Int. J. Anal. Appl. (2022), 20:62 Solvability of the Solution of Superlinear Hyperbolic Dirichlet Problem Iqbal M. Batiha1,2,∗ 1Department of Mathematics, Al Zaytoonah University of Jordan, Queen Alia Airport St 594, Amman 11733, Jordan 2Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE ∗Corresponding author: i.batiha@zuj.edu.jo Abstract. 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. 1. 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 [1–3]. Then the catalog of Partial Differential Equations (PDEs) have been enriched as the science developed and in particular physics [4–7]. 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 [8–11]. 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 Received: Sep. 30, 2022. 2010 Mathematics Subject Classification. 35L03, 30C15. Key words and phrases. nonlinear hyperbolic equation; energy inequality method; existence; uniqueness. https://doi.org/10.28924/2291-8639-20-2022-62 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-62 2 Int. J. Anal. Appl. (2022), 20:62 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 T > 0, Ω ⊂Rn and Q = Ω × (0,T ) = { (x,t) ∈Rn+1 : x ∈ Ω, 0 < t < T } . We consider the nonlinear parabolic problem  utt −a∆u + b(x,t)ut + uq = f (x,t) u(x, 0) = ϕ(x), ut(x, 0) = ψ(x), u(x,t) |Γ= 0 , (P1) in which the nonlinear parabolic equation is given as follows Lu = utt −a∆u + b(x,t)ut + uq = f (x,t), (1.1) with the initial condition lu = u(x, 0) = ϕ(x), (1.2) and the Dirichlet boundary conditions u(x,t) |Γ= 0, ∀t ∈ (0,T ), (1.3) 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: A1. b1 ≤ b(x,t) ≤ b0, (x,t) ∈ Q̄. We establish a priori bound and prove the existence of a solution of problem (1.1)-(1.3). To this aim, let Lu = F, where L = (LF, l1, l2), and F = (f ,ϕ,ψ) be the operator equation corresponding to problem (1.1)-(1.3). The operator L acts from E to F, and the Banach space E consists of all functions u(x,t) with the finite norm ‖u‖2E = max 0≤τ≤T ‖uτ (x,τ)‖2L2(Ω) + max 0≤τ≤T ‖∇u‖2L2(Ω) + ‖ut‖ 2 L2(Q) + max 0≤τ≤T ‖u(x,τ)‖q+1 Lq+1(Ω) . (1.4) The Hilbert space F consists of the vector valued functions F = (f ,u0) with the norm ‖F‖2F = ‖f‖ 2 L2(Q) + ‖ψ‖ 2 L2(Ω) + ‖ϕx‖ 2 L2(Ω) + ‖ϕ‖ q+1 Lq+1(Ω) . (1.5) The associated inner product is given as (F,G)F = (f ,g)L2(Q) + ( ϕx, (g0 )x ) L2(Ω) + (ψ,g1)L2(Ω) . (1.6) Int. J. Anal. Appl. (2022), 20:62 3 We assume that the data functions ϕ and ψ satisfy the conditions of the form (1.3), i.e., ϕ |Γ= ψ |Γ= 0. At the upcoming section, we intend to establish a priori estimate for the solution of problem (1.1)- (1.3). 2. 