Remarks on a Nonlinear Wave Equation in a Noncylindrical Domain^†

ABSTRACT

In this paper we investigate the existence of solution for an initial boundary value problem of the following nonlinear wave equation:

u'' - Δu + |u|^ρ =f in .

where represents a non-cylindrical domain of^{_Rn+ 1}. The methodology, cf. Lions³, consists of transforming this problem, by means of a perturbation depending on a parameter ε > 0, into another one defined in a cylindrical domain Qcontaining . By solving the cylindrical problem, we obtain estimates that depend on ε. These ones will enable a passage to the limit, when ε goes to zero, that will guarantee, later, a solution for the non-cylindrical problem. The nonlinearity |uε|ρ introduces some obstacles in the process of obtaining a priori estimates and we overcome this difficulty by employing an argument due to Tartar(⁸) plus a contradiction process.

Keywords:
nonlinear problem; non-cylindrical domain; hyperbolic equation

RESUMO

Nesse artigo investigamos a existência de solução para um problema de valor inicial e de contorno associado à seguinte equação da onda não linear

u'' - Δu + |u|ρ = f em

onde representa um domínio não cilíndrico do^{_Rn+ 1}. A metodologia, conforme Lions⁽³⁾, consiste em transformar o problema original, por meio de uma perturbação dependendo de um parâmetro ε > 0, em um outro definido em um domínio cilíndrico Q que contêm . Resolvendo o problema no domínio cilíndrico, obtemos estimativas que dependem de ε. Tais estimativas nos permitirão tomar o limite, quando ε tende a zero, garantindo assim a existência de uma solução para o problema não cilíndrico. A não linearidade |uε|ρ introduz alguns obstáculos no processo de obtenção das estimativas, tais dificuldades são superadas por meio de um argumento devido a Tartar(8) combinado com um argumento de contradição.

Palavras-chave:
problema não linear; domínio não cilíndrico; equação hiperbólica

1 INTRODUCTION

Let us consider the general non-cylindrical initial-boundary value problem:

By we represent a bounded increasing domain of Rn × (0, ∞) and its lateral boundary.

This type of question was initially investigated by J.-L. Lions³ by applying a method, called by himself the penalty method, which will be described later. He obtained weak solutions for (1.1) for the special case β(s) = 0 and γ(s) = |s|^ρ, ρ > 0 a real number. Cooper and Bardos¹ extended this result to a larger class of regions by assuming only that there is a smooth mapping φ: Rⁿ × (0, T) → Rⁿ × (0, T) such that Q^* = φ() is monotone increasing and φ preserves the hyperbolic character of (1.1).

Medeiros⁵ generalized the results obtained by Lions³ in another direction considering the case of γ(s) as a real continuous function satisfying the condition γ(s) s ≥ 0, for all s ∈ R, and β(s) = 0.

In Nakao-Narazaki⁷ the authors worked with general real continuous functions, with restrictions, and obtained existence and decay of the solutions.

In the present work, we shall investigate the existence of weak solutions for (1.1) when β(s) = 0 and γ(s) = |s|^ρ.

Let T be a positive real number and let {Ω_t}_{t ∈ [0, T]} be a family of bounded open sets of Rⁿ, with regular boundary Γ_t. We denote by the non-cylindrical domain of R^{n+ 1} defined by

= Ωt × {t}.

with regular lateral boundary

= Γ_t × {t}.

Therefore, we consider the following problem:

where the derivatives are in the sense of the theory of distributions, Δ represents the usual spatial Laplace operator in Rⁿ and ρ is a positive real number satisfying certain conditions.

The methodology, cf. Lions², consists of transforming (1.2), by means of a perturbation depending on a parameter ε > 0, into a problem defined in a cylindrical domain Q. Then we have to solve the cylindrical problem and get estimates to pass to the limit when ε → 0.

Let us consider a bounded open set Ω ⊂ Rⁿ, with C²-boundary Γ and such that → Q (seeFig. 1).

Figure 1
Scheme representing the domains

For each ε > 0, we are looking for u_ε:Q → R solution of the problem:

where M is defined by

, and .

We call attention to the fact that the nonlinearity |uε|ρ in (1.3) generates some obstacles in the process of obtaining a priori estimates for the problem (1.3), by the energy method, because, at a certain point of our proof, we get a term of the type

∫Ω|∇uε(t)|2dx + ∫Ω|∇uε(t)|ρuε(t)dx

whose sign cannot be controlled. At this point of the proof we employ an argument due toTartar(⁸) plus contradiction process cf.(⁶).

