On energy wave equations with non-negative non-linear terms

International Journal of Non-Linear Mechanics 82 (2016) 6–16

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International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm

Review

On energy wave equations with non-negative non-linear terms M. Milla Miranda a, A.T. Louredo a, M.R. Clark b, H.R. Clark c,n a b c

Universidade Estadual da Paraíba, DM, PB, Brazil Universidade Federal do Piauí, DM, PI, Brazil Universidade Federal Fluminense, IME, RJ, Brazil

art ic l e i nf o

a b s t r a c t

Article history: Received 3 January 2015 Received in revised form 22 December 2015 Accepted 22 December 2015 Available online 27 February 2016

This paper deals with the existence of at least one solution and the uniform stabilization of the energy of an initial-boundary value problem for a non-linear wave equation with non-linear boundary condition of the feedback type. The non-linearities in both waves and boundary equations behave as functions of the ϱ type jujR for ϱ 4 1. & 2016 Published by Elsevier Ltd.

Keywords: Existence of solutions Uniform stabilization Non-linear system with non-negative terms

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1. Physical motivation and some previous works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2. Extensions and applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3. Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2. Notations and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1. Some notation, hypotheses and preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3. Proof of Theorems 2.1 and 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1. Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2. Passage to the limit in m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3. Passage to the limit in l. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4. Some remarks on Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5. Proof of Theorem 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5.1. Equivalence of the energies E and Eε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.5.2. Differential inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4. Additional comments and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1. Introduction

n

Corresponding author. Tel.: þ 55 21 2629 2060. E-mail addresses: [email protected] (M. Milla Miranda), [email protected] (A.T. Louredo), [email protected] (M.R. Clark), [email protected] (H.R. Clark). http://dx.doi.org/10.1016/j.ijnonlinmec.2015.12.007 0020-7462/& 2016 Published by Elsevier Ltd.

Let Ω be an open, bounded and connected set of Rn with its boundary Γ of class C 2 . Suppose also that Γ is partitioned into Γ 0 and Γ 1 both with positive measure and Γ 0 \ Γ 1 empty. This paper is concerned with the global existence of solutions and uniform stability of the energy for the following non-linear

M. Milla Miranda et al. / International Journal of Non-Linear Mechanics 82 (2016) 6–16

initial-boundary value problem:
00
u  μΔu þ gð; uÞ ¼ f in Ω  ð0; 1Þ;

u¼0 on Γ 0  ð0; 1Þ;


∂u þ hð; u0 Þ þ qð; uÞ ¼ 0 on Γ 1  ð0; 1Þ;

∂ν

uð0Þ ¼ u0 ; u0 ð0Þ ¼ u1 in Ω;

ð1:1Þ

where ν is the unit outward normal on Γ 1 , the real-valued functions μ ¼ μðtÞ; f ¼ f ðx; tÞ; g ¼ gðx; sÞ; h ¼ hðx; sÞ and q ¼ qðx; sÞ are defined in R þ ; Ω  R þ ; Ω  R; Γ 1  R and Γ 1  R, respectively. ρ Moreover, the functions g and q behave as j sj R and j sj σR , respectively, for ρ 4 1 and σ 41, and h is a continuous and strong monotone function in the variable s. 1.1. Physical motivation and some previous works Problem (1.1) comes from many physical situations. Perhaps the most significant of them is the saturation property of the nuclei of atoms. Thus, the non-linear equation (1.1)1 arises in quantum theory of the meson in its classical aspects. In fact, in Schiff's paper [23], it is developed to account for nuclear saturation and shell structure in terms of many-body forces. These forces are derived from mesons that obey the non-linear wave equation ∂2 u  Δu þ G0 ðuÞ ¼ f F 0 ðuÞ ∂t 2

in Ω  ð0; 1Þ;

where the prime denotes the derivative with respect to u, f ¼ f ðx; tÞ is the nucleon source density, FðuÞ is the non-linear coupling function and GðuÞ is the non-linear field function. In particular, a typical physical example of GðuÞ is an even function, given by GðuÞ ¼ 12u2 þ 14u4 . In this context, Jörgens [5], in the absence of sources ðf ¼ 0Þ, formulated the following equation as a version to the motion of quantum mechanics: ∂2 u  Δu þ μ2 u þ η2 j uj 2 u ¼ 0 ∂t 2

in Ω  ð0; 1Þ;

and proved the existence and uniqueness of solutions for an initial-boundary value problem associated with the above equation. Indeed, in Jörgens [5,6] began a rigorous mathematical research of equations of the type ∂2 u  Δu þ F 0 ðj uj 2R Þu ¼ 0 ∂t 2

in Ω  ð0; 1Þ:

Later, Lions and Strauss [14] developed a large fields of research on non-linear evolution equations which include the models of Schiff and Jörgens. In the above papers the non-linearities in the wave equation behave as j uj ρ u, and thus the energy method works well. When the non-linearity is for instance of the type u2 , we need, however, a totally different approach. Lions [12] analyzed this using the potential well method, which was introduced by Sattinger in [22]. Tartar [25] developed even another method to study the same non-linearity. Recently, Medeiros et al. [17] showed via Tartar and energy methods the existence and uniqueness of global solutions for the problem
″
u  Δu þ j uj ρR ¼ f in Ω  ð0; 1Þ;

u¼0 on Γ  ð0; 1Þ; ð1:2Þ


uðx; 0Þ ¼ u0 ðxÞ; u0 ðx; 0Þ ¼ u1 ðxÞ in Ω; with restrictions on the size of the norms of the initial data u0 and u1 , and ρ 4 1. Observe that Eq. (1.2)1 is obtained when we set 1 ρ FðuÞ ¼ u and GðuÞ ¼ ρ þ 1 j uj u in Schiff 's model. Another significant problem in our research is the initialboundary value problem with both Dirichlet and feedback

boundary conditions
00
u  μΔu ¼ 0

u¼0


∂u
μ þ δu0 ¼ 0
∂ν

uðx; 0Þ ¼ u0 ðxÞ; u0 ðx; 0Þ ¼ u1 ðxÞ

7

in Ω  ð0; 1Þ;

on Γ 0  ð0; 1Þ; on Γ 1  ð0; 1Þ;

ð1:3Þ

in Ω:

When μ is a constant, Komornik and Zuazua [7], Lasiecka and Triggiani [11], Quinn and Russell [21], among others proved by applying semigroups theory the existence and uniqueness of solutions for system (1.3). This method does not work when in (1.3)1 the coefficient is a time-dependent function, i.e., μ ¼ μðtÞ. In this case, Milla Miranda and Medeiros [18] studied the existence and uniqueness of solutions via Faedo–Galerkin–Lions' method, by constructing a special Hilbertian basis that satisfies the feedback boundary condition at t ¼ 0. Furthermore, the asymptotic behavior of the energy was also established. Many non-linear problems with non-linearities in the wave equation and in the boundary equation are related with our problem (1.1). We refer to the reader the works of Komornik [8], Lasiecka and Tataru [10] and Zuazua [29]. There, it is included a qualitative analysis of the properties of the solutions and the solvability of these problems via semigroup theory. In addition to these works, we cite Cavalcanti et al. [4], Milla Miranda and San Gil Jutuca [19], and Vitillaro [26]. All these last authors established existence of solutions via Faedo–Galerkin–Lions' method, and some qualitative properties of the solutions. The proposed problem (1.1) generalizes substantially problems (1.2) and (1.3), since all equations in (1.1) are non-linear. This means that problem (1.1) is physically more realistic, and as a consequence, we must deal with more technical difficulties. In significant works of Lasiecka and Tataru [10] and Cavalcanti et al. [3], the existence of solutions of similar problems to (1.1) is investigated. In [3] the main ideas of [10] are used and they obtained the existence of global solutions. In our paper, the method used to obtain the existence of global solutions of (1.1) is different from that used in [3], and consequently in [10], because there is no a priori control of the sign of some terms of the energy. As we are about to see, our strategy to overcome this problem is to use an idea introduced by Tartar [25]. Moreover our boundary dissipation hð; u0 Þ depends on x, and one part of the potential energy of system (1.1) depends on t. This makes more difficult to obtain a solution for problem (1.1). One way to obtain the blow-up of the solutions in finite time for problem (1.1) is to have hypotheses that let you know the sign of the source terms gð; uÞ and qð; uÞ. Note that these terms are related with Z Z Gð; uÞ dx and Q ð; uÞ dΓ Ω

