Solvability of the Solution of Superlinear Hyperbolic Dirichlet Problem

. In this paper, we aim to study the solutions of superlinear hyperbolic problems with boundary condition of Dirichlet type where we show the existence and the uniqueness of the strong solutions for the superlinear problems by the method of energy inequality.


Introduction and position of the problem
The partial differential equations were probably formulated for the first time during the birth of rational mechanics in the 17th century [1][2][3]. Then the catalog of Partial Differential Equations (PDEs) have been enriched as the science developed and in particular physics [4][5][6][7]. If we only have to remember a few names, we must cite that of Euler, then those of Navier and Stokes, for the equations of fluid mechanics, those of Fourier in the heat equation, Maxwell for those of electromagnetism, Schrodinger and Heisenberg for the equations of quantum mechanics, and of course that of Einstein for the PDEs of the theory of relativity. A giant leap was made by L. Schwartz when he gave birth to the theory of distributions (around the 1950s), and at least comparable progress is due to L.
Hormander for the development of pseudo differential calculus (in the early 1970s). The complexity of nonlinearity and challenges in their theoretical study in have attracted a lot of interest from many mathematicians and scientists see [8][9][10][11].
Many natural phenomena and modern problems of physics, mechanics, biology, and technology can be modeled by nonlinear hyperbolic equations. The method used here is one of the most efficient functional analysis methods in solving partial differential equations, it is called a priori estimate method or the energy-integral method, see [10]. In this work, we study the solutions to hyperbolic problems with boundary conditions of Dirichlet type where we show the existence and uniqueness of the strong solutions for semilinear problems by the method of energy inequality, where we found a difficulty in the choice of the multiplier, and the uniqueness which is emanating from a priori estimate. Let We consider the nonlinear parabolic problem in which the nonlinear parabolic equation is given as follows with the initial condition where a, q are positive odd integers, p ≥ 1, and where f (x, t), ϕ(x) and ψ(x) are given functions and b(x, t) satisfies the following assumption: We establish a priori bound and prove the existence of a solution of problem (1. The Hilbert space F consists of the vector valued functions F = (f , u 0 ) with the norm The associated inner product is given as At the upcoming section, we intend to establish a priori estimate for the solution of problem (1.1)-

A priori bound
In the theory of PDEs, an a priori estimate (also called an apriori estimate or a priori bound) is an estimate for the size of a solution or its derivatives of a PDE. A priori is Latin for "from before" and refers to the fact that the estimate for the solution is derived before the solution is known to exist. One reason for their importance is that if one can prove an a priori estimate for solutions of a differential equation, then it is often possible to prove that solutions exist using the continuity method or a fixed point theorem. Some important definitions and theorems will be next listed in this section.
Theorem 2.1. If assumption A1 is satisfied, then for any function u ∈ D(L), there exists a positive

1)
and D(L) is the domain of definition of the operator L defined by Proof. Taking the scalar product in L 2 (Q) of Eq. (1.1) and the operator Mu The successive integration by parts of integrals on the right-hand side of (2.2) gives and In this regard, we have (2.6) By substituting (2.3)-(2.6) into (2.2), we obtain (2.7) By applying Cauchy inequality with ε, i.e., |ab| ≤ a 2 2ε + εb 2 2 , we can estimate the last term on the right-hand side of (2.7) and get By using assumptions A1 and using the Gronwall's Lemma, the estimate (2.8) becomes Then, by passing to the maximum, we get Now, we let R(L) be the range of the operator L. Since we do not have any information about R(L), except that R(L) ⊂ F , we must extend L so that estimate (1.6) holds for this extension and its range represents the whole space F . For this purpose, we present the next proposition. Moreover, the convergence of Lu n to f in L 2 (Q) gives: As we have the uniqueness of the limit in D (Q), we conclude from (2.11) and (2.12) that f = 0.
Then it is generated from (2.9) that l 1 u n −→ ϕ x and l 2 u n −→ ψ in L 2 (Ω) .
On the other hand, we have Then, we obtain The priori estimate (2.1) can be then extended to strong solution, i.e., we have the estimate (2.13) In light of the estimate given in (2.13), we can infer the next theoretical results. There is then a corresponding sequence u n ∈ D(L) such as z n = Lu n . Immediately, the estimate (2.8) becomes: where p and q tend towards infinity. We can consequently deduce that (u n ) n∈N is a Cauchy sequence in E. So like E is a Banach space, it exists u ∈ E such as Then z ∈ R(L), and so R(L) ⊂ R(L). Also, we conclude here that R(L) is closed because it is Banach (any complete subspace of a metric space, not necessarily complete, is closed). Thus, it remains to show the reverse inclusion either z ∈ R(L), and then it exists a Cauchy sequence (z n ) n∈N in F constituted of the elements of the set R(L) such that lim n−→+∞ z n = z, or z ∈ R(L) because R(L) is closed subset. So R(L) is complete. There is then a corresponding sequence u n ∈ D(L) such that Lu n = z n . Consequently from (2.8), we get where p and q tend towards infinity. We can immediately deduce that (u n ) n∈N is a Cauchy sequence in E, and so like E is a Banach space, it exists u ∈ E such as Once again, there is a corresponding sequel (Lu n ) n∈N ⊂ R(L) such as Lu n = Lu n on R (L) , ∀n ∈ N.
So we have lim n−→+∞ Lu n = z and consequently z ∈ R (L), which implies R L ⊂ R (L).

Existence and uniqueness of solution
In this section, additional results are listed below, which are related to the existence and uniqueness of strong solution for the main Problem (P1). Proof. To prove this result, we should note that we first have where W = (w , w 0 , w 1). So for w ∈ L 2 (Q) and for all we have Q Lu.w dxdt = 0.
By putting w = u t , we obtain This gives Therefore, we have u t = w = 0. Since the range of the trace operators is everywhere dense in the Hilbert space F with the associate norms ϕ x L 2 (Ω) and ψ L 2 (Ω) , then the equality (3.1) implies that ω 0 = 0 and ω 1 = 0. Hence W = 0 implies R(L) = F . Corollary 3.1. If for any function u ∈ D(L), we have the following estimate: Then the solution of the problem (P1), if it exists, is unique.
Proof. Let u 1 and u 2 be two solutions of problem (P1), i.e., As L is linear, we then obtain L (u 1 − u 2 ) = 0.

Conclusion
We have used the method of energy inequality for the super liner problems to show the existence and the uniqueness of the solution. In addition, we have studied the solution of superlinear hyperbolic problems with boundary condition of Dirichlet type.

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