Solvability of Nonlinear Wave Equation with Nonlinear Integral Neumann Conditions

. In this paper, we examine a nonlinear hyperbolic equation with a nonlinear integral condition. In particular, we prove the existence and the uniqueness of the linear problem by the Fadeo Galerkin method, and by applying an iterative process to some signiﬁcant results obtained for the linear problem, the existence and the uniqueness of the weak solution for the nonlinear problem are additionally examined.


Introduction
The nonlinear hyperbolic equations describes important processes in the nonlinear evolution equation basis of mathematical models of diverse phenomena and processes in mechanics, physics, technology, biophysics, biology, ecology, and many other areas [1][2][3][4]. Such ubiquitous occurrence of nonlinear hyperbolic equations is to be explained, first of all, by the fact that they are derived from fundamental laws in the real world [5,6].
Let us remark only that for broad classes of equations, the fundamental questions of solvability and uniqueness of solutions of various boundary value problems have been solved and that the differentiability properties of the solutions have been studied in detail [7][8][9]. General results of the solvability and uniqueness were inferred by different methods such as the energy method, upper lower method and the Fadeo Galerkin methods. The later one is regarded one of the most important methods that were mainly developed in the 1960s, but they are still powerful tools today to deal with nonlinear evolution equations, especially those who are modeled by non-classical boundary conditions that consist of integral conditions [10][11][12][13]. Non-local and integral partial differential equations are used to solve a vast range of current physics and technology challenges [14][15][16][17][18][19]. When it is hard to directly measure the minimum and maximum values on the border, the overall value or average is known. This method might be utilized for modeling where we can model more complicated domain with nonlinear integral condition.
Motivated by the above perspective, we trait in this work to discuss a nonlinear evolution equation with a nonlinear integral condition. In particular, we aim to focus on the solvability of the solution of nonlinear hyperbolic problems with the integral condition of the second type by the method of Fadeo-Galerkin. In the following sections, we present first the existence of the linear problem, and then by applying an iterative process based on the results obtained for the linear problem, we prove the existence and the uniqueness of the weak solution of the nonlinear problem.

The statement of the main problem
In this section, we let Q = (x, t) ∈ R 2 , x ∈ Ω = ]0, l[ and 0 < t < T , besides we consider the main following initial boundary value problem for a nonlinear hyperbolic equation: Assuming that f ∈ L 2 (Q) and ϕ, ψ ∈ L 2 (Ω). The nonlinear hyperbolic equation is given as follows: which satisfies the following identities: • The initial conditions x ∈ (0, l) .
• The boundary conditions are integral conditions of the second type defined as: and for all v ∈ L 2 (Q), we have: in which we define the space V by V = H 1 (Ω).
Actually, the space V is provided with the norm v V = v H 1 (Ω) , and hence it is a Hilbert space.
From this point of view, we are now able to formulate problem (P 2 ) in order to precisely study it.
From this fact, we will need to the following hypothesis: .

Position of problem (P 1 )
In the rectangular area Q = Ω × (0, T ), and T < ∞, we consider the following linear problem (P 2 ): in which the hyperbolic equation is given as follows: with the initial conditions: and with the integral condition of the second type: such that the space V = H 1 (Ω) provided with the norm v V = v H 1 (Ω) is a Hilbert space. We are now able to formulate the problem (P 2 ), precisely to study it, according to the following hypothesis: . Definition 3.1. The weak solution of problem (P 2 ) is a function that satisfies: • u admits a strong derivative ∂u ∂t ∈ L 2 0, T ; L 2 (Ω) .
• The following identity: 3.1. Variational formulation. By multiplying the equation: by an element v ∈ V , and the by integrating the result over Ω, we obtain: By using the boundary conditions and using Green's formula, ( 3.3) becomes where (·, ·) denotes the scalar product L 2 (Ω).

3.2.
Study of the existence of weak solution of problem (P 2 ). The demonstration of the existence of the solution of problem (P 2 ) can be discussed based on the Faedo-Galerkin method which consists of carrying out the next three steps.