2 NOTATIONS AND HYPOTHESES

As usual we represent by L2(Ω) the Lebesgue space of square integrable functions on Ω. The spaces L2(Ωt) are identified, for all t ∈ [0, T], with closed subspaces of L2(Ω). We denote by Lp(0, T; L2(Ωt)) and Lp(0, T; (Ωt)) the following spaces

Lp(0, T; L2(Ωt)) = {υ ∈ Lp(0, T; L2(Ω)); υ(t) ∈ L2(Ωt)}, 1 ≤ p ≤ ∞

and

Lp(0, T; (Ωt)) = {υ ∈ Lp(0, T; (Ω)); υ(t) ∈(Ωt)}, 1 ≤ p ≤ ∞

In the following we will denote by(·, ·), |·| and ((·, ·)), ||·, ·|| the inner products and norms of the Hilbert spaces L2(Ω) and (Ω) respectively.

We will develop our work under the following assumptions:

1. (H1) (Geometric condition) The family {Ωt}t ∈ [0, T] is increasing in the following sense: if t1 ≤ t2 then ⊆ .

2. (H2) (Regularity condition) If υ ∈ (Ω) and υ = 0 a.e. in Ω - Ωt, then υ ∈ (Ωt);

3. (H3)(Immersion condition) 1 < ρ ≤ , for n ≥ 3, and ρ > 1, for n = 1 or n = 2.

Remark 1 By Sobolev embedding theorem, we have H1(Ω) → Lq(Ω), with = - , that is, q = for n>2. In the case n = 1, H¹(Ω) →C(). In the casen = 2, H¹(Ω) →L^∞(Ω). In the proof of our result, we need the embedding of the space L^2ρ(Ω) intoL^{ρ + 1}(Ω). As Ω is bounded and ρ > 1, we have, from (H3), that H¹(Ω) →L^2ρ(Ω) → L^{ρ + 1}(Ω).

Remark 2 If the boundary Γ_t of Ω_t is for t ∈ [0,T] a manifold of class C² and υ = 0 on Ω - Ω_t it implies (H2) . In fact, Γ is of classC². Thus, Γ ∪ Γ_t is of class C². Therefore, by the trace theorem

γ₀:H¹(Ω - Ω_t) →H^1/2(Γ ∪ Γ_t),

since the boundary of Ω - Ω_t is Γ ∪ Γ_t, which is continuous. Thus, for each υ ∈H¹(Ω - Ω_t), we have:

≤C.

But υ = 0 on Ω - Ω_t. Thus,

= 0

Thus, H^1/2(Γ ∪ Γ_t) is a Hilbert space, what implies υ = 0 on Γ ∪ Γ_t.

3 MAIN RESULTS

The main result of this work is contained in the following Theorem:

Theorem 3.1 Given u₀ ∈ (Ω0), u1 ∈ L2(Ω0) and f ∈ L1(0, ∞;L2(Ωt). Set

γ(, ) = (||2 + ||||2 + ∫Ω||ρdx + ) e,

where ,andare extensions of u₀, u₁and f, respectively, and were defined in the previous section.Suppose, in addition to the hypotheses (H1)-(H3),that

and

where C₀is the constant of the embedding of(Ω) into Lρ + 1(Ω). Then, there exists a nonlocal solution for the problem (1.2), satisfying u ∈ L∞ (0, T; (Ωt)) and u' ∈ L∞ (0, T; L2(Ωt)).

Proof. First we will solve, for each ε > 0, the problem (1.3). Let {wv}v ∈ N be an orthonormal basis of (Ω). For ε > 0 fixed and each m ∈ N, we consider uεm(x, t) = gjm(t)wj(x), x ∈Ω and t ∈ [0, T), which is solution of the following approximate problem:

for all w ∈ [w1, w2, ..., wm] = span{w1, w2, ..., wm}.

Remark 3 From (3.1), (H4), (H5) and the Remark 1, there exists such that, for all m ≥ , we have

Replacing, if necessary, and for m <, we can consider, from now on, that = 1.

The local existence, for some T_m > 0, is a consequence of the results about systems of nonlinear ordinary differential equations.

We need estimates which permit to pass to the limit in the approximate solutionu_εm(t) when mgoes to infinity and show that (3.1) has a nonlocal solution.

Estimate 1 Taking w = in (3.1) we obtain

Integrating (3.3) from 0 tot < T_m we have

The main question at this point of the proof is that we don't know the sign of

J(u) = ||u||2 + ∫Ω|u|ρudx,

for u = uεm(t) and u = in the inequality (3.4).

To overcome this difficulty we will do some computation. First, we observe that

since the last inequality is a consequence of the immersion (Ω) → Lρ + 1(Ω), see (H3) and Remark 1.

