On Wave Equations Without Global a Priori Estimates

Ω |u| u dx. To solve this problem we employ techniques of L. Tartar [16], see also D.H. Sattinger [12] and we succeed to prove the existence and uniqueness of global weak solution for an initial boundary value problem for the operator L(u), with restriction on the initial data u0, u1 and on the function f . With this restriction we are able to apply the compactness method and obtain the unique weak solution.


Introduction
Motivated by a nonlinear theory of measons field, cf.L.I. Schiff [14], K. Jörgens [6], [7], initiated the investigation, from mathematical point of view, of a nonlinear model for partial differential equations of the type: for a real function u = u(x, t), x ∈ R n and t ≥ 0.
Under strong hypothesis on f , u 0 and u 1 , it was proved existence of weak solutions for (1.3), cf.J.L. Lions [10] or J.L. Lions and W.A. Strauss [8].The uniqueness of weak solutions was proved when 3), is fundamental when we apply the energy method, and the case u 2 is not included.Thus, J.L. Lions [10] investigated the case u 2 , in (1.3), by an argument idealized by D.H. Satinger [12].L. Tartar analysed the case u 2 by another method, look Section 2 of the present paper.
In this paper we investigate the problem (1.3) but with nonlinearity of the type |u| ρ , ρ > 1, after a remark of N.A. Larkin, UEM, Paraná-Brasil, personal communication.
Therefore, we plan, in this paper, to investigate the initial boundary value problem: This problem with acoustic boundary conditions on a part of Σ and the dissipative term βu ′ in the equation (1.4) 1 , was investigated by G. Antunes et all [1].
In Section 2 we prove the existence and uniqueness of local weak solution for (1.4), that is, the solution is defined only for 0 ≤ t ≤ T 0 , T 0 fixed.In Section 3 with a restriction on the size of u 0 , u 1 and f , we prove the existence of global weak solution for (1.4), that is, the solution is defined for all 0 ≤ t < ∞.In both cases we must have We also can prove uniqueness of weak solutions as in J.L. Lions [10].

Local solutions
We observe that all derivatives we consider are in the sense of the theory of distributions.We employ the notation L p (Ω), 1 ≤ p ≤ ∞, H m (Ω), m ∈ N, for Lebesgue and Sobolev spaces, respectively.We also employ the notation L p (0, T ; X), 1 ≤ p ≤ ∞ where X is a Banach space.
(Ω) and f ∈ L 1 (0, T ; L 2 (Ω)) be given.Then, there exist T 0 , with 0 < T 0 < T , and a unique function , which is a weak solution of the initial boundary value problem (1.4).
Remark 2.2 By Sobolev's embeding theorem, we have: If n = 1 we have continuous functions and when n = 2 the embeding of H 1 (Ω) in L q (Ω) holds for any real number q ≥ 1.
We need in the proof of Theorem 2.1 the embeding of H 1 (Ω) in the spaces L 2ρ (Ω) and L ρ+1 (Ω).Thus, we fixe and we have the case treated by L. Tartar [16].Since 2ρ < 2n n − 2 = q, we have Proof of Theorem 2.1.Since H 1 0 (Ω) is separable it has a "Hilbertien" basis, represented by w 1 , w 2 , . . ., w n , . . .(cf. H. Brezis [2]).Denote by The approximate system for the Galerkin method consists in the following scheme: Here, we employ the notation ( , ) for the inner product in L 2 (Ω) and a(u, v) for the Dirichlet form: Observe that (2.1), for each fixed m ∈ N, is a system of nonlinear ordinary differential equations.It has a local solution u m = u m (t) for 0 < t < t m < T .We will prove the existence of 0 < T 0 < T such that (2.1) has a solution u m (t) for 0 ≤ t ≤ T 0 .Moreover, we obtain uniform estimates for u m (t) in [0, T 0 ], which permits to pass to the limits when m → ∞, obtaining a function u( • , t) which is weak solution of (1.4).We need estimates for u m (t).
Thus, we have: Analysis of the right hand side of (2.2) By Cauchy-Schwarz inequality: Observe that: From Remark 2.2, we obtain: Thus, Returning to (2.2), observing the above inequality, we have: 2 , otherwise we consider ϕ m (t)+1 instead of ϕ m (t).Therefore, from (2.6) we have: and 1 − ρ < 0, integrating (2.7) we get: where > 0, we have: Therefore, we get: Choosing A > 0, sufficiently large such that and defining we can see that for all 0 ≤ t < T * .Let T 0 be fixed such that 0 < T 0 < T * .Hence, from (2.8) we obtain This inequality implies that: The above estimates are sufficient to pass the limit in the approximate system (2.1) as m → ∞.Note also that, as in J. Lions [10], we apply a compacity argument.To pass the limit in the nonlinear term |u m (t)| ρ we apply Lemma 3.2 in J. Leray and J.L. Lions [9].Remark 2.4 About uniqueness of local weak solution, given by Theorem 2.1, we can apply the same argument of J.L. Lions [10].In fact, we have the restriction = q and this is the condition we need to apply Hölder's inequality.