Γ1

which constitute a part of the energy of system (1.1). Here Gðx; sÞ and Q ðx; sÞ denote the anti-derivative with respect to s of the functions gðx; sÞ and qðx; sÞ, respectively. In our case, we have no information on the sign because gðx; sÞ and qðx; sÞ behave like αðxÞ j sj ρ and β ðxÞj sj ρ , respectively, with α A L1 ðΩÞ and β A L1 ðΓ 1 Þ. In fact, if gðx; sÞ ¼ j sj ρ and qðx; sÞ ¼ j sj σ , then the expressions Z Z Z 1 Gð; uÞ dx ¼ j uj ρ u dx and Q ð; uÞ dx ρþ1 Ω Ω Γ1 Z 1 j uj σ u dΓ ¼ σ þ1 Ω do not have the same sign for all t. Consequently, the blow-up properties of the energy in finite time are an open question. However, see Cavalcanti et al. [3], if gðsÞ ¼ j sj ρ s and qðsÞ ¼ j sj σ s, then the blow-up question is answered because the signs of

8

M. Milla Miranda et al. / International Journal of Non-Linear Mechanics 82 (2016) 6–16

the terms Z 1 j uj ρ þ 2 dx ρþ2 Ω

and

1.3. Organization of the paper

Z 1 j uj σ þ 2 dΓ σ þ 2 Γ1

are positive. For more comments about paper [10] and what it concerns on the uniform stabilization, see Section 4 of the present paper. To establish the existence of global solutions for problem (1.1) we have assumed some restrictions on the size of the norms of the initial data and employing the following techniques: (a) the method of Faedo–Galerkin–Lions with a special basis, (b) Strauss approximations of continuous functions by Lipschitz continuous ones, (c) the method of Tartar with a modification, (d) Aubin– Lions' compactness theorem and (e) trace theorems for nonsmooth functions. It is worth noting that with h being just continuous in s and j hð; sÞj does not behave as j sj for j sj big enough, the uniqueness of solutions is an open problem. For more details see Remark 2.1. The exponential decay rate of the energy of problem (1.1) is derived by the multiplier method supposing f ¼ 0; q ¼ 0; gð; sÞ ¼ j sj ρ , and employing the ideas contained in Komornik [8] and Komornik and Zuazua [7]. In Remark 2.2 we give some examples of functions g and q of problem (1.1) that satisfy the hypotheses of Theorem 2.1. 1.2. Extensions and applicability The extensions of the results brought in this work can be used in several physical situations, especially in problems arising from non-linear mechanics, since the tools from Functional Analysis and PDE theory are presented carefully. We emphasize that the general aspects considered in the hypotheses about the functions of system (1.1) make our approach more realistic from physical point of view. The coefficient μ ¼ μðtÞ of system (1.1) has also important physical motivations. To this respect, let us recall some examples of real-world models where they appear naturally: (i) In homogeneous material, the vibrations of membranes or strings produce coefficients of the type: Z  Z  μðtÞ ¼ μ j uðx; tÞj 2 dx or μðtÞ ¼ μ j ∇uðx; tÞj 2 dx Ω

Ω

ð1:4Þ

The paper to follow is divided into three parts. In Section 2 the notations and preliminary results are established, and also the main results, namely Theorems 2.1 and 2.3, are listed. In Section 3 is devoted to prove Theorems 2.1 and 2.3, and other particular results related, and finally Section 4 is dedicated to additional comments and open problems.

2. Notations and main results 2.1. Some notation, hypotheses and preliminary results Regarding the functional spaces, we will use the standard notations for the usual Lebesgue and Sobolev spaces. In particular, the symbols ð; Þ; ð; ÞΓ 1 ; j  j and j  j Γ 1 denote the scalar products and the norms of the Hilbert spaces L2 ðΩÞ and L2 ðΓ 1 Þ, respectively. We will also need to use the Hilbert space V ¼ n o v A H 1 ðΩÞ; γ 0 ðvÞ ¼ 0 a:e: on Γ 0 equipped with the scalar product  n  X ∂u ∂v ; ððu; vÞÞ ¼ and norm J u J ¼ ððv; vÞÞ1=2 : ∂xi ∂xi i¼1 Moreover, let  Δ be the self-adjoint Laplace operator of L2 ðΩÞ defined by the triplet fV; L2 ðΩÞ; ðð; ÞÞg, see Lions [13]. Thus, the domain of operator  Δ is given by   ∂u Dð  ΔÞ ¼ v A V \ H 2 ðΩÞ; ¼ 0 on Γ 1 : ∂ν In order to state the results of this paper for problem (1.1), we will assume to follow some hypotheses on the real-valued functions g; q; μ and h, and also on the constants ρ and σ . Namely, ðR; L1 ðΩÞÞ and q A W 1;1 ðR; L1 ðΓ 1 ÞÞ: ð2:1Þ g A W 1;1 loc loc Rs Rs Being Gðx; sÞ : ¼ 0 gðx; τÞ dτ and Q ðx; sÞ : ¼ 0 qðx; τÞ dτ we suppose


∂gðx; sÞ
ρþ1 ρ
ra2 j sj ρ  1 j Gðx; sÞj R r a0 j sj R ; j gðx; sÞj R r a1 j sj R ;

R ∂s
R ð2:2Þ for all s A R and a.e. in x A Ω, with the real constants ρ 4 1 and a0 ; a1 ; a2 non-negative. σ þ 1; j qðx; sÞj R rb1 j sj σR ; j Q ðx; sÞj R r b0 j sj R


∂qðx; sÞ
σ 1


∂s
r b2 j sj R

which appear in the Carrier and Kirchhoff equations respectively, where the function μ ¼ μðtÞ is positive and (at least) continuous. It is also frequent to find terms of the kind (1.4) in non-linear plates models. (ii) In non-homogeneous material, the motion propagation produces coefficients more general as the following ones:   Z μðx; t; λÞ ¼ μ x; t; j uðx; tÞj 2 dx or   ZΩ μðx; t; λÞ ¼ μ x; t; j ∇uðx; tÞj 2 dx ð1:5Þ

where μ0 , μ1 are real constants.

which reproduce non-linear mechanical structures interesting, realistic and with rich mathematical challenges.

hA C 0 ðR; L1 ðΓ 1 ÞÞ; hðx; 0Þ ¼ 0;   hðx; sÞ  hðx; rÞ ðs  rÞ Zd0 ðs  rÞ2 ;

Ω

The results obtained about system (1.1) can be applied to many problems in this field and related fields, improving results described in the literature or even generalizing models, as for example, considering the aforementioned coefficients (1.4) and (1.5) into Eq. (1.1)1. For other mathematical models where it is possible applies our arguments, the readers can see, for instance, Balachandran and Magrab [1], Nayfeh and Mook [20], Timoshenko et al. [27], and Timoshenko and Young [28].

qðx; 0Þ ¼ 0; ð2:3Þ

R

for all s A R and a.e. in x A Γ , with the real constants σ 41 and b0 ; b1 ; b2 non-negative.

μ A W 1;1 ð0; 1Þ; loc

μ0 A L1 ð0; 1Þ;

0 o μ0 r μðtÞ r μ1 ;

8 t Z0; ð2:4Þ

ð2:5Þ

for all r; s A R and a.e. in x A Γ 1 , with the constant d0 40. Specifically, the real number ρ and σ are chosen as follows: nþ1 n rρr for n Z 3; n n2 2n  1 n1 rσ r for n Z3: and 2ðn  1Þ n2

ρ 4 1 for n ¼ 1; 2 and σ 41 for n ¼ 1; 2

ð2:6Þ

2ðn  1Þ n For n Z 3 we consider pn ¼ n2n  2 and p1 ¼ n  2 . Thus, we have from (2.6)1 and (2.6)2 that the following dense and continuous

M. Milla Miranda et al. / International Journal of Non-Linear Mechanics 82 (2016) 6–16

of solutions for problem (1.1). However, when hðx; sÞ satisfies the condition

embedding 2ρ

ρþ1

V↪L ðΩÞ↪L ðΩÞ↪L pn

Lp ðΩÞ↪Lðρ  1Þn ðΩÞ; n

ρ

ðΩÞ↪L ðΩÞ;

V↪Lρ þ 1 ðΓ 1 Þ;

ð2:7Þ

n

Lp1 ðΓ 1 Þ↪L2ðn  1Þðσ  1Þ ðΓ 1 Þ; n

ð2:8Þ

hold. Furthermore, when ρ 4 1 and either n ¼ 1 or n ¼ 2, the n n continuous embedding (2.7)1 without Lp ðΩÞ and (2.8) without Lp1 ðΓ 1 Þ are true. For use latter, from (2.7) and (2.8), we will denote by ki , r j and ml , for i ¼ 0; …; 5, j ¼ 0; 1 and l ¼ 0; …; 4 positive real constants depend on the measure of Ω, and such that J v J Lρ þ 1 ðΩÞ r k0 J v J ;

J v J Lρ ðΩÞ rk1 J v J ;

J v J Lðρ  1Þn ðΩÞ rk3 J v J ;

J v J L2ρ ðΩÞ r k2 J v J ;

J v J Lpn ðΩÞ r k4 J v J ;

J v J Lρ þ 1 ðΓ 1 Þ r k5 J v J ; ð2:9Þ

j vj rr 0 J v J ;

J v J L2 ðΓ 1 Þ r r 1 J v J

J v J Lσ þ 1 ðΓ 1 Þ r m0 J v J ;

for all v A V;