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 constant c independent of u such that max 0≤τ≤T ‖uτ (x,τ)‖2L2(Ω) + max 0≤τ≤T ‖∇u‖2L2(Ω) + ‖ut‖ 2 L2(Q) + max 0≤τ≤T ‖u(x,τ)‖q+1 Lq+1(Ω) ≤ c ( ‖f‖2L2(Q) + ‖ψ‖ 2 L2(Ω) + ‖ϕx‖ 2 L2(Ω) + ‖ϕ‖ q+1 Lq+1(Ω) ) , (2.1) and D(L) is the domain of definition of the operator L defined by D(L) = {u : u ∈ L∞ ( 0,T,Lq+1(Ω) ) , ut ∈ L∞ ( 0,T,L2 (Ω) ) } satisfying condition (1.3). Proof. Taking the scalar product in L2(Q) of Eq. (1.1) and the operator Mu = ut, where Qτ = Ω × (0,T ), yields (Lu,Mu)L2(Qτ ) = (utt,ut)L2(Qτ ) −a (∆u,ut)L2(Qτ ) + (but,ut)L2(Qτ ) + (u q,ut)L2(Qτ ) = (f ,ut)L2(Qτ ) . (2.2) The successive integration by parts of integrals on the right-hand side of (2.2) gives (utt,ut)L2(Qτ ) = ∫ Qτ utt ·utdxdt = 1 2 ∫ Ω u2t dx − 1 2 ∫ Ω ψ2dx = 1 2 ‖ut(x,τ)‖2L2(Ω) − 1 2 ‖ψ‖2L2(Ω) , (2.3) 4 Int. J. Anal. Appl. (2022), 20:62 besides we have −a (∆u,u)L2(Qτ ) = −a ∫ Qτ ∆u ·utdxdt = a ∫ Ω ∇u2dx − 1 2 ∫ Ω ϕ2xdx = a‖∇u‖2L2(Ω) − 1 2 ‖ϕx‖2L2(Ω) , (2.4) and (bu,u)L2(Qτ ) = ∫ Qτ b(x,t)u2t dxdt. (2.5) In this regard, we have (cuq,ut)L2(Qτ ) = 1 q + 1 ∫ Ω uq+1dx − 1 q + 1 ∫ Ω ϕq+1dx = 1 q + 1 ‖ut(x,τ)‖ q+1 Lq+1(Ω) − 1 q + 1 ‖ϕ‖q+1 Lq+1(Ω) . (2.6) By substituting (2.3)-(2.6) into (2.2), we obtain 1 2 ‖ut(x,τ)‖2L2(Ω) − 1 2 ‖ψ‖2L2(Ω) + a‖∇u‖ 2 L2(Ω) − 1 2 ‖ϕx‖2L2(Ω) + ∫ Qτ b(x,t)u2t dxdt + 1 q + 1 ‖u(x,τ)‖q+1 Lq+1(Ω) − 1 q + 1 ‖ϕ‖q+1 Lq+1(Ω) = (f ,ut) . (2.7) By applying Cauchy inequality with ε, ( i.e., |ab| ≤ a2 2ε + εb2 2 ) , we can estimate the last term on the right-hand side of (2.7) and get 1 2 ‖ut(x,τ)‖2L2(Ω) + a‖∇u‖ 2 L2(Ω) + ∫ Qτ b(x,t)u2t dxdt + 1 q + 1 ‖u(x,τ)‖q+1 Lq+1(Ω) ≤ 1 2ε ‖f‖2L2(Qτ ) + ε 2 ‖u‖2L2(Qτ ) + 1 2 ‖ψ‖2L2(Ω) + 1 2 ‖ϕx‖2L2(Ω) + 1 q + 1 ‖ϕ‖q+1 Lq+1(Ω) . By using assumptions A1 and using the Gronwall’s Lemma, the estimate (2.8) becomes ‖ut(x,τ)‖2L2(Ω) + ‖∇u‖ 2 L2(Ω) + ∫ Qτ u2t dxdt + ‖u(x,τ)‖ q+1 Lq+1(Ω) ≤ max { 1 2 , 1 2ε ,b0, 1 q + 1 } min { 1 2 ,a,b1, 1 q + 1 } exp (ε 2 T ) × [ ‖f‖2L2(Q) + ‖ψ‖ 2 L2(Ω) + ‖ϕx‖ 2 L2(Ω) + ‖ϕ‖ q+1 Lq+1(Ω) ] . Then, by passing to the maximum, we get max 0≤τ≤T ‖uτ (x,τ)‖2L2(Ω) + max 0≤τ≤T ‖∇u‖2L2(Ω) + ‖ut‖ 2 L2(Q) + max 0≤τ≤T ‖u(x,τ)‖q+1 Lq+1(Ω) ≤ c [ ‖f‖2L2(Q) + ‖ψ‖ 2 L2(Ω) + ‖ϕx‖ 2 L2(Ω) + ‖ϕ‖ q+1 Lq+1(Ω) ] , Int. J. Anal. Appl. (2022), 20:62 5 where c = max { 1 2 , 1 2ε ,b0, 1 q + 1 } min { 1 2 ,a,b1, 1 q + 1 } exp (ε 2 T ) . So, we have ‖u‖E ≤ √ c ‖Lu‖F . (2.8) � 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. Proposition 2.1. The operator L : E −→ F has a closure. Proof. Let (un)n∈N ⊂ D (L) be a sequence where un −→ 0 in E, and Lun −→ (f ; ϕx,ψ) in F. (2.9) Now, we must prove that f ≡ 0 and (ϕ,ψ) ≡ (0, 0) . The convergence of un to 0 in E drives: un −→ 0 in D′ (Q) . (2.10) According to the continuity of the derivation of D′ (Q) in D′ (Q) and the continuity the distribution of the function uq, the relation (2.10) involve Lun −→ 0 in D′ (Q) . (2.11) Moreover, the convergence of Lun to f in L2 (Q) gives: Lun −→ f in D′ (Q) . (2.12) 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 l1un −→ ϕx and l2un −→ ψ in L2 (Ω) . 