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Computational & Applied Mathematics

On-line version ISSN 1807-0302

Comput. Appl. Math. vol.28 no.2 São Carlos  2009

http://dx.doi.org/10.1590/S1807-03022009000200001 

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: silvano@ufpa.br

 

 


ABSTRACT

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 ƒ 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 [6]. 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 [4], 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 [10] 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 [10] was further developed by Lions [6] with theaid of elliptic regularization associated to the theory of monotonous operator, cf. Browder [3].

More precisely, Lions [6] investigate periodic solutions of the problem

with .

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 ƒ is assumed. Arguments within this paper are inspired by the work by Lions [6].

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 [6], we use an approach due to Prodi [10] 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 [6] investigate the problem (1.2) by elliptic regularization, reducing the problem to the theory of monotonous operators, cf. Lions [6].

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 [3]. 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 [6]. Subsequently we apply Theorem 2.1, p. 171 of Lions [6] 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 [6] in the context of thetelegraph equation.

Indeed, following the same type of reasoning cf. Lions [6], 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 [6], the existence and uniqueness of a function uµ W such that

follows immediately.

The Eq. (2.2) is called of elliptic regularization of problem (1.7).

Our main result is as follows

Theorem 2.1. Suppose ƒ 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 [6] or Medeiros [8].

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 [6], 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)

Setting

it implies

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

.

Thus

.

Since

,

as µ 0, we obtain

.

This implies that

(u(T), θ(T)v) - (u(0), θ(0)v) = 0,

that is,

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

.

Thus

.

Therefore,

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

.

Therefore

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 [2], p. 67, we to show that

In fact, we have

As

,

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

We define

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.

Thus,

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

Setting

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

or equivalently,

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

Therefore

It follows from (2.62) that

Hence, we deduce from (2.62) that,

with ψ given in (2.57).

Thus

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

Indeed,

Therefore,

because ξ' and ρµ are periodic.

Similarly

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

.

Therefore

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.                                                                                                 

 

REFERENCES

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[6] J.L. Lions, Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaires, Dunod, Paris (1969).         [ Links ]

[7] J. Mawhin, Periodic solution of nonlinear telegraph equations, in: A.R. Bedlarek, L. Cesari (Eds.), Dynamical Systems, Academic Press, New York (1977).         [ Links ]

[8] L.A. Medeiros, Applications of Monotony Method. Instituto de Matemática - UFRJ, Lectures Notes (2005).         [ Links ]

[9] R. Ortega and A.M. Robles-Pérez, A maximum principle for periodic solutions of the telegraph equations. J. Math. Anal. Appl., 221 (1998), 625-651.         [ Links ]

[10] G. Prodi, Soluzioni periodiche dell'equazione delle onde com termine dissipativo nonlineare. Rend. Sem. Mat. Pandova, 35 (1965).         [ Links ]

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Received: 27/XI/07.
Accepted: 16/IV/08.

 

 

#743/07.