J v J L ðΓ 1 Þ r m1 J v J ; J v J pn1 rm4 J v J σ

J v J L2ðn  1Þðσ  1Þn ðΓ 1 Þ r m3 J v J ;

L

ðΓ 1 Þ

for all v A V: ð2:10Þ

We also consider λ1 and N1 real positive constants given by 8" #ρ 1 1 " #σ 1 1 9 < = μ0 μ0 n λ1 ¼ min ; ; ð2:11Þ : 4ðρ þ 1Þa0 kρ þ 1 ; 6ðσ þ1Þμ1 b0 m0σ þ 1 0 n

1 1 ρþ1 N 1 ¼ j u1 j 2 þ μð0Þ J u0 J 2 þ a0 k0 J u0 J ρ þ 1 þ μð0Þb0 m0σ þ 1 J u0 J σ þ 1 : 2 2 ð2:12Þ 2.2. Main results

u0 A Dð  ΔÞ \ H 10 ðΩÞ; f

A L1loc ð0; 1; L2 ð

u1 A H 10 ðΩÞ;

1

ð2:13Þ

qð; u0 Þ A L1

μ0

μ0

0

1

0

 n j μ0 ðtÞj dt o λ1 :

ð2:14Þ

u0 A L1 ð0; 1; L2 ðΩÞÞ \ L1 loc ð0; 1; VÞ;

2 u″ A L 1 loc ð0; 1; L ðΩÞÞ;

u A L ð0; 1; L ðΓ 1 ÞÞ; 0

1

2

u

″

2 A L1 loc ð0; 1; L ð

Γ 1 ÞÞ;

ð2:15Þ

H 2 ðΓ 1 Þ

:

∂u A L1 ð0; 1; L2 ðΓ 1 ÞÞ: ∂ν Therefore, in this case, the above Green formulae make sense and applying the energy method we obtain the uniqueness of solutions of problem (1.1) under the hypotheses of Theorem 2.1. The uniqueness of solutions with hðx; sÞ satisfying other hypotheses than those mentioned above have not been investigated. The uniform stability of the energy of system (1.1), it will be obtained supposing additional hypotheses. Namely, we consider f ¼q¼0

and

gðx; sÞ ¼ j sj ρ for ρ 4 1:

ð2:17Þ

The sets Γ 0 and Γ 1 are connected and disjoints with positive measure, and defined by

Γ 0 ¼ fx A Γ ; mðxÞ  νðxÞ r 0g and Γ 1 ¼ fx A Γ ; mðxÞ  νðxÞ 4 0g ð2:18Þ n

where mðxÞ : ¼ x  x for some fixed x A R where x  y denotes the usual scalar product of Rn . Moreover, we set hðx; sÞ ¼ ½mðxÞ  νðxÞpðsÞ with 0

0

½pðsÞ  pðrÞðs  rÞ Z p0 ðs rÞ2

pð0Þ ¼ 0;

ð2:19Þ

μ0 ðtÞ r 0 a:e: in ð0; 1Þ:

ð2:20Þ

Furthermore, we introduce the following constants: 2

ρþ1

M ¼ Lþ

μ0 nkρ0 þ 1 þ ρþ1

3k0 ; 2ðρ þ 1Þ

2

ρþ1

λn2 ¼

μ0 Rkρ5 þ 1 þ μ0 ðn 1Þkρ0 þ 1 ;



ρ 1 1

μ0

4ðρ þ1ÞM

;

ρþ1

ð2:21Þ

where R≔maxf J x J ; x A Ω g. We now can show the following result. Theorem 2.2. In addition to the hypotheses (2.17)–(2.21), we assume u0 A Dð  ΔÞ \ H 10 ðΩÞ and u1 A H 10 ðΩÞ such that  Z 1   1=2 N2 2 n n J u0 J o λ2 ; 2 exp j μ0 ðtÞj dt o λ2 : ð2:22Þ

such that u″  μΔu þgð; uÞ ¼ f in L2loc ð0; 1; L2 ðΩÞÞ; ∂u þhð; u0 Þ þ qð; uÞ ¼ 0 in L1loc ð0; 1; L1 ðΓ 1 ÞÞ; ∂ν uð0Þ ¼ u0 ; u0 ð0Þ ¼ u1 ;

r C2 J u J σ 1

ð0; 1; L2 ðΓ 1 ÞÞ:

k 1 1 N2 ¼ j u1 j 2 þ μð0Þ J u0 J 2 þ 0 J u0 J ρ þ 1 ; 2 2 ρþ1

Then there exists at least a function u in the class u A L1 ð0; 1; VÞ;

ðΓ 1 Þ

These above results and equality (2.16)2 yield

L¼

n

 Z 2 j f ðtÞj dt exp

n1

From this and regularity (2.15) for u, we have

and that 1

L

for all s; r A R and p0 is a positive real constant. In addition to hypothesis (2.4) on the function μ one assumes

f A L1 ð0; 1; L2 ðΩÞÞ;

ΩÞÞ;

J u0 J o λ1 ;  12 Z 1 2 ð2N 1 Þ2 þ

8 s A R;

J qð; uÞ J L2 ðΓ 1 Þ r b1 J u J σL2σ ðΓ Þ r C 1 J u J σ2ðn  1Þ

p A C 0 ðRÞ;

Theorem 2.1. Suppose 0

j hðx; sÞj r d1 j sj

see (3.31), we have from (2.15)3 that hð; u0 Þ A L1 ð0; 1; L2 ðΓ 1 ÞÞ. Moreover, from the restrictions in (2.6), for σ , we obtain

V↪H 2 ðΓ 1 Þ↪Lp1 ðΓ 1 Þ↪L2σ ðΓ 1 Þ↪Lσ þ 1 ðΓ 1 Þ↪Lσ ðΓ 1 Þ; 1

9

μ0

ð2:16Þ

provided the hypotheses in (2.1)–(2.6) hold. Remark 2.1. Expressions (2.15) and (2.16)1 imply, respectively that 1=2 Γ 1 ÞÞ and u0 A L1 ðΓ 1 ÞÞ. Thus the Green loc ð0; 1; H formulae  ∂uðtÞ 0 ð  ΔuðtÞ; u0 ðtÞÞ ¼ ððuðtÞ; u0 ðtÞÞÞ  ; u ðtÞ ∂ν

1 1 ∂u ∂ν A Lloc ð0; 1; L ð

does not make sense because the duality paring 〈; 〉 is not defined. Therefore, the energy method does not work to obtain uniqueness

μ0

0

Then there exists at least a function u in the class (2.15) such that u″  μΔu þ j uj ρ ¼ 0 in L2loc ð0; 1; L2 ðΩÞÞ; ∂u þ ðm  νÞpðu0 Þ ¼ 0 in L1loc ð0; 1; L1 ðΓ 1 ÞÞ; ∂ν uð0Þ ¼ u0 ; u0 ð0Þ ¼ u1 ; provided the hypothesis (2.6) holds. The proof of Theorem 2.2 carries over to Theorem 2.1 with minor modifications. Therefore, in the next section we will present only the proof of Theorem 2.1.

10

M. Milla Miranda et al. / International Journal of Non-Linear Mechanics 82 (2016) 6–16

To show that the energy associated with the solutions established in Theorem 2.2 is asymptotically stable, we still need to assume that j pðsÞj rp1 j sj

for all s A R ðp1 positive constantÞ

and

μ0 Z 1; ð2:23Þ

where μ0 was fixed in (2.4). The energy of system (1.1) is given by Z 1 μðtÞ 1 J uðtÞ J 2 þ j uðtÞj ρ uðtÞ dx; EðtÞ ¼ j u0 ðtÞj 2 þ 2 2 ρþ1 Ω

where the constant d0 was introduced in (2.5).

Theorem 2.3. Let u be the solution obtained in Theorem 2.2 and suppose that (2.23) holds, then   1 for all t Z 0; ð2:25Þ EðtÞ r 3Eð0Þexp  ηt 3

η ¼ min



 1 τ0 μ0 p0 ; 4 0; 2S P

pffiffiffi 1=2 pffiffiffi 1=2 S ¼ 4 2Rμ0 þ 2 2ðn  1Þr 0 μ0 ; ð2:26Þ

ð2:28Þ

Remark 2.2. To make the hypotheses of Theorem 2.1 consistent we present the following functions g and q which satisfy the required conditions: and qðx; sÞ ¼ bðxÞj sj σ , with a A L1 ðΩÞ, 1. gðx; sÞ ¼ aðxÞj sj ρ b A L1 ðΓ 1 Þ; 2. gðx; sÞ ¼ aðxÞg M ðsÞ for M Z 1 and g M ðsÞ ¼ j sj ρ for j sj r M and g M ðsÞ ¼ j sj þ M ρ  M for j sj 4 M. Analogously, the functions qðx; sÞ ¼ bðxÞqM ðsÞ for b A L1 ðΓ 1 Þ. 3. gðx; sÞ ¼ aðxÞg M ðsÞ for M Z 1 and g M ðsÞ ¼ j sj ρ for j sj r M and g M ðsÞ ¼ M ρ for j sj 4 M. Analogously, the functions qðx; sÞ ¼ bðxÞqM ðsÞ for b A L1 ðΓ 1 Þ. Remark 2.3. Employing similar arguments to those in the proof of Theorem 2.3, we are able to show an exponential decay rate of the ∂g ðx; sÞ satisfying suienergy of problem (1.1) for f  0, q  0 and ∂x i table hypotheses.