6 Int. J. Anal. Appl. (2022), 20:62 On the other hand, we have ‖u‖2E = max 0≤τ≤T ‖uτ (x,τ)‖2L2(Ω) + max 0≤τ≤T ‖∇u‖2L2(Ω) + ‖ut‖ 2 L2(Q) + max 0≤τ≤T ‖u(x,τ)‖q+1 Lq+1(Ω) ≥‖ux (x, 0)‖2L2(Ω) + ‖ut(x, 0)‖ 2 L2(Ω) ≥‖ϕx‖2L2(Ω) + ‖ψ‖ 2 L2(Ω) . Now, due to un −→ 0 in E, then ‖u‖2E −→ 0 in R. Consequently, we get 0 ≥‖ϕx‖2L2(Ω) + ‖ψ‖ 2 L2(Ω) . Then, we obtain ϕx = 0 and ψ = 0. Let L be the closure of this operator with the domain of definition D(L), and hence the result holds. � Definition 2.1. A solution of the operator equation L̄u = F is called a strong solution to problem (1.1)-(1.3). The priori estimate (2.1) can be then extended to strong solution, i.e., we have the estimate max 0≤τ≤T ‖uτ (x,τ)‖2L2(Ω) + max 0≤τ≤T ‖∇u‖2L2(Ω) + ‖ut‖ 2 L2(Q) + max 0≤τ≤T ‖u(x,τ)‖q+1 Lq+1(Ω) ≤ c ( ‖f‖2L2(Q) + ‖ψ‖ 2 L2(Ω) + ‖ϕx‖ 2 L2(Ω) + ‖ϕ‖ q+1 Lq+1(Ω) ) , ∀u ∈ D(L̄). (2.13) In light of the estimate given in (2.13), we can infer the next theoretical results. Corollary 2.1. The range R(L̄) of the operator L̄ is closed in F and is equal to the closure R(L) of R(L), i.e. R(L̄) = R(L). Proof. Let z ∈ R(L) such that there is a Cauchy sequence (zn)n∈N in F constituted of the elements of the set R(L) such as lim n−→+∞ zn = z. There is then a corresponding sequence un ∈ D(L) such as zn = Lun. Immediately, the estimate (2.8) becomes: ‖up −uq‖E ≤ C‖Lup −Luq‖F → 0, where p and q tend towards infinity. We can consequently deduce that (un)n∈N is a Cauchy sequence in E. So like E is a Banach space, it exists u ∈ E such as lim n−→+∞ un = u in E. By virtue of the definition of L̄ ( lim n−→+∞ un = u in E, if lim n−→+∞ Lun = lim n−→+∞ zn = z, and then lim n−→+∞ L̄un = z as L̄ is closed, and so L̄u = z), the function u satisfies: u ∈ D ( L̄ ) , L̄u = z. Int. J. Anal. Appl. (2022), 20:62 7 Then z ∈ R(L̄), and so R(L) ⊂ R(L̄). Also, we conclude here that R(L̄) 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(L̄), and then it exists a Cauchy sequence (zn)n∈N in F constituted of the elements of the set R(L̄) such that lim n−→+∞ zn = z, or z ∈ R(L̄) because R(L̄) is closed subset. So R(L̄) is complete. There is then a corresponding sequence un ∈ D(L̄) such that L̄un = zn. Consequently from (2.8), we get ‖up −uq‖E ≤ C ∥∥L̄up − L̄uq∥∥F → 0, where p and q tend towards infinity. We can immediately deduce that (un)n∈N is a Cauchy sequence in E, and so like E is a Banach space, it exists u ∈ E such as lim n−→+∞ un = u in E. Once again, there is a corresponding sequel (Lun)n∈N ⊂ R(L) such as L̄un = Lun on R (L) ,∀n ∈N. So we have lim n−→+∞ Lun = z and consequently z ∈ R (L), which implies R ( L̄ ) ⊂ R (L). � 3. 