GLOBAL SOLUTIONS TO A QUASILINEAR HYPERBOLIC EQUATION

. This article concerns the existence and decay of solutions of a mixed problem for a quasilinear hyperbolic equation which has its motivation in a mathematical model that describes the nonlinear vibrations of the cross-section of a bar.


Introduction
Milla Miranda et al [16] presented a mathematical model for the small longitudinal vibrations of the cross sections of a bar of length L which is clamped on one end and the other end is glued in a mass M .This model has the form u (x, t) − ∂ ∂x σ(u x (x, t)) = 0, 0 < x < L, t > 0; u(0, t) = 0, M u (L, t) + σ(u x (L, t)) = 0, t > 0; u(x, 0) = u 0 (x), u (x, 0) = u 1 (x), 0 < x < L, where u(x, t) denotes the displacement of the cross section x of the bar at time t, and u = ∂u ∂t .To obtain (1.1) we use Hooke's law τ (x, t) = σ(u x (x, t)) in which τ (x, t) and u x (x, t) are the tension and the deformation of the bar at (x, t), respectively, and σ(s) is a real function.The linear version of Problem (1.1) can be found in Timoshenko et al [18, p 387].
Then Dafermos [3] obtained the existence and decay of solutions.EJDE-2020/100 We focus our attention on Problem (1.1) with σ(s) = |s| p s, with p > 0, ( u(x, 0) = u 0 (x), ∂u ∂t (x, 0) = u 1 (x), 0 < x < L. (1.3) We observe that the function σ(s) given in (1.2) is different from the σ(s) considered in the above papers.Note also that the existence of global solutions of (1.3) with zero Dirichlet boundary conditions and without internal damping is an open problem (cf.J. L. Lions [9]).This justifies the introduction of the internal damping for obtain the existence of global solutions of (1.3).
Tsutsumi [19] and Giorgio and Matarazzo [4] considered Problem (1.3) with zero Dirichlet boundary conditions.They obtain global solutions for in an n-dimensional case.Later Maia and Milla Miranda [13] analyzed Problem (1.3) with zero Dirichlet boundary conditions in an abstract framework.The authors obtained global solutions and decay of solutions for this problem and generalized the papers [4,19].
Maia and Milla Miranda [13] found an estimate for (u m ), where u m is an approximate solution of (1.3), to apply the theory of monotone operators.For that, the eigenvectors of a positive self-adjoint operator of a Hilbert space and the projection method are used.This approach does not work in Problem (1.3) because of the boundary conditions (1.3) 2 To overcome the above difficulty, the authors in [16] introduced in equation (1.3) 1 the internal damping u xxxx to obtain the existence and decay of solutions of (1.3).
Our objective in this article is not introduce new internal damping in (1.3) 1 , but decrease the class of functions σ(s) given in (1.2) to obtain global solutions of (1.3).More precisely, considering the truncated of functions |s| p s (see Examples in Section 6), we succeed in to obtain the existence, uniqueness and exponential decay of solutions of Problem (1.3) in an n-dimensional case.
In our approach to prove the existence of solutions, we use the Faedo-Galerkin method with a special basis, the theory of monotone operators (cf.J. L. Lions [9] and Medeiros and Pereira [15]) and results on the trace of non-smooth functions.The estimate for (u m ) is obtained thanks to the truncation of the functions |s| p s and the special basis.In the decay of solutions is used a Liapunov functional (cf.Komornik and Zuazua [8] and Komornik [7]) We note that it is not usual for hyperbolic problems to have an equation at the boundary which contains a nonlinear term of the normal derivative and the second derivative with respect to t, respectively, of the solution.
As far as we know, the only results on the existence of global solutions of (1.3) are given in the present paper and in Milla Miranda et al [16].In this case the existence of solution for the linear case can also be obtained using semigroup theory as in Goldstein [6].

Notation and main results
Let Ω be open bounded set of R n whose boundary Γ is constituted of two parts Γ 0 and Γ 1 such that Γ = Γ 0 ∪ Γ 1 and Γ 0 ∩ Γ 1 = ∅.With ν(x) is denoted the unit exterior normal at x ∈ Γ 1 .
We consider the functions σ i : R → R (i = 1, 2, . . ., n) such that σ i is globally Lipschitz, σ i is increasing and σ i (0) = 0, i = 1, 2, . . ., n. (2.1) With the above notation, we introduce the quasilinear hyperbolic problem Here, u = ∂u ∂t .We obtain the following results.Theorem 2.1.Assume hypotheses (2.1) hold and Then, there exists an unique function u with and the initial conditions To state the estimates on the decay of E(t), we introduced some notation and consider one more hypothesis on σ i .We set the notation in which a 1 and a 2 are positive constants.We assume that there exist positive constants b i (i = 1, 2, . . ., n) such that (2.9) ) ) Theorem 2.2.Let u be the solution obtained in Theorem 2.1.Assume that (2.9) is satisfied.Then To prove Theorem 2.1, we need some previous results.