3. Proof of Theorems 2.1 and 2.3 In order to prove Theorem 2.1 we need of the following two lemmas: Lemma 3.1. Let m and n be functions in L1 ð0; TÞ with mðtÞ Z0 and nðtÞ Z 0 a:e: in ð0; TÞ, and a Z 0 a real constant. Let φ : ½0; T-R be a continuous function, φðtÞ Z 0 for all t A ½0; T and such that Z t Z t 1 2 1 φ ðtÞ r a2 þ mðτÞφðτÞ dτ þ nðτÞφ2 ðτÞ dτ for all t A ½0; T: 2 2 0 0 Then

φðtÞ r a þ

Z



T

mðtÞ dt 0

Z exp

t 0

nðτÞ dτ

3.1. Proof of Theorem 2.1 The proof of the existence of solutions will be carried out through the method of Faedo–Galerkin–Lions with a special basis of V \ H 2 ðΩÞ. Initially, let ðhl Þ be a sequence of vectors of Lemma 3.2 and let ðu1l Þ be a sequence of vectors of C 1 0 ðΩÞ such that in H 10 ðΩÞ:

ð3:1Þ

Note that the equations ð2:27Þ



The proof of Lemma 3.2 is done in Louredo and Milla Miranda [15].

u1l -u1

"  #2   2 3 1 3p1 2 r 1 ðn  1Þ ; P ¼ μ0 μð0ÞR3 p1 þ μð0ÞR þ μ0 μð0Þ R 2 2 2

τ0 ¼ minfmðxÞ  νðxÞ; x A Γ 1 g 4 0:

1. hl ðx; 0Þ ¼ 0 a.e. in Γ 1 ; 2. ½hl ðx; sÞ  hl ðx; rÞðs  rÞ Zd0 ðs  rÞ2 for all s; r A R and a:e: in Γ 1 ; 3. For each l there exists a function cl A L1 ðΓ 1 Þ satisfying j hl ðx; sÞ  hl ðx; rÞj r cl ðxÞj s rj for all s; r A R and a:e: in Γ 1 ; 4. ðhl Þ converges to h uniformly on bounded sets of R and a:e: in Γ 1 ,

ð2:24Þ

for all t Z0 and we have the following result.

where

Lemma 3.2. Let h be a function satisfying hypotheses in (2.5). Then there exists a sequence ðhl Þ with hl A C 0 ðR; L1 ðΓ 1 ÞÞ for each l A N, and such that

∂u0 þhl ð; u1l Þ þ qð; u1l Þ ¼ 0 ∂ν

1

þ μðtÞðqð; ulm ðtÞÞ; wÞΓ 1 ¼ ðf ðtÞ; wÞ

Lemma 3.1 is a consequence of Lemma A.5 of Brezis [2, p. 157].

for all w A W m ;

u0lm ð0Þ ¼ u1l :

ulm ð0Þ ¼ u0 ;

ð3:3Þ

The above finite-dimensional system has solution ulm defined in ½0; t lm Þ. The following estimates allow us to extend this solution to the whole half-line ½0; 1Þ and to pass to the limits as m-1 and as l-1. Estimate I: Setting w ¼ u0lm in (3.3)1, we obtain i dZ 1 dh 0 j ulm ðtÞj 2 þ μðtÞ J ulm ðtÞ J 2 þ Gð; ulm ðtÞÞ dx 2 dt dt Ω   Z   d þ μðtÞ Q ð; ulm ðtÞÞdΓ þ μðtÞ hl ð; u0lm ðtÞÞ; u0lm ðtÞ Γ1 dt Γ1 Z 1 0 ¼ ðf ðtÞ; ulm ðtÞÞ þ μ0 ðtÞ J ulm ðtÞ J 2 þ μ0 ðtÞ Q ð; ulm ðtÞÞ dΓ : 2 Γ1 Integrating on ½0; t with 0 o t ot lm , we get Z 1 0 1 j ulm ðtÞj 2 þ μðtÞ J ulm ðtÞ J 2 þ Gð; ulm ðtÞÞ dx 2 2 Ω Z Z t   Q ð; ulm ðtÞÞ dΓ þ μðτÞ hl ð; u0lm ðτÞÞ; u0lm ðτÞ Γ dτ þ μðtÞ Γ1

Z

t 0

for all t A ½0; T:

ð3:2Þ

hold, since u0 A Dð  ΔÞ; hl ðx; 0Þ ¼ 0 and qðx; 0Þ ¼ 0 a.e. in Γ 1 . Now we fix l A N and consider the basis fwl1 ; wl2 ; wl3 ; …g of V \ 2 H ðΩÞ such that u0 and u1l belong to the subspace ½wl1 ; wl2  spanned by w1 and w2 . Thus for each m A N we built the subspace W m ¼ spanfwl1 ; wl2 ; …; wlm g such that the approximate solutions P l ulm ðx; tÞ ¼ m j ¼ 1 g jlm ðtÞwj ðxÞ of problem (1.1) satisfy the approximate problem:   ðu″lm ðtÞ; wÞ þ μððulm ðtÞ; wÞÞ þ ðgð; ulm ðtÞÞ; wÞ þ μðtÞ hl ð; u0lm ðtÞÞ; w Γ

¼



on Γ 1 for all l

ðf ðτÞ; u0lm ðτÞÞ dτ þ

Z

þ 0

t

Z Γ1

1 2

1

0

Z

t 0

μ0 ðτÞ J ulm ðτÞ J 2 dτ 1 2

μ0 ðτÞQ ð; ulm ðτÞÞ dΓ dτ þ j u1l j 2

Z Z 1 þ μð0Þ J u0 J 2 þ Gð; u0 Þ dx þ μð0Þ Q ð; u0 Þ dΓ : 2 Ω Γ1

ð3:4Þ

M. Milla Miranda et al. / International Journal of Non-Linear Mechanics 82 (2016) 6–16

From (2.12), (3.1) and (3.6) there is λ0 A N such that N 1l o N 1 for λ Z λ0 , where N 1 is fixed in (2.12). Now, setting μ φ2 ðtÞ ¼ j u0lm ðtÞj 2 þ 0 J ulm ðtÞ J 2 þ γ 2 J ulm ðtÞ J σ þ 1

From Assumption (2.2) and notations (2.9), it yields
Z



Gð; ulm ðtÞÞ dx
r a0 kρ þ 1 J ulm ðtÞ J ρ þ 1 ;

0 Ω
Z
R


Gð; u0 Þ dx
r a0 kρ þ 1 J u0 J ρ þ 1 ;

0 Ω

2

R

and from hypotheses (2.3), (2.4) and notations (2.10), it yields

Z


μðtÞ Q ð; ulm ðtÞÞ dΓ

r μ1 b0 m0σ þ 1 J ulm ðtÞ J σ þ 1 ;
Γ1

R Z

0
μð0Þ
r μ b0 mσ þ 1 J u0 J σ þ 1 : Q ð; u Þ d Γ 1 0

Γ1

R

1 0 1 ρþ1 j u ðtÞj 2 þ μ0 J ulm ðtÞ J 2  a0 k0 J ulm ðtÞ J ρ þ 1 2 lm 2 Z t  μ1 b0 m0σ þ 1 J ulm ðtÞ J σ þ 1 þ μ0 d0 j u0lm ðτÞj 2Γ 1 dτ r

t

j f ðτ

0

Z

t

þ 0

1 τÞj dτ þ 2

Þj j u0lm ð

jμ

0

μ0

Therefore, for all l Z l0 ,

0

Z

t 0

j μ ðτÞj J ulm ðτÞ J dτ 0

0

ð3:11Þ  J ulm ðtÞ J r

and

2

1=2

μ0 n for all t A ½0; t lm Þ and J ulm ðtÞ J o λ1 :

K0 ð3:12Þ

2

ðτÞj b0 m0σ þ 1 J ulm ðτÞ J σ þ 1 dτ þN 1l ;

ð3:5Þ

Lemma 3.3. Inequalities (3.12) are true for all t A ½0; 1Þ. Precisely, let ½0; t lm Þ be an interval of the existence of the solutions ulm of (3.3). n Then J ulm ðtÞ J o λ1 for all t A ½0; 1Þ, for all l Z l0 and for all m A N. n