From (3.5) we have

∫Ω|uεm(x, t)|ρuεm(x, t)dx ≥ ||uεm(t)||ρ + 1

and thus,

This functional will be employed for u = uεm(t) and u = later.

Therefore, the sign of both sides of (3.4) is related to the sign of the function

P(λ) = - λρ + 1

for λ ≥ 0 and ρ > 1.

From the definition of P(λ), we observe that it is increasing in the open interval

(0,)

and has a maximum value at . See an example for the graph of P(λ), at Figure 2 bellow, when ρ = 3 and C0 = 0.3.

Figure 2
Graph of P(λ) for ρ = 3 and C0 = 0.3.

As J(u) = ||u||2 +∫Ω|u|ρudx ≥ P(||u||), we can conclude that

Then, from the equation (3.2), we conclude that

Thus, the right hand side of (3.4) is non negative.

To analyze the left hand side of (3.4) we need the following Lemma:

Lemma 3.1 From the hypotheses (H4) and (H5), it follows that the approximate solution uεm satisfies

for all t ∈ [0, T), m ∈ N and ε > 0 fixed.

Proof. Let us apply a contradiction argument. In fact, suppose there exists m0 ∈ N such that

for some 0 < t < Tm0 and ε > 0 fixed. From (3.2) we have

From (3.11) and the continuity of ||uεm0(t)||, we conclude that there exists t0 > 0 such that

0 ≤ ||uεm0(t)|| < for all t ∈ (0, t₀).

From (3.10), the set

{t > 0; ||u_εm0(t)|| ≥ }

is non-empty, closed and bounded below. Thus, there exists a minimum for this set that we will call t*. By continuity of ||uεm0(t)|| we have

Hence, from (3.12) and (3.7), we observe that, for all t ∈ [0, t*)

So, integrating (3.3) from 0 to t ≤ t*, we obtain the inequality bellow

From (3.8) and (3.13), both sides of (3.14) are positive.

For the last term of the right side of (3.14), by Young's inequality, we observe that

Therefore, using (3.15) and (3.13) in (3.14) we get, for all t ∈ [0, t*),

where K1 - ||2 + ||||2 +∫Ω||ρdx +.

Thus, using Gronwall's inequality, we can conclude

Applying (3.17) in (3.16) for t = t*, we obtain

|(t*)|2 +|| uεm0(t*)||2 < K1

and it follows

||uεm0(t*)||2 ≤ (|||2 + ||||2 +∫Ω||ρdx + ||||)

The right hand side is equal to γ(,) and, combining it with (3.2), allows us to conclude that

|uεm0(t*)||2 < .

and this contradicts (3.12). So, the proof of the Lemma 3.1 is finished.

From (3.9), we have that for all t ∈ [0, t*)

||uεm(t)||2 +∫Ω|uεm(t)|ρuεm(t)dx ≥ 0,

thus, from (3.4), it follows

Applying Gronwall's inequality to (3.18) we conclude that there exists a constant K2 > 0, independent of ε, m and t, such that

Passage to the limit

From the inequality on (3.19) we obtain a subsequence, still denoted (uεm)m ∈ N, such that

To take the limit in the nonlinear term, we define

W(0, T) = {u ∈ L∞(0, T; (Ω)); u' ∈ L∞(0, T; L2(Ω))}

and observe that, since (Ω) L²(Ω), from compactness theorem of Lions-Aubin²,

W(0, T) L²(0, T;L²(Ω))

Observe that, from (3.20) and (3.21), we have

(u_εm)_{m ∈ N} is bounded inW(0, T)

and, thus, we can extract a subsequence, still denoted by (u_εm)_{m ∈ N}, such that

u_εm → u_ε strongly inL²(0, T;L²(Ω)) ≡L²(Q).

Then, there is a subsequence such that

On the other hand, by the hypothesis H3, we have that(Ω)L²(Ω) and, therefore, by (3.19), we obtain

From (3.22) and (3.23), thanks to Lions², Lemma 1.3, we have

From (3.20), (3.21) and (3.24) it follows that, for all υ ∈ (Ω),

((uεm, υ)) ((u_ε, υ)) in L^∞(0,T)

(|u_εm|^ρ, υ) (|u_ε|^ρ, υ) inL^∞(0, T)

(M,υ) (M,υ) inL^∞(0, T).

The convergences obtained above allow us to take the limit in the approximate equation, when m goes to infinity, and get

Applying (3.24) to θ ∈D(0, T) it follows that

-((t), υ)θ'(t)dt -〈Δuε(t), υ〉θ(t)dt +(|uε(t)|ρ, υ)θ(t)dt + (M(t)(t), υ)θ(t)dt = ((t), υ)θ(t)dt,

for all υ ∈ (Ω).

