EXISTENCE AND UNIFORM DECAY ESTIMATES FOR THE FOURTH ORDER WAVE EQUATION WITH NONLINEAR

. In this paper, we consider the initial boundary value problem for the fourth order wave equation with nonlinear boundary velocity feedbacks f 1 ( u νt ), f 2 ( u t ) and internal source | u | ρ u . Under some geometrical conditions, the existence and uniform decay rates of the solutions are proved even if the nonlinear boundary velocity feedbacks f 1 ( u νt ), f 2 ( u t ) have not polynomial growth near the origin respectively. By the combination of the Galerkin approximation, potential well method and a special basis constructed, we ﬁrst obtain the global existence and uniqueness of regular solutions and weak solutions. In addition, we also investigate the explicit decay rate estimates of the energy, the ideas of which are based on the construction of a special weight function φ ( t ) (that depends on the behaviors of the functions f 1 ( u νt ), f 2 ( u t ) near the origin), nonlinear integral inequality and the Multiplier method. Γ

For the linear second order wave equations with nonlinear boundary feedback, there is an abounding literature about its initial boundary value problem. In [43], Zuazua studied the following second order wave equation , (x, t) ∈ Γ 1 × (0, ∞), u(x, 0) = u 0 , u t (x, 0) = u 1 , x ∈ Ω, (1.2) where x 0 is a fixed point in R n , and m(x) = x − x 0 . When f (y) = |y| p on [0,1] for some p ≥ 1, he proved that the energy decays exponentially if p = 1 and polynomially if p > 1. In the later case, he gave that there exists a positive constant C such that When the nonlinear boundary velocity feedback f (y) is weaker than any polynomial near the origin, for instance, ∀ y ∈ (0, 1), f (y) = e −1/y . Lasiecka and Tataru [20] showed that the energy of solutions decays with the following rate: where S(t) is the solutions (contraction semigroup) of the differential equation d dt S(t) + q(S(t)) = 0, S(0) = E(0), (1.5) and q is closely related to the behavior of the feedback f (y) near the origin. They were the first to consider that the energy decay rate estimates associated to the solutions of some differential equation and without assuming that the feedback has a polynomial behavior near the origin. Martinez [26] complemented Lasiecka and Tataru's work in [20] concerning the linear wave equation subject to nonlinear boundary feedback. He proved that the energy of problem (1.5) decays to zero with an explicit decay rate estimates. The process of the proof relies on the construction of some special weight functions and some nonlinear integral inequalities. The method presented in [26], gives us a variety of explicit decay rate estimates, although in some simple cases a direct application of the above method doesn't give us optimal decay rates. For instance, when f (y) = y p , p > 1, by the method of [26], the energy decay is given by E(t) ≤ C(1 + t) −2/p , which is less good estimate than the estimate of (1.3). In spite of this, it is possible to obtain optimal decay rate estimates by this method for some other example, see [26] for details.
(1. 6) He showed that the presence of the superlinear damping term −|u t | m−2 u t , when 2 ≤ p ≤ m, implies the global existence of solutions for arbitrary initial data, in opposition with the nonexistence phenomenon occurring when m = 2 < p. Zhang and Hu [42] proved the asymptotic behavior of the solutions of problem (1.6), where the initial data is inside a stable set. The blow up phenomenon of the solutions occurs when the initial data is inside an unstable set. More results on the second order wave equations with nonlinear boundary source and damping terms, the reader can see [2,3,25] and papers cited therein.
It is worth mentioning that the potential well theory (stable or unstable sets) is a very important and popular way to study the qualitative properties of nonlinear evolution equations. This method was first introduced by Sattinger [30] to investigate the global existence of solutions for nonlinear hyperbolic equations. Hence, it has been widely used and extended by many authors to study different kinds of evolution equations, we refer the reader to see [6,7,30,[34][35][36][38][39][40] and references therein.
Let us mention some known results about the second order wave equations with nonlinear internal damping and source terms u tt − u + g(u t ) = f (u), (x, t) ∈ Ω × (0, ∞). (1.7) Geogev and Todorova [13] investigated the initial boundary value problem of equation (1.7), where g(u t ) = |u t | m−1 u t , f (u) = |u| p−1 u. They proved the existence of global solutions under the condition 1 < p ≤ m. When p ≥ m > 1, they also obtained the finite time blow up of solutions for sufficient large initial data. Ikehata [15] studied the initial boundary value problem of equation (1.7), where g(u t ) = δ|u t | m−1 u t and f (u) = |u| p−1 u. He proved that 1 ≤ m < p < ∞ if n = 1, 2, and 1 ≤ m < p ≤ n n−2 if n ≥ 3, the problem has a global solution for sufficiently small initial data. When g(u t ) = au t (1 + |u t | m−2 ), f (u) = b|u| p−2 u, Messaoudi [1,27] investigated the global existence and exponential decay behavior of solutions respectively.
