Local and Global Existence of Solution for Love Type Waves with Past History

: In this paper, we consider an initial boundary value problem for nonlinear Love equation with inﬁnite memory. By combining the linearization method, the Faedo–Galerkin method, and the weak compactness method, the local existence and uniqueness of weak solution is proved. Using the potential well method, it is shown that the solution for a class of Love-equation exists globally under some conditions on the initial datum and kernel function.


Introduction
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 was developed in 1888 by Love. This kind of system appears in the models of nonlinear Love waves or Love type waves. It is a generalization of a model introduced by [1][2][3].
In order to completely study an evolutionary mathematical equation, a quantitative and qualitative study must be approached. An initial boundary value problem for a nonlinear Love equation with infinite memory has been considered by Zennir and et al. in [4] and the finite time blow up of weak solution has been shown under a relationship between the relaxation function g and nonlinear sources, i.e., u → ∞ when t → T * (T * is a finite time). Next, a very general decay rate for solution of the same problem, by certain properties of convex functions combined with some estimates, has been obtained in [5]. These two last results are considered as a qualitative studies. Obviously, in order to complete the study, we have to address the problem in quantitative terms. This is the subject of our present article, from a different angle, where we proved in detail, with the use of the most modern methods, the local existence (on small temporal period [0, T max ]) and global existence of solution on (0, ∞).
Investigations on the propagation of surface waves of Love-type are 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 few results for damped equations of Love waves or Love type waves. However, the existence of solutions or blow up results, with different boundary conditions, have been extensively studied by many authors.
To begin with, one must goes to the origins of Love's equation. It is derived in [6,7] by the energy method. Under the assumptions that the Kinetic energy per unit of length is (1) and the potential energy per unite of length is where F is an area of cross-section, w is a cross-section radius of gyration about the central line.
Using in (2) the corrected form of tension, we have Then, 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, κ 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 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, is obtained by using the Fourier method and method of small parameter.
In this article, Love-equation is considered as follows subject to the homogeneous Dirichlet boundary conditions and the following initial conditions where y = y(t, x), y x = ∂y ∂x (t, x), y xx = ∂ 2 y ∂x 2 (t, x), λ, λ 0 , λ 1 , λ 2 , L > 0 are constants. The past history in (6) is t −∞ µ(t − s)y xx (s)ds, which is considered as a damping term. It is well known that the damping terms play an important role in the studying the propagation mechanism of wave. It shows a behavior which is something between that of elastic solids and Newtonian fluids. Indeed, the stresses in these media depend on the entire history of their deformation, not only on their current state of deformation or their current state of motion.
Equation (6) is a generalization of a class of symmetric regularized long wave equations, known in abbreviation as (SRLWEs), given by Equation (10) 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. The functions µ and f satisfy Hypothesis 1. µ ∈ C 1 (R + , R + ) is a nonincreasing function such that: . . , 4. Our interest in this paper arose in the first place in consequence of a query for existence of unique solution.
Equations of Love waves or Love-type waves have been studied by many authors, we refer to [8][9][10][11][12][13], and references therein. In [8] a higher order iterative scheme is established for a Dirichlet problem for a class nonlinear Love-type equations ∂ tt y − y xx − ∂ tt y xx = f (x, t, y), and the authors get a recurrence sequence that converges at a rate 1 to a local unique weak solution of the above mentioned equation. In [8] is considered the following nonlinear Love equation with initial conditions and homogeneous Dirichlet boundary conditions and the authors established the existence of a unique local weak solution, a blow-up result for solutions with negative initial energy, the global existence and exponential decay of weak solution. In [9] is investigated the following Love equation with initial conditions and boundary conditions of two-point type and the authors proved existence of a weak solution, uniqueness, regularity, and decay properties of solution. In [13] is investigated the following nonlinear Love equation with initial conditions and homogeneous Dirichlet boundary conditions and the authors proved existence and uniqueness of a solution, a blow-up of the solution with a negative initial energy and the exponential decay of weak solution.
The existence/nonexistence, exponential decay of solutions, and blow-up results for viscoelastic wave equations, have been extensively studied and many results have been obtained by many authors (see [14][15][16]).
In this paper, the attention is focused on the local and global existence of weak solution of the problem (6)- (8). In Section 2, combining the linearization method, the Faedo-Galerkin method, and the weak compactness method, the local existence and uniqueness of the weak solution of the problem (6)-(8) is proved. In Section 3, using the potential well method, it is shown that the solution for class of Love-equation exists globally under some conditions on the initial datum.
The following technical result will play an important role in the sequel. [4] For Now, we will prove the existence of a unique local solution for (6)- (8). Our main result is as follows. Theorem 1. Let y 0 , y 1 ∈ H 1 0 (Ω) ∩ H 2 (Ω) be given. Assume that (Hypothesis 1) -(Hypothesis 4) hold. Then there exists a T * ∈ (0, T] such that the problem (6)-(8) has a unique local solution y for which y, ∂ t y, ∂ tt y ∈ L ∞ ((0, T * ); H 1 0 (Ω) ∩ H 2 (Ω)). (14) Proof of Theorem 1. Firstly, we will construct a sequence {y m } m∈N . Then, the Faedo-Galerkin method combined with the weak compactness method shows that {y m } m∈N converges to y which is exactly a unique local solution of (6)-(8).
Step 1. Let T > 0 be fixed and M > 0 be arbitrarily chosen. We set Let For some T * ∈ (0, T] and M > 0, we put Take y 0 ≡ 0 and define the sequence {y m } m∈N as follows where Let {w j } ∞ j=1 be an orthonormal basis of H 1 0 (Ω), formed by the eigenfunctions of the operator − ∂ 2 ∂x 2 . Let also, V k = span{w 1 , w 2 , . . . , w k }. We have for Note that We seek k functions ϕ solves the problem This leads to a system of ODE's for unknown functions ϕ mi . Based on standard existence theory for ODE, the system (22) admits a unique solution ϕ Step 2. Now we will prove that there exist constants M > 0 and for all m, k ∈ N. We partially estimate the terms of the associated energy. We replace y m and w with y (17) and we get Using (13), we obtain Let us denote the left hand side of (23) as m (t) = e (k) (y m ) + e (k) (y xm ) + e (k) (∂ t y m ).
So, we obtain y m ∈ W 1 (M, T * ). Step 3. Consider the Banach space We will show the convergence of {y m } m∈N to the solution of our problem. Let w m = y m+1 − y m . Then w m satisfies By (Hypothesis 2)-(Hypothesis 4), (15) and (36), we have Then where By (41), using Gronwall's Lemma, we get From here, it follows that {y m } m∈N is a Cauchy sequence in W 1 (T * ). Therefore there exists y ∈ W 1 (T * ) such that y m → y strongly in W 1 (T * ).