3.2.1.
Step 1: Construction of the approximate solutions. As the space V is separable, then there exists a sequence w 1 , w 2 , · · · , w m having the following properties: (3.5) In particular, we can say: Now, the Faedo Galerkin's approximation aims to search about a function in which and (3.9) In addition, we have We obtain consequently a system of first-order nonlinear differential equations: From this view, we consider the vector: coupled with the matrices: . Now, we can immediately write problem (P 4 ) in the matrix form as: As the matrix entries B m are linearly independent (because it is a diagonal matrix), then det B m = 0.
So, it is invertible, and then g m is the solution of the following states: Now, it is easy to verify that this ordinary differential system has a solution where the matrix: is of constant coefficients and the vector B −1 m f m are continuous functions and majorized by integrable functions on (0, T ). Consequently, we can conclude that there exists a t m that depends only on |α im | and |β im |.

3.2.2.
Step 2: A priori estimate. Herein, we intend to begin this step with state and prove the next result.
where c 1 and c 2 are two positive constants independent of m.
Proof. By multiplying the equation of (P 3 ) by g km (t), and then by summing the result over k, we obtain: So, we obtain Thus, we get Integrating the above equality over 0 to t coupled wit using the Cauchy inequality with ε, i.e.

3.2.3.
Step 3: Convergence and the result of existence.
Proof. From Lemma 1.2, we might deduce that there are subsequences denoted respectively by u m k , ∂u m k ∂t of (u m ) and (u m ) t such that and ∂um k ∂t w in L 2 0, T ; L 2 (Ω) .
(3. 16) We know that according to Relikh-Kondrachoff's theorem the injection of H 1 (Q) into L 2 (Q) is compact. In addition, like the results of Rellich's theorem, any weakly convergent sequence in H 1 (Q) has a subsequence which converges strongly in L 2 (Q) . So, we can assert: On the other hand, from Lemma 1.3, there is a subsequence of u m k k , which is still denoted by u m k , converges almost everywhere to u such that It is still essential to demonstrate that w = ∂u ∂t . This actually suffices to prove: To this aim, we note that as then the proof of (3.19) is equivalent to prove that which means lim u m k − ϕ − χ, v L 2 (0,T ;L 2 (Ω)) = 0, ∀v ∈ L 2 (0, T ;L 2 (Ω)), as In fact, by using the equality with the help of using u m k ∈ L 2 0, T ; V m k and u m k t ∈ L 2 0, T ; V m k , we can get: for all t ∈ [0, T ]. On the one hand, we have for t ∈ [0, T ]. Besides, we have: which implies: Proof. From Theorem (3.1), we have shown that the limit function u satisfies the first two conditions of the definition 3.1. Now we will demonstrate (iii). According to the Theorem 3.1, we have: On the other hand, we have which implies: From the uniqueness of the limit, we get u (0) = ϕ. By using the same previous steps, we demonstrate u t (0) = ψ. It remains to demonstrate (iv). To this aim, we have: Integrating (P 3 ) over (0, T ), we obtain: for all k = 1, m and for all t ∈ [0, T ]. Using (3.13) and that V m dense in V and passing to the limit in (3.22), we get: This immediately implies that (iv) is verified.

Weak solution of the nonlinear problem
Initially, we present the considered solution's concept. For this purpose, we let v = v (x, t) be any By multiplying by v and integrating the result over Q τ , we obtain Now, by using integration by parts and the conditions on y and v , we get (4.1) It then results from (4.1) that or Thus, it is the time to build a recurring sequence starting with y (0) = 0. The sequence y (n) n∈N is defined as follows: Given the element y (n−1) , then for n = 1, 2, 3, · · · , we can solve the following problem: According to the last linear problem, we fix the n each time. Problem (P 4 ) admits then a unique solution y (n) (x, t), which can be given by the Fadeo-Galarkin method. In this regard, we assume As a result, we have the following new problem: Integrating over t yields: .
Then (y (n) ) n converges to an element of L 2 (0, T, H 1 (Ω)), say y . Now, we will show that lim n−→∞ is a solution to the problem (P 5 ) by showing that y satisfies: A (y , v ) = Qτ G(x, t, y , y x ) · v (x, t)dxdt.

Conflicts of Interest:
The authors declare that there are no conflicts of interest regarding the publication of this paper.