For the second order wave equations with nonlinear internal source and boundary velocity feedback, Cavalcanti et al. [4] studied the following initial boundary value problem          u tt − u = |u| p u, (x, t) ∈ Ω × (0, ∞), u = 0, (x, t) ∈ Γ 0 × (0, ∞), u ν = −f (u t ), (x, t) ∈ Γ 1 × (0, ∞), u(x, 0) = u 0 , u t (x, 0) = u 1 , x ∈ Ω. (1.8) They proved the existence of global solutions and uniform decay rate estimates of the energy provided that the nonlinear boundary feedback f (u t ) has not a polynomial growth near the origin by using the potential well method and the Galerkin approximation. When f (u t ) = −α(x)|u t | m−2 u t or f (u t ) = −α(x)(|u t | m−2 u t + |u t | µ−2 u t ), 1 ≤ µ ≤ m, and α(x) ∈ L ∞ (Γ 1 ), α(x) ≥ 0, Vitillaro [31] extended the potential well theory. He obtained the local existence, blow up and global existence results of solutions. More results on the initial boundary value problem for the wave equations with nonlinear internal source and boundary velocity feedback, we refer readers to see (Di and Shang [9], Feng and Li [11,12], Liu, Sun and Li [24]) and the papers cited therein.
There are some literature on the initial boundary value problem or Cauchy problem for the fourth order wave equations with source and damping terms in the interior of Ω Guesmia [14] investigated the initial boundary value problem of equation (1.9). He obtained a global existence and a regularity result and proved that the solutions decay exponentially if g(y) behaves like a linear functions. For more results on the qualitative problem of the fourth order wave equations with interior source and damping terms, the reader is referred to see [8,37,41] and references therein.
When people studied the small transversal vibrations of a thin plate (Lagnese and Lions [17], Lagnese [18]) and the strong or uniform stabilization of different plate and beam models (Lasiecka [19], Puel and Tucsnak [28]), some nonlinear evolution equations with the main part u tt + 2 u = 0 and different nonlinear boundary feedbacks were obtained. For example, Komornik [16] studied the following evolutionary problem: where µ ∈ (0, 1), l ∈ C 1 (Γ 1 ), and f : R → R is a non-decreasing, continuous function. The subscripts ν and τ stand for the normal and tangential derivatives to Γ 0 and Γ 1 . He proved the global existence, regularity results and gave some stabilization properties for problem (1.10) by using the Multiplier method. It is worth mentioning that the Multiplier method has already been used by many authors for different reasons, we also refer to the related papers [5,16,21] about the Multiplier method. Motivated by the above results, in the present work we study the initial boundary value problem of the fourth order wave equation with an internal nonlinear source |u| ρ u, and nonlinear boundary velocity feedbacks f 1 (u νt ), f 2 (u t ). As far as we know, there is little information on the well-posedness and energy decay estimates for problem (1.1). Naturally, our attention of this paper is paid to the study of the related qualitative properties to problem (1.1). Here, when the boundary velocity feedbacks f 1 (u νt ), f 2 (u t ) have not the polynomial behaviour near the origin for wave equation supplemented with an interior source |u| ρ u acting in the domain, we first investigate the global existence, uniqueness of regular solutions and weak solutions by the combination of Galerkin approximation, potential well method and a special basis constructed. In addition, we also prove that the energy of problem (1.1) decays uniformly to zero, which is based on a weight function φ(t) constructed, Multiplier method and nonlinear integral inequality.
Our paper is organized as follows. In Section 2, we introduce some potential wells, basic definitions, important lemmas, and main results of this paper. In Section 3-4, we show the global existence and uniqueness of the regular solutions and weak solutions respectively. In the last Section, we investigate the explicit decay rate estimates of the energy.

Preliminaries and main results
In order to state our results precisely, we first introduce some notations, basic definitions, important lemmas and some functional spaces.
Let Ω be a bounded domain of R n with C 4 boundary Γ and x 0 be a fix point in R n . We shall define and introduce a partition of the boundary Γ such that Throughout this paper, the following inner products and norms are used for precise statement: and the Hilbert space Since Γ 0 has positive (n − 1) dimensional Lebesgue measure, by Poincaré inequality, we can endow V with the equivalent norm u V = u 2 (see [22]). To obtain the results of this paper, let us consider the potential energy and total energy associated to the solutions of problem (1.1). We may define the (positive) number which is also called the depth of the potential well. Moreover, the value d is shown to be the Mountain pass level associated to the elliptic problem Here, let B 1 > 0 be the optimal constant of Sobolev imbedding from V into L ρ+2 (Ω), which satisfies the inequality u ρ+2 ≤ B 1 u 2 , ∀ u ∈ V . From this inequality, we discover that and the function We can easily see (the simple proof can be founded in [32]) that where λ 1 is the absolute maximum point of function f . Now, we will give some basic hypotheses to establish the main results of this paper.
