Local solutions for a hyperbolic equation

Let Ω be an open bounded set of Rn with its boundary Γ constituted of two disjoint parts Γ0 and Γ1 with Γ0 ∩ Γ1 = ∅. This paper deals with the existence of local solutions to the nonlinear hyperbolic problem ∣∣∣∣∣∣∣ u′′ −4u + |u| = f in Ω× (0, T0), u = 0 on Γ0 × (0, T0), ∂u ∂ν + h(·, u′) = 0 on Γ1 × (0, T0), (∗) where ρ > 1 is a real number, ν(x) is the exterior unit normal at x ∈ Γ1 and h(x, s) (for x ∈ Γ1 and s ∈ R) is a continuous function and strongly monotone in s. We obtain existence results to problem (∗) by applying the Galerkin method with a special basis, Strauss’ approximations of continuous functions and trace theorems for non-smooth functions. As usual, restrictions on ρ are considered in order to have the continuous embedding of Sobolev spaces.


Introduction
Motivated by a nonlinear theory of mesons field introduced by L. I. Schiff [27], K. Jörgens in [5,6] began a rigorous mathematical investigation, from a mathematical point of view, of equations of the type ∂ 2 u ∂t 2 − u + F (|u| 2 )u = 0.
(1.1) Specifically, K. Jörgens [6] proved the existence and uniqueness of solutions for the equation where Ω is a bounded open set of R n with boundary Γ.This equation is of the type (1.1) when Motivated by the works of K. Jörgens [5,6], the authors J.-L.Lions and W. A. Strauss [28] initiated and developed a large field of research on nonlinear evolution equations that includes K. Jörgens' model.See also, F. E. Browder [1], J. A. Goldstein [3,4], L. A. Medeiros [14], I. E. Segal [26], W. A. Strauss [28] and von Wahl [30].
Medeiros et al. [15] proved the existence and uniqueness of global solutions of the nonlinear hyperbolic problem where ρ > 1 is a real number with restrictions given by the continuous embedding of Sobolev spaces and the initial data u 0 and u 1 do not have restrictions on their norms.
Considering the boundary Γ of Ω constituted of two disjoint parts Γ 0 and Γ 1 such that Γ 0 ∩ Γ 1 = ∅ and denoting by ν(x) the unit exterior normal vector at x ∈ Γ 1 , Milla Miranda and Medeiros [20] studied the existence and uniqueness of solutions of the problem u(x, 0) = u 0 (x), u (x, 0) = u 1 (x) in Ω. ( When µ > 0 is constant, existence and uniqueness of global strong solutions for (1.3) has been proved by Komornik and Zuazua [7], Quinn and Russell [25] applying semigroup theory.This method does not work for (1.3) because the boundary condition (1.3) 3 depends on µ(t).For this reason Milla Miranda and Medeiros [20] constructed a special basis where lie approximations of the initial data, so the Galerkin method works well with this basis.Using this approach they proved the well-posedness for (1.3).The existence of solutions of problem (1.3) with nonlinear boundary conditions has been obtained, by using the theory of monotone operators by Zuazua [7], Lasiecka and Tataru [18], and applying the Galerkin method by Lourêdo and Milla Miranda [12].
Motivated by (1.2) and (1.3) we consider in this paper the following problem: (1.4) With restrictions on the real number ρ > 1 due to the continuous embedding of Sobolev spaces, we obtain the existence of local solutions to problem (1.4) in two cases: first, h(x, s) = δ(x)p(s) with p Lipschitzian and strongly monotone.In the second case h(x, s) is only continuous in s and strongly monotone in s, and the initial data belong to a class more regular Local solutions for a hyperbolic equation 3 than in the first case.In our approach, we apply the Galerkin method with a special basis, the Strauss' approximations of continuous functions and trace theorems for non-smooth functions.
It is worth noting that the term Ω |u| ρ u dx does not have a definite sign.This fact brings serious difficulties to obtain global solutions to problem (1.4) without considering restrictions on the norms of the initial data.
Hereafter, this paper is organized in three sections, namely, Section 2 is devoted to the notations and statements of the two main results.In Section 3 we present the proof of Theorem 2.1 in which the case h = δp is considered, where p is a Lipschitz continuous function.In Section 4, Theorem 2.2 is proved which contains the case where s → h(•, s) is only a continuous function in s.
The scalar product and norm of the space L 2 (Ω) will be denoted by (•, •) and | • |, respectively.We represent by V the Hilbert space V = {v ∈ H 1 (Ω); v = 0 on Γ 0 }, equipped with the scalar product and norm respectively.All scalar functions considered in this paper are real-valued.
In what follows, we introduce necessary hypotheses on some objects of problem (1.4) in order to state our first result.
Let p : R → R be a function satisfying: p is Lipschitz-continuous and strongly monotone in the second variable, i.e. (p(s) The function δ : The real number ρ is chosen according to the spatial dimension n.
satisfies the compatibility condition Then there exist a real number T 0 with 0 < T 0 ≤ T and a unique function u in the class A. T. ) and the initial conditions u(0) = u 0 , u (0) = u 1 . (2.8) Moreover, T 0 is explicitly given by where (2.10) and k 1 > 0 is the constant of the continuous embedding of V in L 2 (Ω), defined in inequality (3.2).
To state our second result we make the following considerations: let A = − be the selfadjoint operator of L 2 (Ω) defined by the triplet {V, L 2 (Ω); ((•, •))}.Then the domain of − is given by and it is known that D(− ) is dense in V, see this statement for instance, in Lions [10].
We suppose the function h : h is strongly monotone in the second variable, i.e.
[h(x, r) − h(x, s)] (r − s) ≥ d 0 (r − s) 2 , ∀r, s ∈ R, for almost all x ∈ Γ 1 , (d 0 is a positive constant). (2.12) Then there exist a real number T 0 > 0 (the same T 0 given in Theorem 2.1) and at least one function u in the class ) )

Case p Lipschitz
We begin by making some considerations.Since p : R → R is a Lipschitz continuous function then p(v) ∈ H 1/2 (Γ 1 ) for v ∈ H 1/2 (Γ 1 ) and the mapping p : is continuous, for this result we refer to Marcus and Mizel [13].
Remark 3.1.The regularity of the trace mapping of order zero, γ 0 : V → H 1/2 (Γ 1 ) ensures that the mapping p = p • γ 0 with p : V → H 1/2 (Γ 1 ) is continuous.Remark 3.2.Throughout this section, in order to facilitate the notation, the mapping p(v) for v ∈ V will be denoted just by p(v).
, and the linear operator δ : In fact, using the theory of interpolation for Hilbert spaces (see for instance the reference [11]) it can be shown that the linear operators 1/2 are equivalent.
Under the restrictions (2.3) on ρ, we have (ρ − 1)n ≤ 2ρ ≤ 2n n−2 = q for n ≥ 3, and this implies In (3.1) we mean by X → Y that the spaces X, Y satisfy X ⊂ Y and the injection of X in Y is continuous.We denote by k 0 , k 1 and k 2 the constants immersion that satisfying We now can proceed to the proof of our first result.
Proof of Theorem 2.1.Proposition 3.6 provides us sequences (u 0 l ) and (u We now construct a special basis of V ∩ H 2 (Ω) in the following way: for l ∈ N we consider the basis {w l 1 , w l 2 , . . ., w l j , . . .
, where [w l 1 , w l 2 ] denotes the subspace generated by w l 1 , w l 2 .According to this basis we determine approximate solutions u lm (t) of problem (3.4) with h = δp, that is, u lm (t) = ∑ m j=1 g jlm (t)w l j , where g jlm (t) is defined as the solutions of the approximate problem where V l m is the subspace generated by w l 1 , w l 2 , . . ., w l m .The solution u lm of (3.4) is defined on [0, t lm ) with 0 < t lm ≤ T 0 .The next estimate enables us to extend u lm to the whole interval [0, T 0 ].

First estimate: Setting
By usual inequalities and (3.2) we get Taking and combining (3.6), (3.7), (3.5), and using hypotheses (2.1) and (2.2) on p and δ, we get Observing that lm (t), and together with ϕ lm (t) ≥ 1, we find Combining this inequality with (3.8), we derive Local solutions for a hyperbolic equation and recalling the identity We obtain by (3.9) that d dt ϕ which implies In the above expression L was defined in (2.10).By the convergences in (3.3) we find , ∀l ≥ l 0 , ∀m.
Thus, from (3.10) it follows that By hypothesis (2.9), we obtain Then (3.11) provides With this limitation and taking into account (3.7), we find where these limitations are independent of l ≥ l 0 and m.
ρ−1 L (ρ−1)n (Ω) u lm (t) L q (Ω) |u lm (t)| From this, notations (3.2) and using the constant N introduced in (3.12) that bounds ϕ lm (t), we get Combining the last inequality with (3.14) and considering hypothesis (2.1) 3 , we obtain Remark 3.7.To apply Gronwall's lemma in inequality (3.15) we need to derive an upper bound for (u lm (0)).This is the key point of the proof of Theorem 2.1.We get this limitation thanks to the choice of the special basis of V ∩ H 2 (Ω), previously built in this section.We make t = 0 in the approximate system (3.4) 1 , take v = u lm (0), and after that apply Gauss' theorem, to obtain Using (3.3) 3 , we find  where the constant P is independent of l ≥ l 0 , m and t ∈ [0, T 0 ].
With the above estimate, we obtain ). (3.18) Passage to the limit in m.The constants N and P in (3.12) and (3.17) are independent of l ≥ l 0 , m and t ∈ [0, T 0 ].Thus, the estimates (3.13) and (3.18), allow us to find a subsequence of (u lm ), which still will be denoted by (u lm ), and a function u l such that The convergence (3.19) 1 , (3.19) 2 and the Aubin-Lions theorem provide the convergence u lm → u l in L 2 (0, T 0 ; L 2 (Ω)).Therefore, We also have where k 1 and N are introduced in (3.2) and (3.12), respectively.From this ∀l ≥ l 0 and ∀m.

Case h(x, s) continuous in s
Initially note that, since h is a continuous function, the following Strauss' approximations were shown by Louredo and Milla Miranda [12] .
Proposition 4.1.Assume that h satisfies hypotheses (2.11).Then there exists a sequence (h l ) of functions of C 0 (R; L ∞ (Γ 1 )) satisfying the following conditions: (i) h l (x, 0) = 0 for almost all x in Γ 1 ; (ii) [h l (x, s) − h l (x, r)] (s − r) ≥ d 0 (s − r) 2 , ∀s, r ∈ R and for almost all x in Γ 1 ; (iii) there exists a function c l ∈ L ∞ (Γ 1 ) such that (iv) (h l ) converges to h uniformly on bounded sets of R for almost all x in Γ 1 .
Proof of Theorem 2.1.We proceed as in Theorem 2.1, changing the function δ(x)p(s) into h l (x, s).Let (u 1 l ) be a sequence of functions of D(Ω) such that In a similar way as we made to obtain the estimates (3.12), (3.16) and (3.17) of Section 3, we find where the constants N, D and P are independent of l ≥ l 0 , m and t ∈ [0, T 0 ].The estimates (4.4) provide a subsequence of (u lm ), which still will be denoted by (u lm ), such that In a similar way as in the convergence (3.22) and (3.23), we get Convergence (4.5) and (4.6) allow us to pass to limit in m in the approximate equations of (4.3).Therefore, for v ∈ V and θ ∈ L 2 (0, T 0 ).By analogous arguments used to obtain (3.25) and (3.26), we find ) Estimates (4.4) imply in the existence of a subsequence of (u l ), which still will be denoted by (u l ), and a function u such that Local solutions for a hyperbolic equation 13As in (4.6) 1 , we derive |u l | ρ → |u| ρ weak star in L ∞ (0, T 0 ; L 2 (Ω)).