1 1 ρþ1 N 1l ¼ j u1l j 2 þ μ1 J u0 J 2 þ a0 k0 J u0 J ρ þ 1 þ μ1 b0 m0σ þ 1 J u0 J σ þ 1 : 2 2 ð3:6Þ We now need to control the sign of the following sum of (3.5): 1 μ J u ðtÞ J 2 a0 kρ0 þ 1 J ulm ðtÞ J ρ þ 1  μ1 b0 m0σ þ 1 J ulm ðtÞ J σ þ 1 : 2 0 lm 1 2 ρþ1 3 σ þ1  γ2 λ JðλÞ ¼ μ0 λ  γ 1 λ 4 2 ρþ1

where γ 1 ¼ a0 k0

for λ Z0;

ð3:7Þ

; γ 2 ¼ μ1 b0 mσ0 þ 1 , and whose differential is

1 ρ 3 σ J ðλÞ ¼ μ0 λ  ðρ þ 1Þγ 1 λ  ðσ þ 1Þγ 2 λ : 2 2 0

Therefore, as Jð0Þ ¼ 0 and λ Z 0 then JðλÞ Z0. Thus, J 0 ðλÞ Z 0, i.e., ρ1

ðρ þ 1Þγ 1 λ

3 1 σ 1 þ ðσ þ 1Þγ 2 λ r μ0 : 2 2

r

μ0

σ 1

0rλ

and

4ðρ þ 1Þγ 1

r

μ0

6ðσ þ 1Þγ 2

1 3 μ J u ðtÞ J 2  γ 1 J ulm ðtÞ J ρ þ 1  γ 2 J ulm ðtÞ J 2 Z 0; 4 0 lm 2 for J ulm ðtÞ J o λ1 and t A ½0; t lm Þ. From this n

1 1 1 μ J u ðtÞ J 2 þ γ 2 J ulm ðtÞ J σ þ 1 r μ0 J ulm ðtÞ J 2  γ 1 J ulm ðtÞ J ρ þ 1 4 0 lm 2 2  γ 2 J ulm ðtÞ J σ þ 1 : ð3:9Þ Plugging (3.9) into (3.5), we obtain

0

þ N1 l :

0

j μ ðτÞj J ulm ðτÞ J dτ þ 0

Z

2

0

t

ð3:13Þ

and independent of m A N. Estimate II: To simplify the notation we will not write the variable t and neither the subscripts l and m. Thus, differentiating with respect to t Eq. (3.3)1 and setting w ¼ u″ , we obtain i   1 dh ″ 2 j u j þ μ J u0 J 2 þ μ0 ððu; u″ ÞÞ þðg 0 ð; uÞu0 ; u″ Þ þ μ0 hð; u0 Þ; u″ Γ 1 2 dt  0      þ μ h ð; u0 Þ; ½u″ 2 Γ þ μ0 qð; uÞ; u″ Γ þ μ q0 ð; uÞu0 ; u″ Γ 1

0

″

¼ ðf ; u Þ þ

μ0 2

0

Ju J :

Taking w ¼ μμ u″ in approximate equation (3.3)1, we find



ð3:10Þ

1

0



j μ0 ðτÞj b0 mσ0 þ 1 J ulm ðτÞ J σ þ 1 dτ

1

2

μ0 ððu; u″ ÞÞ þ μ0 hð; u0 Þ; u″

1 0 1 1 j u ðtÞj 2 þ μ0 J ulm ðtÞ J 2 þ γ 2 J ulm ðtÞ J σ þ 1 2 lm 4 2 Z t Z t þ μ0 d0 j u0lm ðτÞj 2Γ 1 dτ r j f ðτÞj j u0lm ðτÞj dτ 0

μ0

for all t A ½0; 1Þ; for all l Zl0 and for all m A N:

0

0

t

Therefore, Lemma 3.3 ensures that (3.12) is true for all t Z 0, that is  1=2 2 j u0lm ðtÞj r K 0 and J ulm ðtÞ J r K0

Using (3.13) in (3.10) we obtain a real constant K 1 4 0 such that Z 1 j u0lm ðtÞj 2Γ 1 dt rK 1 for all t A ½0; 1Þ; for all l Zl0 ; ð3:14Þ

we have that these λ s satisfy the inequality (3.8). Note that these n n inequalities are equivalent to 0 r λ r λ1 , where λ1 is fixed in (2.11). n Thus JðλÞ Z0 for λ A ½0; λ1 . Therefore, from (3.7) we have

Z

 1=2 n K 0 . But this fact conand consequently λ1 ¼ J ulm ðt n Þ J r 2=μ0 tradicts the hypothesis (2.14)2. This concludes the proof.□

ð3:8Þ

Choosing λ A R with ρ1

Proof. Hypothesis (2.14)1 ensures that J ulm ð0Þ J ¼ J u0 J o λ1 . Suppose that affirmation is not true. Thus, there exists t 1 A ð0; t lm Þ n such that J ulm ðt 1 Þ J ¼ λ1 . Let t n be such that t n ¼ inf ft 1 A n ð0; t lm Þ; J ulm ðt 1 Þ J ¼ λ1 g. By the continuity of the norm one has n J ulm ðt n Þ J ¼ λ1 , which ensures 0 o t n o t lm . Let t A ½0; t n Þ. Then n J ulm ðtÞ J o λ1 . Inequality (3.12) provides  1=2 2 K 0 for all t A ½0; t n Þ; J ulm ðtÞ J r

μ0

To this, we consider the function

1 þ 2

This, (2.4), (2.13) and Lemma 3.1, it yields

 Z 1  Z 1 2 φðtÞ r ð2N 1 Þ1=2 þ j f ðtÞj dt exp j μ0 ðtÞj dt ¼ : K 0 :

j u0lm ðtÞj r K 0

where

0rλ

in (3.10) and observing that 1=μ1 r 1=μ0 , we get Z t i2 Z t 1 2 1h 2 φ ðtÞ r ð2N 1 Þ1=2 þ j f ðτÞj j φðτÞj dτ þ j μ0 ðτÞj φ2 ðτÞ dτ: 2 2 μ0 0 0

0

Taking into account the last four inequalities in (3.4), and using hypotheses (2.4) and (2.5), we find

Z

11

 μ0  gð; uÞ; u″ : μ







0 ″ Γ 1 þ μ qð; uÞ; u Γ 1 ¼

μ0  ″  μ0  ″ ″  f;u  u ;u μ μ

Combining the last two equalities, we have i  0  μ0 1 d h 00 2 μ0 j u j þ μ J u0 J 2 þ μ0 d0 j u00 j 2Γ 1 r f ; u00 þ J u0 J 2  ðf ; u00 Þ 2 dt 2 μ

12

M. Milla Miranda et al. / International Journal of Non-Linear Mechanics 82 (2016) 6–16

þ

μ0

00

2 μ0 u þ ðgð; uÞ; u00 Þ  ðg 0 ð; uÞu0 ; u00 Þ  μðq0 ð; uÞu0 ; u00 ÞΓ 1 : μ μ

2 u00lm -u00l weak star in L1 loc ð0; 1; L ðΩÞÞ;

ð3:15Þ Next, let us derive a bound above for the last three terms on the right-side of identity (3.15). First of all, let T 4 0 be a real fixed number and C ¼ CðTÞ 4 0 will denote positive constants that do not depend on l and m. From inequalities (2.2), (2.9), estimates in (3.13) and usual inequalities, it yields
0

μ
j μ0 j 00 j μ0 j 00 ρ 

ðgð; uÞ; u00 Þ

r a1 k1 J u J ρ j u j rC ju j:

μ

μ0

μ0

Again, from the inequalities (2.2), (2.9), estimates in (3.13), usual inequalities and observing that 1=n þ 1=pn þ 1=2 ¼ 1, we find

C ρ1 
ðg 0 ð; u0 Þ; u00 Þ
ra2 k3 k4 J u J ρ  1 J u0 J j u00 j r C J u0 J j u00 j r J u0 J 2 2 C þ j u00 j 2 : 2 We get, from the inequalities (2.3), (2.10), estimates (3.13), usual inequalities, and 1=2ðn  1Þ þ 1=pn1 þ 1=2 ¼ 1, that


μðq0 ð; uÞu0 ; u00 Þ
Γ1 σ 1

r μ1 b2 J u J L2ðσ  1Þðn  1Þ ðΓ Þ J u0 J 1

pn L 1ð

Γ1 Þ

j u00 j Γ 1

r μ1 b2 mσ3  1 m4 J u J σ  1 J u0 J j u00 j Γ 1 r C J u0 J j u00 j Γ 1 μ d0 r C J u0 J 2 þ 0 j u00 j 2Γ 1 : 2