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). Theorem 3.1. Let assumption A1 be satisfied. Then for all F = (f ,ϕ) ∈ F, there exists a unique strong solution u = L̄−1F = L−1F of problem (1.1)-(1.3). Proof. To prove this result, we should note that we first have (Lu,W )F = ∫ Q Lu.wdxdt + ∫ Ω l1u.w0dx + ∫ Ω l2u.w1dx, (3.1) where W = (w,w0,w1). So for w ∈ L2 (Q) and for all u ∈ D0(L) = {u, u ∈ D (L) : l1u = 0, l2u = 0} , we have ∫ Q Lu.wdxdt = 0. By putting w = ut, we obtain∫ Qτ uttut + ∫ Qτ b(x,t)u2t dxdt + ∫ Qτ uq+1dxdt = a ∫ Qτ ∆u.ut 1 2 ‖ut(x,t)‖2L2(Ω) + ∫ Qτ b(x,t)u2t dxdt + 1 q + 1 ‖u(x,τ)‖q+1 Lq+1(Ω) = −a‖∇u‖2L2(Ω) . 8 Int. J. Anal. Appl. (2022), 20:62 This gives 1 2 ‖ut(x,t)‖2L2(Ω) + ∫ Qτ b(x,t)u2t dxdt + 1 q + 1 ‖u(x,τ)‖q+1 Lq+1(Ω) ≤ 0, max 0≤τ≤T ‖uτ (x,τ)‖2L2(Ω) + b1 ∫ Qτ u2t dxdt + 1 q + 1 ‖u(x,τ)‖q+1 Lq+1(Ω) ≤ 0. Therefore, we have ut = w = 0. Since the range of the trace operators is everywhere dense in the Hilbert space F with the associate norms ‖ϕx‖L2(Ω) and ‖ψ‖L2(Ω) , then the equality (3.1) implies that ω0 = 0 and ω1 = 0. Hence W = 0 implies R(L) = F. � Corollary 3.1. If for any function u ∈ D(L), we have the following estimate: ‖u‖E ≤ √ c ‖F‖F , Then the solution of the problem (P1), if it exists, is unique. Proof. Let u1 and u2 be two solutions of problem (P1), i.e.,{ Lu1 = F Lu2 = F =⇒ Lu1 −Lu2 = 0. As L is linear, we then obtain L (u1 −u2) = 0. According to (2.8), we obtain ‖u1 −u2‖2E ≤ c ‖0‖ 2 F = 0, which gives u1 = u2. � 4. 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 publi- cation of this paper. References [1] G. Bahia, A. Ouannas, I.M. Batiha, Z. Odibat, The Optimal Homotopy Analysis Method Applied on Nonlinear Time- fractional Hyperbolic Partial Differential Equations, Numer. Methods Partial Differ. Equ. 37 (2020), 2008–2022. https://doi.org/10.1002/num.22639. [2] T.E. Oussaeif, B. Antara, A. Ouannas, et al. Existence and Uniqueness of the Solution for an Inverse Problem of a Fractional Diffusion Equation with Integral Condition, J. Funct. Spaces. 