in (6), in which F[y] = −y∂ t y x − u x ∂ t u.
Our research falls within the scope of the modern Time-partial differential equations interests; it is considered among the issues that have wide applications in science and engineering that are related to the energy systems. The importance of this research, although it is theoretical, lies in the following: we found that viscoelastic damping term causing the decrease in energy and decreasing followed the infinite memory, depending on initial data. It will be very interesting if one considers numerical studies. It will be our next research project. Acknowledgments: For any decision, the authors would like to thank the anonymous referees and the handling editor for their careful reading and for relevant remarks/suggestions to improve the paper.

Conflicts of Interest:
The authors declare no conflict of interest.