Taking into account the last three inequalities in (3.15), integrating on ½0; t and coming back to the notation with variable t and subscripts l and m, we obtain Z Z t 1 00 1 μ d0 t 00 2 00 j ulm ðtÞj 2 þ μðtÞ J u0lm ðtÞ J 2 þ 0 j f ðτ Þj j u ðτÞj Γ 1 dτ r 2 2 2 0 0

j μ0 ðτÞj C j μ0 ðτÞj j u00lm ðτÞj dτ j f ðτÞj þ þ μ0 μ0

Z t 0 j μ ðτÞj C þ þ j ulm ″ðτÞj 2 dτ 2 μ0 0 Z  1 t 0 1 μð0Þ 1 2 J ul J : þ j μ ðτÞj þ 2C J u0lm ðτÞ J 2 dτ þ j u00lm ð0Þj 2 þ 2 0 2 2 ð3:16Þ For this inequality provides an estimate for the terms on the leftside, we need bounded j u00lm ð0Þj . This is possible thanks to the choice of the special basis of V \ H 2 ðΩÞ and (3.2). Note that j u00lm ð0Þj is bounded. In fact, setting t ¼ 0 and v ¼ u00lm ð0Þ in Eq. (3.3), we obtain by the Gauss' lemma and Eq. (3.2), that j u00lm ð0Þj 2 þ μð0Þð  Δu0 ; u00lm ð0ÞÞ þ ðgð; u0 Þ; u00lm ð0ÞÞ ¼ ðf ð0Þ; u00lm ð0ÞÞ: This implies j u00lm ð0Þj 2 r C for all l Z l0 and for all m A N. From this, (3.16) and Lemma 3.1 we obtain for all l Zl0 and for all m A N, that Z t J u00lm ðτÞ J 2Γ 1 dτ rC in ½0; T: ð3:17Þ J u0lm ðtÞ J rC; j u00lm ðtÞj r C; 0

3.2. Passage to the limit in m From estimates (3.13), (3.14), (3.17) and diagonal process allow us to find a function ul and subsequences of ðulm Þ, which will still be denoted by ðulm Þ, such that ulm -ul weak star in L1 ð0; 1; VÞ; u0lm -u0l weak star in L1 ð0; 1; L2 ðΩÞÞ; u0lm -u0l weak star in L1 loc ð0; 1; VÞ;

u0lm -u0l weak in L2 ð0; 1; L2 ðΓ 1 ÞÞ; u00lm -u00l weak in L2loc ð0; 1; L2 ðΓ 1 ÞÞ:

ð3:18Þ

The convergence on the linear terms of (3.3) are directly obtained from (3.18). The limit on the non-linear terms are analyzed as follows. From the convergence (3.18)1, (3.18)2 and Aubin– Lions' compactness theorem, we obtain ulm ðx; tÞ-ul ðx; tÞ a.e. in Ω  ð0; TÞ. This and the regularity of g ensure that gðx; ulm ðx; tÞÞgðx; ul ðx; tÞÞ a:e: in Ω  ð0; TÞ. R Now, the inequalities (2.2), (2.9) and (3.13) imply Ω j gð; ulm Þj 2 dx r a1 k2 J ulm J 2ρ rC. Therefore, from Lions' lemma [12, Chapter I, Lemma 1.3] and diagonal process, it yields gð; ulm Þ-gð; ul Þ weak in L2loc ð0; 1; L2 ðΩÞÞ:

ð3:19Þ

Due to (3.18)1 we have ulm -ul weak star in L ð0; 1; H ðΓ 1 ÞÞ. From this, (3.18)5 and Aubin–Lions' compactness theorem, ulm ðx; tÞ -u R l ðx; tÞ a.e. 2in Γ 1  ð0; TÞ. Clearly, (2.3) and (2.10)2 imply Γ 1 j qð; ulm ðtÞÞj dΓ r C, and so we conclude 1

qð; ulm Þ-qð; ul Þ weak in L2loc ð0; 1; L2 ðΓ 1 ÞÞ:

1=2

ð3:20Þ

From the convergence (3.18)3 and proceeding as in (3.20), we find hl ð; u0lm Þ-hl ð; u0l Þ weak in L2loc ð0; 1; L2 ðΓ 1 ÞÞ:

ð3:21Þ

Therefore, convergence (3.18), (3.19)–(3.21) allows us to pass to the limit in approximate equation (3.3), and as V \ H 2 ðΩÞ is dense in V, we obtain Z 1 Z 1 ðu00l ðtÞ; vÞθðtÞ dt þ μðtÞððul ðtÞ; vÞÞθðtÞ dt 0 0 Z 1 Z 1   þ ðgð; ul ðtÞÞ; vÞθðtÞ dt þ μðtÞ hl ð; u0l ðtÞÞ; v Γ1 θðtÞ dt 0 0 Z 1 Z 1 μðtÞðqð; ul ðtÞÞ; vÞΓ1 θðtÞ dt ¼ ðf ðtÞ; vÞθðtÞ dt; ð3:22Þ þ 0

0

1 for all v A V and for all θ A C 1 0 ðΩÞ. If in (3.22) we take v A H 0 ðΩÞ and observe the regularities of ul ″, gð; ul Þ and f , we find

u00l  μΔul þ gð; ul Þ ¼ f

in L2loc ð0; 1; L2 ðΩÞÞ:

ð3:23Þ

Using (3.23) and (3.18)1, we have Δul A L ð0; 1; L ðΩÞÞ and ul A L1 ð0; 1; VÞ respectively, and thus 1

∂ul  1=2 A L1 ðΓ 1 ÞÞ: loc ð0; 1; H ∂ν

2

ð3:24Þ

Multiplying both sides of (3.23) by vθ with v A V and θ A C 1 0 ð0; 1Þ, integrating on Ω  ð0; 1Þ and using (3.24), we obtain Z 1 Z 1 ðu00l ðtÞ; vÞθðtÞ dt þ μðtÞððul ðtÞ; vÞÞθðtÞ dt 0 0  Z 1 Z 1 ∂ul ðtÞ ðgð; ul ðtÞÞ; vÞθðtÞ dt  μðtÞ ; v θðtÞ dt þ ∂ν 0 0 Z 1 ðf ðtÞ; vÞθðtÞ dt; v A V; 8 θ A C 1 ¼ 0 ðΩÞ; 0

where 〈; 〉 denotes the duality paring between H  1=2 ðΓ 1 Þ and H 1=2 ðΓ 1 Þ. Comparing the above equation with (3.22), and observing the regularities of hl ð; u0l Þ and qð; ul Þ, we find ∂ul þhl ð; u0l Þ þ qð; ul Þ ¼ 0 ∂ν

in L2loc ð0; 1; L2 ðΓ 1 ÞÞ:

ð3:25Þ

3.3. Passage to the limit in l From the estimates (3.13), (3.14), (3.17) and convergence (3.18), we get for all l Z l0 that  1=2 2 K 0 for all t A ½0; 1Þ; j u0l ðtÞj r K 0 ; J ul ðtÞ J r

μ0

M. Milla Miranda et al. / International Journal of Non-Linear Mechanics 82 (2016) 6–16

Z

1 0

j u0l ðtÞj 2Γ 1 dt r K 1

13

without restrictions on the sizes of the norms of the initial data. Therefore, to obtain an estimate like (3.11) we had to do this.

for all t A ½0; 1Þ;

J u0l ðtÞ J r C; j u00l ðtÞj r C for all t A ½0; T; Z t j u00l ðτÞj 2Γ 1 dτ r C for all t A ½0; T:

3.4. Some remarks on Theorem 2.2 ð3:26Þ

0

These estimates allow to obtain similar convergence as those obtained in (3.18). So there exists a function u and subsequences of ðul Þ, which will still be denoted by ðul Þ such that the convergence in (3.18) hold true if ulm is replaced by ul and ul by u. Employing the same arguments as in (3.19) and (3.20), we get gð; ul Þ-gð; uÞ weak in L2loc ð0; 1; L2 ðΩÞÞ; qð; ul Þ-qð; uÞ weak in L2loc ð0; 1; L2 ðΓ 1 ÞÞ:

ð3:27Þ

To prove Theorem 2.2 we apply similar arguments used in the proof of Theorem 2.1. In this case, we replace the function J of (3.7) by 1 2 ρþ1 J 1 ðλÞ ¼ μ0 λ  M λ 4

for λ Z 0;

ð3:35Þ

and M is the same constant of (2.21). In the sequel, we list some results obtained in the proof of Theorem 2.2 which will be used in the study of the uniform stability of the energy. Similar arguments used in the proof of Lemma 3.3, which yield

In the sequel, we refer the convergence (3.18), but understanding that we are using ul and u in the place of ulm and ul , respectively. The convergence (3.27)1, (3.18)4 and Eq. (3.23) imply

J 1 ðλÞ Z0 for λ A ½0; λ1 

Δul -Δu weak in L2loc ð0; 1; L2 ðΩÞÞ;

Let ðpl Þ be the approximations [see Lemma 3.2] of the function p, with p satisfying hypothesis (2.23). Under hypothesis (2.23), we obtain (cf. in [15])

ð3:28Þ

and so u00  μΔu þ gð; uÞ ¼ f

in L2loc ð0; 1; L2 ðΩÞÞ:

ð3:29Þ

We also have from the convergence (3.18)1 and (3.28) that ∂ul ∂u - weak in L2loc ð0; 1; H  1=2 ðΓ 1 ÞÞ: ∂ν ∂ν

ð3:30Þ

Next, we analyze the convergence of the term hl ð; ul Þ by means of Strauss' approximations, see [24]. In fact, we fix T 4 0 and by convergence (3.18)3, 5 and Aubin–Lions' compactness theorem, we obtain u0l ðx; tÞ-u0 ðx; tÞ a.e. in Γ 1  ð0; TÞ. Note that, if we suppose ðx; tÞ fix in Γ 1  ð0; TÞ then the sets fu0l ðx; tÞ; l Z l0 g are bounded, and thus the properties on hl imply that hl ðx; u0l ðx; tÞÞ-hðx; u0 ðx; tÞÞ a:e:

in Γ 1  ð0; TÞ:

ð3:31Þ

Furthermore, from Eqs. (3.23) and (3.25), we obtain Z Z T Z T   1 T 0 1 μ hl ð; u0l Þ; u0l Γ1 dt ¼ ðf ; u0l Þ dt þ μ J ul J 2 dt  j u0l ðTÞj 2 2 2 0 0 0 1 1 2 μðTÞ μ ð0Þ J ul ðTÞ J 2 þ J u0 J 2 þ j ul j  2 2 2 Z T Z T    ðgð; u0l Þ; u0l Þ dt  μ qð; ul Þ; u0l Γ dt: 0

0

1

Combining (3.31), (3.32), Strauss' approximations and diagonal process hl ð; u0l Þ-hð; u0 Þ

in L1loc ð0; 1; L1 ðΓ 1 ÞÞ:

ð3:33Þ

Taking the limit in Eq. (3.25) and using convergence (3.27), (3.30) and (3.33), we obtain ∂u þhð; u0 Þ þ qð; uÞ ¼ 0 ∂ν

in L1loc ð0; 1; L1 ðΓ 1 ÞÞ:

and

J ul ðtÞ J o λ1 for all t A ½0; 1Þ: n

ð3:36Þ

j pl ðsÞj r

3p1 j sj 2

for all s A R:

ð3:37Þ

As in the proof of Theorem 2.1 [see Section 3.2], we obtain ul in the class (2.15), and that ul ″  μΔul þ j ul j ρ ¼ 0 in L2loc ð0; 1; L2 ðΩÞÞ; ∂ul þ ðm  νÞpl ðu0l Þ ¼ 0 in L2loc ð0; 1; L2 ðΓ 1 ÞÞ: ∂ν

ð3:38Þ

As u0l A L2loc ð0; 1; H 1=2 ðΓ 1 ÞÞ we have ðm  νÞpl ðu0l Þ A L2loc ð0; 1; H 1=2 ðΓ 1 ÞÞ (see [16] and results of interpolation in Hilbert spaces). This and Eqs. (3.38) imply that ul is solution of the problem  Δu l ¼ b f in Ω  ð0; 1Þ; ul ¼ 0 on Γ 0  ð0; 1Þ; ∂ul b on Γ 1  ð0; 1Þ; ¼g ∂ν b A L2loc ð0; 1; H 1=2 ðΓ 1 ÞÞ. Therefore, where b f A L2loc ð0; 1; L2 ðΩÞÞ and g using results on regularity of the solutions of elliptic problems, we obtain ul A L2loc ð0; 1; V \ H 2 ðΩÞÞ:

ð3:39Þ

1

Taking into account, this identity, estimates (3.26), hypotheses (2.4) and (2.13), and convergence (3.1), we find Z T   hl ð; u0l Þ; u0l Γ dt r C for all l Z l0 : ð3:32Þ 0

n

ð3:34Þ

The initial conditions in (2.16) are obtained directly using the convergences (3.18). The results obtained in (3.29) and (3.34) complete the proof of Theorem 2.1. Remark 3.1 (Local existence in time). It is worth noting that the R term Ω gð; uÞu0 dx does not have a definite sign. This fact brings serious difficulties to obtain global solutions for problem (1.1)

3.5. Proof of Theorem 2.3 The strategy to prove Theorem 2.3 is to show that the energy Z 1 μðtÞ 1 J ul ðtÞ J 2 þ j u ðtÞj ρ ul ðtÞ dx ð3:40Þ El ðtÞ ¼ j u0l ðtÞj 2 þ 2 2 ρþ1 Ω l associated with the solutions ul of the equations in (3.38) satisfying the inequality (2.25). Consequently, the uniform stability of EðtÞ (which is defined in (2.24)) will be obtained by taking the inferior limit in El ðtÞ as l-1. We suppose a special geometry on the subsets Γ 0 and Γ 1 . Throughout this section we assume that both partitions Γ 0 and Γ 1 of Γ have positive measure, Γ 0 \ Γ 1 ¼ ∅, and that there exists x0 A Rn such that

Γ 0 ¼ fx A Γ ; mðxÞ  νðxÞ r 0g and Γ 1 ¼ fx A Γ ; mðxÞ  νðxÞ 4 0g; where mðxÞ : ¼ x  x0 . The hypotheses (2.4), (2.20) and (2.23) imply 1 r μ0 r μðtÞ r μð0Þ

for all t A ½0; 1Þ:

ð3:41Þ

Now, we introduce the function

θðtÞ ¼ 2μ0 ðu0l ðtÞ; m  ∇ul ðtÞÞ þ μ0 ðn  1Þðu0l ðtÞ; ul ðtÞÞ

ð3:42Þ

14

M. Milla Miranda et al. / International Journal of Non-Linear Mechanics 82 (2016) 6–16

and, for ε 4 0, it defines the perturbed energy Eε ðtÞ : ¼ El ðtÞ þ εθ ðtÞ

for all t Z0:

Moreover, we consider the function Z μðtÞ 1 J ul ðtÞ J 2 þ j u ðtÞj ρ ul ðtÞ dx: En ðtÞ ¼ 2 ρþ1 Ω l

ð3:43Þ

ð3:44Þ

In the sequel, to simplify the notation we will not write the variable t and neither the subscript l. 3.5.1. Equivalence of the energies E and Eε We take the scalar product of L2 ðΩÞ on both sides of (3.38)1 with u0 , using Eq. (3.38)2 and hypotheses on p and m  ν, we obtain E0 r  τ0 μ0 p0 j u0 j 2Γ 1 :

ð3:45Þ

We derive from (3.42) that j θ j R r 2μ0 Rj u0 j J u J þ μ0 r 0 ðn  1Þj u0 j J u J :

ð3:46Þ

The inequalities in (3.36) imply 1 μ J u J 2  M J u J ρ þ 1 Z0 4 0

for all t A ½0; 1Þ;

ð3:47Þ

and from this, we have ρþ1

k 1 μ JuJ2  0 JuJρþ1 Z0 4 0 ρþ1

for all t A ½0; 1Þ:

ρþ1

k 1 1 μ J u J 2 r μ0 J u J 2  0 J u J ρ þ 1 : 4 0 2 ρ þ1 Note also that ρþ1

k 1 μ J u J 2  0 J u J ρ þ 1 r En : 2 ρ þ1 These last two inequalities imply

2

μ1=2 0

ðEn Þ1=2 r

2

μ1=2 0

E1=2

Thus

Z Z ∂u ðm  ∇uÞ dΓ  μ0 μ ðm  νÞj ∇uj 2 dΓ þ 2μ0 μ Γ Γ ∂ν   Z Z ∂u 2 ¼  μ0 μ ðm  νÞ dΓ  μ0 μ ðm  νÞj ∇uj 2 dΓ ∂ν Γ0 Γ1  2 Z Z ∂u ∂u þ2μ0 μ ðm  νÞ dΓ þ2μ0 μ ðm  ∇uÞ dΓ : ∂ν Γ0 Γ 1 ∂ν

From Eq. (3.38)2 and inequality (3.37), we obtain
Z

Z




∂u 0
2
¼
2
ðm  ∇uÞ d Γ ðm  ν Þpðu Þðm  ∇uÞ d Γ



∂ ν Γ1 Γ1  2 Z Z 3p1 j u0 j 2 dΓ þ ðm  νÞj ∇uj 2 dΓ : r R3 2 Γ1 Γ1

for all t A ½0; 1Þ:

On the other hand, from (3.40) we get j u0 j r 21=2 E1=2 for all t A ½0; 1Þ. Combining the two preceding inequalities with (3.46) it yields j θ j r SE, where S is defined in (2.26), and this implies  εSE r εθ r εSE. And thus we get the desired inequalities

3.5.2. Differential inequality Differentiating the function θ, it yields

θ0 ¼ 2μ0 ðu00 ; m  ∇uÞ þ 2μ0 ðu0 ; m

The last three expressions above provide

ð3:48Þ

 ∇u0 Þ þ μ0 ðn  1Þðu00 ; uÞ þ μ0 ðn 1Þj u0 j 2 :