Therefore,

for all υ ∈ (Ω), that is,

where g(t) = (t) + Δuε(t) - |uε(t)|ρ - M(t)(t).

Thus, =g in the sense of the distributions.

But,

g = ( + + Δu_ε + |u_ε(t)|^ρ -M(t)) ∈ L1(0, T; H-1(Ω)

and therefore,

We will study the initial conditions on (3.28). From the convergences (3.20) and (3.21) we obtain that uεm(0) u_ε(0) as m → ∞ inL²(Ω). Then, using the approximate problem (3.1), we conclude that

From the approximate problem (3.1) we have, for all υ ∈V_m and almost every t ∈ [0,T],

((t), υ) = ( - |u_ε(t)|^ρ -M(t)(t), υ) - (∇uεm, ∇u).

As (t) ∈ Vm, we can conclude, for almost every t ∈ [0, T],

where C1 is the constant of the embedding of (Ω) into L2(Ω).

From (3.19) and (3.30), we obtain that |||| ≤K₃, not depending on m and ε and, then, inL¹(0, T;H^-1(Ω)). This convergence, (3.21) and (3.1) imply that

Now, to finish the proof of theorem 3.1, we need take the limit when ε goes to zero.

In fact, note that from (3.19), we have the same estimates in ε as those obtained for m. That is,

Thus, there exists a subsequence, still denoted by (u_ε)_{ε > 0}, such that

Applying, as above, the same arguments of compactness, it follows

Also, by (3.32), we have

The limitation in (3.36), combined with the convergence in (3.34), allows us to conclude that

M(x, t)w'(x, t) = 0 in Q

and thus, see the definition of M in (1.4),

As w(x, 0) = ũ₀(x), then w(x, 0) = 0 in Ω - Ω₀ and, therefore, by (3.37) and the geometric condition (H1),

By (3.38) and the fact that w(t) ∈ (Ω), it follows, from the regularity condition (H2), that w(t) ∈ (Ωt). That is,

w ∈ L∞(0, T; (Ωt)).

From (3.32), we observe that there is a subsequence of M(t)(t) that converges weakly to some function in L2(0, T; (Ω) when ε tends to zero. Then, passing to the limit the equation (3.28), we observe that w could not be a weak solution for the PDE u'' - Δu + |uρ| = f over all the extended domain Q. But, as M vanishes at , we can consider the restriction of w to and the theorem will be proved.

If we denote by u, the restriction of w to, we have that

u ∈L^∞(0, T;(Ω_t))

u' ∈L^∞(0, T;L²(Ω_t)).

On the other hand, from (3.28), by restriction to , we obtain, in the sense of distributions

where û_ε denote the restriction of u_ε to.

Finally, from (3.33) - (3.35) we can pass to the limit in (3.39), when ε goes to zero and we can obtain, in the sense of distributions,

u'' - Δu + |u^ρ| =f

The initial conditions for the problem (1.2) follow from (3.29) and (3.31) and the same kind of arguments used to show these equations.

REFERENCES

• ¹
  J. Cooper & C. Bardos. A non linear wave equation in time dependent domains. Journal of Mathematics Analysis and Application, 42 (1972), 47-58.
• ²
  J.-L. Lions. "Quelques Méthodes de Résolution des Problemes aux Limites Non Linéaires", Dunod, Paris (1969).
• ³
  J.-L. Lions. Une remarque sur les problémes de évolution nonlineares dans domains non cilìndriques. Revue Roumaine de Mathématique Pure et Appliquées, 9 (1964), 11-18.
• ⁴
  J.-L. Lions & W.A. Strauss. Some nonlinear evolution equations. Bol. Soc. Math. de France, 93 (1965), 43-96.
• ⁵
  L.A. Medeiros. Nonlinear wave equations in domains with variable boundary. Archive for Rational Mechanics and Analysis, 47(1) (1972).
• ⁶
  L.A. Medeiros, J. Lìmaco & C.L. Frota. On wave equations without global a priori estimates. Bol. Soc. Paranaense de Matemática, 30(2) (2012), 12-32.
• ⁷
  M. Nakao & T. Narazaki. Existence and decay of solutions of some nonlinear wave equations in non cylindrical domains. Mat. Report, XI(2) (1978).
• ⁸
  L. Tartar. "Topics in Nonlinear Analysis". Un. Paris Sud. Dep. Math., Orsay, France (1978).

Publication Dates

• Publication in this collection
  Dec 2015

History

• Received
  16 June 2014
• Accepted
  29 Nov 2015