Global Solution
In this section, we consider the equation subject to the boundary conditions (7) and to the initial conditions (8). Here p > 2. We use methods introduced in [17][18][19][20][21][22][23][24]. Assume that f ∈ L 2 (Ω × R + ). We introduce the energy functional E(t) associated with Equation (53) where Now, we introduce the stable set as follows where and d = inf{ sup In addition, we introduce the "Nehari manifold" It is readily seen that the potential depth d is also characterized by This characterization of d shows that Suppose that (Hypothesis 1) holds. Let y be a solution of Equation (53). Then the energy functional (54) is a nonincreasing function and for all t ≥ 0, > 0, we have Proof. Multiplying (53) by ∂ t y(x, t) and integrating over [0, L], we obtain which completes the proof.
We will prove the invariance of the set W. That is, if for some t 0 > 0 and y(t 0 ) ∈ W, then y(t) ∈ W, ∀t ≥ t 0 . d is a positive constant.

Proof. We have
Using (Hypothesis 1), we get By differentiating the second term in the last equality with respect to ν, we obtain For ν 1 = 0 and , we have By Sobolev-Poincare's inequality, we deduce that K(ν 2 ) > 0. Then Then, by the definition of d, we conclude that d > 0. This completes the proof.
Proof. For y ∈ W and y = 0, we have By (Hypothesis 1), we get Consequently, for any y ∈ W, we have y ∈ B, where This completes the proof. 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 prove the following theorem.
Proof. Since y 0 (0) ∈ W, then Consequently, by continuity, there exists T m ≤ T such that This gives By (Hypothesis 1), we easily see that We then exploit (Hypothesis 1), p > 2 and the embedding H . This means, by the definition of l, Therefore I(t) > 0 for all t ∈ [0, T m ], in view of the following relation This shows that the solution y ∈ W, for all t ∈ [0, T m ] . By repeating this procedure we extend T m to T. This completes the proof.
The next Theorem shows that the local solution is global in time.
Proof. Now, it is enough to show that the following norm is bounded independently of t. To achieve this, we use (54), (55) and (61). We get Since I(t) and λ L 0 ∞ 0 µ(s)|y x (t) − y x (t − s)| 2 dsdx are positive, we conclude that where C is a positive constant depending only on p and l. This completes the proof.

Conclusions
By imposing less conditions with the help of some special results, we obtained local and global existence results extending some earlier results known in the existing literature. The main results in this manuscript are the following. Theorem 1 for local existence of solution and Theorem 3 for the global existence in time based on the potential depth.
This article is considered as an essential link in a series of articles by the same authors in the same type of equations. Our research falls within the scope of interests of many researchers in the modern era, according to the general objectives and broad scope of its application areas.
The importance of this research, although it is theoretical, lies in the following: