On-line version ISSN 1807-0302
Comput. Appl. Math. vol.28 no.2 São Carlos 2009
Periodic solutions for nonlinear telegraph equationvia elliptic regularization
G.M. de AraújoI; R.B. GúzmanI; Silvano B. de MenezesII
IUniversidade Federal do Pará, FM, PA, Brasil
IIUniversidade Federal do Ceará, DM, CE, Brasil, E-mail: firstname.lastname@example.org
In this work we are concerned with the existence and uniqueness of T -periodic weak solutions for an initial-boundary value problem associated with nonlinear telegraph equations typein a domain . Our arguments rely on elliptic regularization technics, tools from classical functional analysis as well as basic results from theory of monotone operators.
Mathematical subject classification: 35Q60, 35L15.
Key words: Telegraph equation, periodic solutions, elliptic regularization, Faedo - Galerkin method.
1 Introduction and description of the elliptic regularization method
In this paper we deal with the existence of time-periodic solutions for the nonlinear telegraph equation
Ω being a bounded domain in with a sufficiently regular boundary Ω.
All derivatives are in the sense of distributions, and by ξ' it denotes . The function we will be assumed as regular as necessary.
We shall use, throughout this paper, the same terminology of the functional spaces used, for instance, in the books of Lions . In particular, we denote by and H = L2(Ω). The Hilbert space V has inner product ((. , .)) and norm given by . For the Hilbert space H we represent its inner product and norm, respectively, by (. , .) and |.|, defined by .
The telegraph equation appears when we look for a mathematical model for the electrical flow in a metallic cable. From the laws of electricity we deduce a system of partial differential equations where the unknown are the intensity of current i and the voltage u, cf. Courant-Hilbert , p. 192-193, among others.
By algebraic calculations we eliminate i and we get the partial differential equation:
called Telegraph Equation. In this case the coefficients C, α, β are constants.
Motivated by this model, Prodi  investigated the existence of periodic solution in t for the equations
in a bounded open set Ω of RN with Dirichlet zero conditions on the boundary.
The problem posed by Prodi  was further developed by Lions  with theaid of elliptic regularization associated to the theory of monotonous operator, cf. Browder .
More precisely, Lions  investigate periodic solutions of the problem
Because of this important physical background, the existence of time-periodic solutions of the telegraph equations with boundary condition for space variable x has been studied by many authors, see [7, 8, 9, 11] and the references therein.
We consider the existence of the solutions w(x, t) of Eq. (1.1), which satisfy the time-periodic (or T -periodic) condition
subject to the Dirichlet condition
Based on physical considerations, we restrict our analysis to the two dimensional space and standard hypothesis on is assumed. Arguments within this paper are inspired by the work by Lions .
However, the classical energy method approach cannot be employed straightly, giving raise to a new mathematical difficulty. In fact, multiplying both sides of the equation (1.1) by w' and integrating on Q, we have, using the periodicity condition, that
In this way we obtain only estimates for (1.1).
which is not sufficient to obtain solution for (1.1).
In view of this, as in Lions , we use an approach due to Prodi  which relies heavily on the following set of ideas: we investigate solutions for (1.1) of the type
Substituting w given by (1.5) in (1.1), we obtain
which contains a new unknown u0, independent of t by definition.
To eliminate u0 in (1.6) we consider the derivative of (1.6) with respect to t obtaining
Suppose that we have found u by (1.7). Observe that by (1.7)1,
Thus u is solution of
g0 independent of t, in which g0 is a known function.
Then u0 is obtained as the solution of the Dirichlet problem:
Therefore, w = u + u0 is the T - periodic solution of (1.1). We are going to resolve problem (1.7) by using elliptic regularization.
Observe that Lions  investigate the problem (1.2) by elliptic regularization, reducing the problem to the theory of monotonous operators, cf. Lions .
In this work we consider the time - periodic problem (1.1), (1.3) and (1.4) and solve it by elliptic regularization as an application of the monotony type results, cf. Browder . Thus our proof is a simpler alternative to the earlier approaches existing in the current literature.
In fact, we consider the periodic problem
Thus for w = u + u0, the function u is determined by (1.7).
We begin the functional space
The Banach structure of W is defined by
In the sequel by (.,.) we will represent the duality pairing between X and X', X' being the topological dual of the space X, and by c (sometimes c1, c2, ...) we denote various positive constants.
Motivated by (1.7) we define the bilinear form b(u, v) for u, v W by
where A = -Δ and γ (u') =|u'|p-2u'.
Then the weak formulation of (1.7) is to find u W such that
for all v W.
Let us point out that the main difficulty in applying standard techniques from classical functional is due to the fact that the bilinear form b(u, v) is not coercive. To resolve this issue, we perform an elliptic regularization on b(u, v), following the ideas of Lions . Subsequently we apply Theorem 2.1, p. 171 of Lions  to finally establish existence and uniqueness of solution to elliptic problem (1.12).
2 Main result
As we said in the Section 1, the method developed in this article is a variant ofthe elliptic regularization method introduced in Lions  in the context of thetelegraph equation.
Indeed, following the same type of reasoning cf. Lions , to obtain the elliptic regularization, given µ > 0 and u, v W we define
where A = -Δ and γ (u') =|u'|p-2u'.
It is easy to see, cf. Lemma 2.2, that the application v πµ(u, v) is continuous on W. This allows to build a linear operator Bµ : W W', (Bµ(u), v)= πµ(u, v).
As we shall see, the linear operator Bµ satisfies the following properties:
(a) Bµ is a strictly monotonous operator; (Bµ(v) - Bµ(z), v - z) > 0 for all v, z W , v z;
(b) Bµ is a hemicontinuous operator; λ (Bµ(v + λz), w) is continuous in ;
(c) Bµ(S) is bounded in W' for all bounded set S in W;
(d) Bµ is coercive;
In view of these properties and as consequence of Theorem 2.1, p. 171 of Lions , the existence and uniqueness of a function uµ W such that
The Eq. (2.2) is called of elliptic regularization of problem (1.7).
Our main result is as follows
Theorem 2.1. Suppose Lp'(0, T ; Lp' (Ω)), with and p > 2. Then there exists only one real function w = w(x, t), (x, t) Q, w W, such that
and wsatisfying (1.1) in the sense of L2(0, T ; V') + Lp' (0, T ; Lp' (Ω)).
Now, we begin by stating some lemmas that will be used in the proof of the Theorem 2.1.
Lemma 2.1. then
for u derivable with respect to t in [0, T ] and u L2(0, T ; V ),u' L2(0, T ; V ) Lp(0, T ; Lp(Q)).
Proof. The proof of Lemma 2.1 can be obtained with slight modifications from Lions  or Medeiros .
Lemma 2.2. The form v πµ(u, v) defined in (2.1) is continuous on W .
Proof. By Cauchy-Schwarz inequality and Young's inequality we have
where cµ is a constant positive that depend of µ. Then the result follows.
Lemma 2.3. The operator Bµ : W W', (Bµ(u), v)= πµ(u, v) is hemicontinuous, bounded, coercive and strictly monotonous from W W'.
Proof. It follows of (2.6) that Bµ(u) is bounded. From Lemma 2.1 and equality , we obtain
because . Thus Bµ is W -coercive. The hemicontinuity of the operator v |v|p-2 v allow us to conclude that the operator Bµ is hemicontinuous. Finally, the proof that the operator Bµ is strictly monotonous follows as in Lions , p. 494.
Proof of Theorem 2.1. The arguments above show that there exists a unique solution uµ W of the elliptic problem (2.2).
Explicitly the Eq. (2.2) has the form:
We need let µ goes to zero in order to obtain uµ u for the solution. Then we need estimates for uµ.
In fact, setting v = uµ in (2.7) and observing that uµ and u'µ are periodic since they belongs to W , we obtain
This implies that
(u'µ) is bounded in L2(0, T ; H) when µ 0 (2.9)
(u'µ) is bounded in Lp(0, T ; Lp(Ω)) when µ 0 (2.10)
Since , we have by Lemma 2.1 that
(uµ) is bounded in Lp(0, T ; Lp(Ω)) (2.12)
In fact, integrating both sides of the equation (2.14) on [0, T ], we obtain
On the other hand,
Therefore, we reach our aim (2.15).
Thus, taking into account (2.14) in (2.2) we get
By using periodicity of uµ, u'µ W , we obtain
On the other hand,
From (2.17), (2.18) and estimate (2.9), we have
Also, from (2.10) and (2.12) we obtain
Combining (2.17), (2.19) and (2.20) with (2.16) we deduce
It follows from (2.21) and (2.10) that there exists a subsequence from (uµ),still denoted by (uµ), such that
uµ u weak in L2(0, T ; V ) (2.22)
u'µ u'weak in Lp(0, T ; Lp(Ω)) (2.23)
γ (u'µ) χ weak in Lp' (0, T ; Lp' (Ω)). (2.24)
Our next goal is tho show that u verifies (1.7)2- (1.7)3.
Indeed, it follows from (2.22) and (2.23) that uµ Cº([0, T ]; H) and
Setting = θv into (2.25) with θ C1([0, T ]; ), θ(0) = θ(T) and v V , we have
Again, by using periodicity of uµ and u'µ we obtain
as µ 0, we obtain
This implies that
(u(T), θ(T)v) - (u(0), θ(0)v) = 0,
u(T) = u(0). (2.29)
The proof that u'(0) = u'(T) will be given later. Now, we go to prove that .
Taking the scalar product on H of with we find
as µ 0.
It follows from (2.30) that
From (2.9), (2.10), (2.11) and (2.13), we deduce
It follows from (2.34) that
Hence, taking = v", v W, in (2.37), we find
By analogy, we prove that
By using periodicity of uµ, v W , we obtain
This implies that
It follows of (2.9) that
Taking = v" L2(0, T ; H) in (2.41) we obtain
From (2.2), we can write
From (2.9), (2.10), (2.22), (2.38), (2.39), (2.40) and (2.42), we can pass to the limit in (2.43) when µ 0 and obtain
Let (ρν) be a regularizing sequence of even periodic functions in t, with period T .
Denote by = u * ρν * ρν , where * is the convolution operator. Integrating by parts, we find u'* ρν * ρν = u * ρ'ν * ρν .
Observe by (2.12) and (2.21) that , v and ' periodic in t.
As in Brézis , p. 67, we to show that
In fact, we have
due to periodicity of u'and ρν , it follows (2.45).
Similarly, we show that
From (2.44) to (2.48) we obtain
Now, let us prove that χ = γ(u').
In fact, from (2.2) and (2.1) we get
It follows from (2.50) and (2.51) that
From the convergences above, we get
Taking into account (2.53) into (2.49) yields
Combining (2.53) and (2.54), we obtain
Since Xµ > 0, for all , then X > 0.
Since γ : Lp(0, T ; Lp(Ω)) Lp' (0, T ; Lp' (Ω)), γ(u') =|u'|p-2u', is hemicontinuous operator, the inequality above implies χ = γ (u'). It is sufficient to set (t) = u'(t) - λw(t), λ > 0, w Lp(0, T ; Lp(Ω)) arbitrarily and let λ 0.
We consider ψ C([0, T ]; V Lp(Ω)) satisfying
in (2.44), yields
because v'(T) = ψ(T), v''(T) = ψ'(T).
In particular, choosing ψ = θ' v, with θ ] 0, T [ and v V Lp(Ω), in (2.59) we get
for all v V L4(Ω) and θ ]0, T [.Hence,
Consequently, there exists a function g0 independent of t such that
We verify that
for all ]0, T [, because u' Lp(0, T ; Lp(Ω)).
Thus, from (2.63) to (2.68) and (2.62), we can write
It follows from (2.62) that
Hence, we deduce from (2.62) that,
with ψ given in (2.57).
Substituting (2.72) into (2.57) we obtain
Note that u'(0) and u'(T) make sense because u' belongs to Lp(0, T ; Lp (Ω)) L2 (0, T; V') + Lp' (0, T; Lp'(Ω)), respectively.
Let u0 be defined by
We recall that because n < 2 and p > 2, we have
where each space is dense in the following one and the injections are continuous.
This and (2.69) implies that g0 H-1(Ω) = V '.
Finally, we apply the Lax-Milgram Theorem to find a unique solution u0 H01(Ω) of the Dirichlet problem (2.74).
Thus, w = u + u0 L2(0, T ; V ) with w' Lp(0, T ; Lp(Ω)) satisfies
that is, w is a T-periodic weak solutions of problem (1.1).
Uniqueness. Let us consider w1 and w2 be two functions satisfying Theorem 2.1 and let ξ = w1 - w2.
We subtract the equations (1.1)1 corresponding to w1 and w2 and we obtain
Denoting by (ρµ) the regularizing sequence defined above, by a similar argument used in the proof of existence of solutions for Theorem 2.1 we obtain
Hence, by using (2.3) and (2.4), we can write
Also, from (2.76) we get
Thus, we have by (2.5) that ψ' Lp(0, T ; Lp(Ω)). Therefore ξ ' * ρµ * ρµ is periodic and
Then by (2.70) we can write
This and (2.79) show that dt make sense and
because ξ' and ρµ are periodic.
Consequently, it follows from (2.75), (2.82), (2.83), (2.84) and (2.85) that
Hence using (2.86), letting µ tend to zero, we have
that is, w'1 = w'2.
This implies that
ξ = w1 - w2 = θ, θ independent of t.
Integrating the last equality on [0, T ] and observing that wi = ui + u0i yields
because . Thus θ V .
It follows from (2.83) that
This implies that, when µ 0
Employing Green's Theorem, we find
Taking into account (2.89) into (2.88) yields θ = 0, which proves the uniqueness of solutions of problem (1.2). Thus, the proof of Theorem 2.1 is complete.
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