2022 (2022), 7667370. https: //doi.org/10.1155/2022/7667370. [3] Z. Chebana, T.E. Oussaeif, A. Ouannas, I. Batiha, Solvability of Dirichlet Problem for a Fractional Partial Differential Equation by Using Energy Inequality and Faedo-Galerkin Method, Innov. J. Math. 1 (2022), 34–44. https://doi. org/10.55059/ijm.2022.1.1/4. https://doi.org/10.1002/num.22639 https://doi.org/10.1155/2022/7667370 https://doi.org/10.1155/2022/7667370 https://doi.org/10.55059/ijm.2022.1.1/4 https://doi.org/10.55059/ijm.2022.1.1/4 Int. J. Anal. Appl. (2022), 20:62 9 [4] R.B. Albadarneh, A.K. Alomari, N. Tahat, I.M. Batiha, Analytic Solution of Nonlinear Singular Bvp With Multi- Order Fractional Derivatives in Electrohydrodynamic Flows, TWMS J. Appl. Eng. Math. 11 (2021), 1125-1137. https://hdl.handle.net/11729/3261. [5] I.M. Batiha, Z. Chebana, T.-E. Oussaeif, A. Ouannas, I.H. Jebril, On a Weak Solution of a Fractional-order Temporal Equation, Math. Stat. 10 (2022), 1116–1120. https://doi.org/10.13189/ms.2022.100522. [6] N. Anakira, Z. Chebana, T.E. Oussaeif, I.M. Batiha, A. Ouannas, A Study of a Weak Solution of a Diffusion Problem for a Temporal Fractional Differential Equation, Nonlinear Funct. Anal. Appl. 27 (2022), 679–689. https: //doi.org/10.22771/NFAA.2022.27.03.14. [7] T. Hamadneh, A. Zraiqat, H. Al-Zoubi, M. Elbes, Sufficient Conditions and Bounding Properties for Control Functions Using Bernstein Expansion, Appl. Math. Inf. Sci. 14 (2020), 1-9. [8] O. Taki-Eddine, B. Abdelfatah, A Priori Estimates for Weak Solution for a Time-Fractional Nonlinear Reaction- Diffusion Equations With an Integral Condition, Chaos Solitons Fractals. 103 (2017), 79–89. https://doi.org/ 10.1016/j.chaos.2017.05.035. [9] T.E. Oussaeif, A. Bouziani, Solvability of Nonlinear Viscosity Equation With a Boundary Integral Condition, J. Nonlinear Evol. Equ. Appl. 3 (2015), 31-45. [10] T.E. Oussaeif, A. Bouziani, Solvability of Nonlinear Goursat Type Problem for Hyperbolic Equation with Integral Condition, Khayyam J. Math. 4 (2018), 198–213. https://doi.org/10.22034/kjm.2018.65161. [11] S. Dhelis, A. Bouziani and T.-E. Oussaeif, Study of Solution for a Parabolic Integro-Differential Equation With the Second Kind Integral Condition, Int. J. Anal. Appl. 16 (2018), 569-593. https://doi.org/10.28924/ 2291-8639-16-2018-569. [12] T.E. Oussaeif, A. Bouziani, Existence and Uniqueness of Solutions to Parabolic Fractional Differential Equations With Integral Conditions, Electron. J. Differ. Equ. 2014 (2014), 179. https://hdl.handle.net/11729/3261 https://doi.org/10.13189/ms.2022.100522 https://doi.org/10.22771/NFAA.2022.27.03.14 https://doi.org/10.22771/NFAA.2022.27.03.14 https://doi.org/10.1016/j.chaos.2017.05.035 https://doi.org/10.1016/j.chaos.2017.05.035 https://doi.org/10.22034/kjm.2018.65161 https://doi.org/10.28924/2291-8639-16-2018-569 https://doi.org/10.28924/2291-8639-16-2018-569 1. Introduction and position of the problem 2. A priori bound 3. Existence and uniqueness of solution 4. Conclusion References