Global solutions
In this section we restrict the size of u 0 , u 1 and f in order to obtain global estimates for approximate solutions u m given in Section 2, system (2.1).These estimates permit us to obtain weak solution for (1.4), defined for all 0 ≤ t < ∞ and x ∈ Ω. Theorem 3.1 (Global Solutions).Let ρ and n be as in Theorem 2.1.For each where with C 0 the constant of embeding of H 1 0 (Ω) in L 2ρ (Ω), L ρ+1 (Ω), and then, there exists a unique function u : Ω × [0, ∞) → R, in the class: which is a weak solution of (1.4).
Proof: We will obtain global estimates for u m (t), solution of the approximate system (2.1), under the assumptions (3.2) and (3.3) for u 0 , u 1 and f . 2 Estimate 3.2 Set w = u ′ m (t) in (2.1).We obtain: Substituting it follows: Observe that for T > 0 arbitrarily fixed, there exist T 0 ∈ (0, T ) such that (3.4) holds for all 0 ≤ t ≤ T 0 by Theorem 2.1.
Integrating (3.4) on [0, t] for 0 ≤ t ≤ T 0 , we obtain: which can be written as: where J : H 1 0 (Ω) → R is defined by: The main question, in this point of the proof, is to show that under the assumptions (3.2) and (3.3), we can control the sign of J(u), for u = u m ( • , t) approximate solution of (2.1), 0 ≤ t ≤ T 0 and at u = u 0 , in the inequality (3.5).
We note that: where in the last inequality we employed We go back to J(u) and we get: which we employ for u = u m ( • , t) and u = u 0 .Whence, the sign of both sides of (3.5) depends of the sign of the function: Analysis of P (λ), λ ≥ 0 • P (λ) has zeros at λ 0 = 0 with order two and at • The derivative of P (λ) has zeros at λ 0 = 0 and , and its maximum value in the interval The approximate graphic of P (λ) is: Fig. 1 Now, let us go back to (3.5).By hypothesis (3.2) and inequality (3.7) we have: For the sign of J(u m (t)) in the left hand side of (3.5) we need the following result: Lemma 3.3 Suppose u 0 , u 1 and γ satisfying the conditions (3.2) and (3.3) of Theorem 3.1.Then the approximate solution (u m ) m∈N satisfies , for all t ∈ [0, T 0 ] and m ∈ N. (3.8) Proof: We argue by contradiction method.In fact, suppose there exists m 0 ∈ N and some t ∈ (0, T 0 ] such that Since ||u m0 (t)|| is continuous in [0, T 0 ], there exists t 0 ∈ (0, T 0 ) such that , for all 0 ≤ t < t 0 . (3.10) Now, we consider the subset τ of (0, T 0 ) defined by: • It is not empty, because of (3.9).
• It is a closet set because the function ||u m0 (t)|| is continuous on [0, T 0 ].
On Wave Equations Without Global a Priori Estimates 29 Thus, by the properties of τ , it has a minimum t * ∈ (0, T 0 ), which satisfies Now let us go back to (3.4) and integrate on (0, t * ).We obtain: , P (λ) has a maximum strictly positive.Then, it implies that J(u m0 (t * )) > 0, and the left hand side of (3.12) is strictly positive.

L
. A. Medeiros, J. Limaco and C. L. Frota then the right hand side of (3.5) is positive.