Results
We denote by k i the Lipschitz constants of σ i (i = 1, 2, . . ., n) and by k = max{k i ; i = 1, 2, . . ., n}.In rest of this article we use the notation.
This inequality proves (i) and (ii).Item (iii) follows from the fact that each σ i is an increasing function.Item (iv) is proved by using the continuity of each σ i and the Lebesgue Dominated Convergence Theorem.
The following result is concerned with the trace of non-smooth functions.Consider the Hilbert space ) is the unit outward normal at x ∈ Γ 1 .The above motivates the following result.
Proof.Consider f ∈ (D(Ω)) n and z ∈ H 1/2 (Γ 1 ).By the trace Theorem there exists w ∈ H 1 Γ0 (Ω) such that γ 0 w = z and in which C is a positive constant independent of w and z.We have This inequality and (3.1) provide γ ν f ∈ H −1/2 (Γ 1 ) and where C 2 > 0 is a constant independent of f ∈ E(Ω).The proposition follows by the denseness of (D(Ω)) n in E(Ω).

Proof of Theorem 2.1
We used the Faedo-Galerkin method with a special basis of H 1 Γ0 (Ω).Consider a basis {w 1 , w 2 , . . ., } of H 1 Γ0 (Ω) such that u 0 , u 1 ∈ [w 1 , w 2 ] where [w 1 , w 2 ] is the subspace generated by w 1 and w 2 .Let u m be an approximate solution of Problem (2.2), that is, g jm (t)w j and u m be solution of the system (u m , w) First estimate.Setting w = u m in (4.1) 1 , we obtain Integrating on [0, t], 0 < t < t m , we obtain Remark 4.1.We have Therefore, Taking into account Remark 4.1 in (4.2), we obtain We denote by C > 0 the various constants independent of m and t ∈ [0, ∞).
Second estimate.Differentiate the approximate equation (4.1) 1 with respect to t then set w = u m .We obtain We have Thus Combining this inequality with (4.4), then integrating on [0, t] and using estimate (4.3), we obtain Next, we estimate the two last terms of the second member of (4.5).

This implies
In a similar way, we obtain Taking into account (4.7) and (4.8) in (4.6), we obtain This inequality and (4.5) provide By estimate (4.3) and the equality u m (t) = t 0 u m (τ )dτ + u 0 , we obtain that (u m ) is bounded in L ∞ loc (0, ∞; H 1 Γ0 (Ω)).This estimate, Proposition 3.1 and part (ii) imply Estimates (4.3), (4.9)-(4.10)provide a subsequence of (u m ), still denoted by (u m ), and a function u such that The above convergences allow us to pass the limit in the approximate equation (4.1) 1 and obtain for all z ∈ L 2 loc (0, ∞; H 1 Γ0 (Ω)), z with compact support.Convergence of (Au m ).In this part, we use the method of the monotone operator (cf.J.L. Lions [9] and Medeiros and Pereira [15]).Fix an arbitrary T > 0. As A is monotone, we have Then by convergence (4.11), we find that By the approximate equation (4.1) 1 , we obtain By convergences (4.11) 1 , (4.11) 4 and noting that the embedding of The las two convergences provide From (4.14), (4.15), (4.16) and (4.17) we obtain lim sup Make z = u1 (0,T ) in (4.12),where 1 (0,T ) is the characteristic function of the interval (0, T ).We obtain Comparing this equality with (4.18), we derive lim sup Taking into account the last inequality in (4.13), we find This inequality and the hemicontinuity of A provide . By diagonalization process and noting that T > 0 was arbitrary, this equality implies 19) and noting that u belongs to L 2 (0, ∞; H 1 Γ0 ), we obtain equation (2.5).
Convergence (4.11) say us that u belong to class (2.4).The verification of the initial conditions (2.7) follows by convergences (4.11).Thus the proof of the existence of solutions is concluded.

Proof of Theorem 2.2
Let u be the solution given by Theorem 2.1.Multiplying both sides of equation (2.5) by u , we obtain (5.1) Also multiply both sides of equation (2.5) by u.We find where Consider ε > 0. We introduce the perturbed energy Relation between E ε (t) and E(t).We have We obtain (5.9) Consider η > 0 defined in (2.12).Then by (5.9) and (5.5), we obtain and therefore This inequality and (5.5) provide inequality (2.13).

Examples
In what follows we will give examples of functions that satisfy the hypotheses considered in Section 1.Consider real numbers p and L i with p ≥ 1 and L i > 1.The function