I 2 r 2μ0

From Eq. (3.38)1, we find

n

ρ þ1

ρþ1

k0

J u J ρ þ 1 þ 2μ0

R

ρþ1

ð3:49Þ

ρþ1

k5

3. From Green's theorem and since obtain

θ0 ¼ 2μ0 μðΔu; m  ∇uÞ  2μ0 ðj uj ρ ; m  ∇uÞ þ 2μ0 ðu0 ; m  ∇u0 Þ þ μ0 ðn  1ÞμðΔu; uÞ  μ0 ðn  1Þðj uj ρ ; uÞ þ μ0 ðn  1Þj u0 j 2 ≕I 1 þ ⋯ þ I6 :

∂mj ∂xj

I 3 r  μ0 nj u0 j 2 þ μ0 Rj u0 j 2Γ 1 :

Our goal is to derive a bound above for each terms on the righthand side of (3.49). 1. From Rellich's identity, see [7], it yields Z I 1 ¼ μ0 μðn 2Þ J u J 2  μ0 μ ðm  νÞj ∇uj 2 dΓ

J u J ρ þ 1:

ð3:54Þ

¼ 1 and u0 ¼ 0 on Γ 0 , we ð3:55Þ R

4. From (3.38)2 we get ðΔu; uð ¼  J u J þ Γ 1 ðm  νÞpðu Þ u dΓ . As
Z
 Z

3
ðm  νÞpðu0 Þ dΓ

r R p1 j u0 j u dΓ
2 Γ1 Γ1 R  2 1 3 1 J u J 2; r R2 p1 r 21 ðn  1Þμð0Þj u0 j 2Γ 1 þ 2 2 2ðn  1Þ 2

Γ

ð3:52Þ

2. Since u ¼ 0 on Γ 0 , we have Z n Z X 1 ∂ j uj ρ ðm  ∇uÞ dΓ ¼ mi ð j uj ρ uÞ dx ∂x ρ þ 1 i Ω Ω i¼1 Z Z n 1 ¼ j uj ρ u dx þ ðm  νÞj uj ρ u dΓ : ρþ1 Ω ρ þ 1 Γ1 Using the inequalities of (2.9)1 and (2.9)2,

Z
n
n ρþ1 ρþ1 ρ

;
ρ þ 1 j uj u dx
r ρ þ 1k0 J u J Ω R

Z
1
R ρþ1
k ðm  νÞj uj ρ u dΓ

r J u J ρ þ 1:
ρ þ 1 ρþ1 5 Γ1 R

1 3 1 E r Eε r E for all t A ½0; 1Þ and 0 o ε r : 2 2 2S

ð3:51Þ

Thus, observing that m  ν r 0 on Γ 0 and reducing similar terms, we get   3p1 2 0 2 I 1 r μ0 μðn  2Þ J u J 2 þ μ0 μð0ÞR3 j u j Γ1 : ð3:53Þ 2

for all t A ½0; 1Þ:

Thus, JuJ r

and  2 Z Z Z ∂u ∂u ∂u ðm  ∇uÞ dΓ ¼ ðm  νÞ dΓ þ ðm  ∇uÞ dΓ : ∂ν Γ ∂ν Γ0 Γ 1 ∂ν

Plugging (3.51) and (3.52) in (3.50), it yields  2 Z ∂u ðm  νÞ dΓ I 1 r μ0 μðn  2Þ J u J 2  μ0 μ ∂ν Γ0  2 Z Z ∂u  μ0 μ ðm  νÞj ∇uj 2 dΓ þ 2μ0 μ ðm  νÞ dΓ ∂ν Γ1 Γ0   Z Z 3p1 2 þ μ0 μR3 j u0 j 2 dΓ þ μ 0 μ ðm  νÞj ∇uj 2 dΓ : 2 Γ1 Γ1

Consequently,

1 μ J u J 2 r En 4 0

Z ∂u þ 2μ0 μ ðm  ∇uÞ dΓ : ð3:50Þ Γ ∂ν  2 Since j ∇uj 2 ¼ ∂u and m  ∇u ¼ ðm  νÞ∂u ∂ν ∂ν on Γ 0 ,  2 Z Z Z ∂u ðm  νÞj ∇uj 2 dΓ ¼ ðm  νÞ dΓ þ ðm  νÞj ∇uj 2 dΓ ∂ν Γ Γ0 Γ1

0

M. Milla Miranda et al. / International Journal of Non-Linear Mechanics 82 (2016) 6–16

15

4. Additional comments and open problems

then  

2 μ μð0Þ 3 μ μ R p1 r 1 ðn  1Þ j u0 j 2Γ 1 þ 0 J u J 2 : I 4 r  μ0 μðn  1Þ J u J 2 þ 0 2 2 2 ð3:56Þ 5. From (2.9) it follows that ρþ1

I 5 r μ0 ðn  1Þk0

J u J ρ þ 1:

ð3:57Þ

Taking into account (3.53)–(3.57) in (3.49) and reducing similar terms, we obtain 1 2

θ0 r  μ0 j u0 j 2  μ0 μ J u J 2 þ L J u J ρ þ 1 þ Pj u0 j 2Γ1 ; where the constants L and P were introduced in (2.21) and (2.27), respectively. From (3.41) we have  μ0 r  1=4. Thus 1 4

1 2

θ0 r  j u0 j 2  μ0 μ J u J 2 þ L J u J ρ þ 1 þ Pj u0 j 2Γ 1 :

ð3:58Þ

Next, we will show that 1 1  μ0 μ J u J 2 þ L J u J ρ þ 1 r  E n ; 2 2 that is, 1 2

1 4

ψ : ¼ μ0 μ J u J 2  μ J u J 2  L J u J ρ þ 1 

ð3:59Þ Acknowledgment Z 1 j uj ρ u dx Z 0: 2ðρ þ 1Þ Ω

In fact, since μ0 Z 1 then 14μ0 þ 14μ r 12μ0 μ. Therefrom, 1 1 1 μ μ J u J 2  μ J u J 2 Z μ0 J u J 2 : 2 0 4 4 Furthermore, Z ρþ1 k 1 J u J ρ þ 1:  j uj ρ u dx Z  0 2ðρ þ 1Þ Ω 2ðρ þ 1Þ

1 4

ρþ1

k0 J u J ρ þ 1; 2ðρ þ 1Þ

ð3:60Þ

and inequality (3.47) provides ρþ1

k 1 J u J ρ þ 1 Z 0: μ JuJ2 LJuJρþ1  0 4 0 2ðρ þ 1Þ

ð3:61Þ

Finally, combining (3.60) and (3.61) we obtain (3.59). Now, inequalities (3.58) and (3.59) permit us to write 1 4

1 2

θ0 r  j u0 j 2  En þ Pj u0 j 2Γ1 : 0

As 14 j u0 j 2 þ 12En ¼ 12E then θ r  12E þ Pj u0 j 2Γ 1 . From this and (3.45), it follows

ε

E0ε r  τ0 p0 j u0 j 2Γ 1  E þ εPj u0 j 2Γ 1 ; 2 and thus

ε

E0ε r E 2

for all 0 o ε r

τ0 μ0 p0 P

:

ð3:62Þ

Note that the choice of η in (2.26) implies that (3.48) and (3.62) hold simultaneously for this η. Thus, from (3.48) we have  2ηE r  η3Eη . Consequently, using (3.62), we obtain E0η r  η3Eη . This η

give us that Eη ðtÞ r3e  3t Eη ð0Þ. From this inequality and (3.48) we have the desired inequality η

El ðtÞ r 3e  3t El ð0Þ

for all t A ½0; 1Þ:

With this we conclude the proof of Theorem 2.3.

(a) We would like to thank the professor L.A. Medeiros by gentle suggestion of this problem and also for his important comments. (b) We also would like to thank the anonymous referees by the careful reading and helpful suggestions which led to a substantial improvement of our original manuscript.

References

Therefore,

ψ Z μ0 J u J 2  L J u J ρ þ 1 

(a) The uniform stability of the energy of system (1.1) – Theorem 2.3 – perhaps might be proved by using Lasiecka's strategies contained in [9]. In fact, if in system (1.1) the function μðtÞ ¼ 1 for all t we think that Lasiecka's technique is applicable, with this it is possible to use the approach via theory of non-linear semigroups done in [9]. If all goes well, the probable results could be improved supposing geometric restriction only on the set Γ 0 , as it was done in Lasiecka and Tataru [10]. This approach provides a good work. (b) A natural question that arises is: is it possible to remove the hypotheses on the sizes of the norms of the initial data and still establish global solutions for system (1.1)? So far, this is an unanswered question. (c) It is natural to try to extend the known results to systems like (1.1) considering the coefficients μ mentioned in (1.4) and (1.5). They are more challenging problems of what have been considered in the literature and well physically consistent.

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