(A1) Suppose that 0 < ρ < 4 n−4 , if n ≥ 5 and ρ > 0, if n = 1, 2, 3, 4. Then, we have the following Sobolev imbedding Assumptions on the functions f i (i = 1, 2) : f i : R → R are nondecreasing C 1 functions such that f i (0) = 0. In addition, there exist some strictly increasing and odd functions g i of C 1 class on [−1, 1] satisfy where g −1 i (s) denote the inverse functions of g i (s) and C i1 , C i2 are positive constants.
In order to obtain the global existence of regular solutions, we shall need the following additional hypotheses.
(A3) Assumptions on the initial data: let us consider satisfying the compatibility conditions Moreover, assume that (A4) E(0) < d and u 0 2 < λ 1 . The next lemma will play an essential role for proving the global existence of regular (weak) solutions of problem (1.1).
Multiplying the equation in (1.1) by u t (t), a direct computation gives that By the hypotheses that f i are nondecreasing C 1 functions such that f i (0) = 0, we know that f i (s)s > 0 for s = 0. Hence, from the definition of E(t), it follows that , from the hypotheses (A4) we have λ 0 < λ 1 . Furthermore, by (2.14), we have f (λ 0 ) ≤ E(0), which together with f is increasing in [0, λ 1 ) and f (λ 2 ) = E(0), it is easy to see that λ 0 = u 0 2 < λ 2 . Next, we prove that u(t) 2 ≤ λ 2 for all t ≥ 0. In deed, by contradiction, suppose that u(t 0 ) 2 > λ 2 for some t 0 ≥ 0. Using the continuity of u(t) 2 , we also may suppose that u(t 0 ) 2 < λ 1 . Thus, by (2.14) again, we see that which contradicts (2.16). This completes the proof of Lemma 2.1.
The following two technical lemmas are very crucial to derive the asymptotic behavior of the energy to problem (1.1).
Lemma 2.2. Let E : R + → R + be a non-increasing function and φ : R + → R + a strictly increasing function of C 1 class such that φ(0) = 0 and φ(t) → +∞ as t → +∞. (2.18) Suppose that there exist σ > 0, σ ≥ 0 and C > 0 such that Then, there exists C > 0 such that Remark 2.1. Note that the above integral inequality was first introduced in Martinez [26], was used in Cavalcanti et al. [4] to prove the decay rate estimates of energy. Lemma 2.3. There exists a strictly increasing function φ : R + → R + of C 2 class on (0, +∞), and such that the following conditions hold where the functions g −1 i (s) (i = 1, 2) were introduced in assumption (A2). Proof. These properties of the function φ are closely related to the behaviors of f i (i = 1, 2) near 0. We will present the construction method of a special weight function φ in Section 5. Now, we are ready to state the main results of this paper.
for all t ≥ 0. Further, the following energy identity holds where the total energy E(t) has been defined by (2.2).
for all t > 0. Besides, the weak solution has the same energy identity given as (2.24).
where the function G(y) = y g1(y)g2(y) g1(y)+g2(y) and the constant C only depending on the initial data E(1) in a continuous way.
Remark 2.2. By a direct calculation, we can show that the G(y) = y g1(y)g2(y) g1(y)+g2(y) is an increasing function. Remark 2.3. we also extend the decay rate estimate of regular solutions to the weak solutions of problem (1.1) by using the standard arguments of density.

Existence, uniqueness of regular solutions
In this section, we study the global existence and uniqueness of regular solutions of problem (1.1) by using the combination of the Galerkin approximation, potential well method and a special basis constructed.
The proof of Theorem 2.1 is divided into five steps. Proof.
The main idea is to use the Galerkin's method. To do this, let us take a basis {w * j } to V . We construct a special basis {w j } from basis {w * j } which are associated with problem (1.1).
If u 0 , u 1 are linearly independent, we take w 1 = u 0 , w 2 = u 1 , and w i , i ≥ 3 of {w * j }, which are chosen to be linearly independent with u 0 , u 1 . If u 0 , u 1 are linearly dependent, we define w 1 = u 0 , and w i , i ≥ 2 of {w * j }, which are chosen to be linearly independent with u 0 . Thus, we represent by V m a subspace of {w j } generated by [w 1 , · · · , w m ].
Next, we construct an approximate solution of problem (1.1) by According to Galerkin's method, these coefficients d j m (t) need to satisfy the following initial value problem of the nonlinear ordinary differential equation Note that we can solve system (3.2) by Picard's iteration method. In fact, the ordinary differential equation (3.2) has a local solution on the interval [0, T m ). The extension of these solutions to the whole interval [0, +∞) is a consequence of a priori estimate which we are going to prove below.
Step 2. The first estimate.
Multiplying (3.2) by d j m (0), summing for j = 1, 2, · · · · · · , and considering t = 0, then we have Using the generalized Green Theorem, it follows that By Hölder inequality and the compatibility condition (A3), we discover that Differentiating equation in (3.2) with respect to t, and substituting w j by u m tt , we deduce that 1 2 We will give the estimate of K 1 = (ρ + 1) Ω |u m | ρ |u m t ||u m tt |dx. From now on, we will denote by C various positive constants which may be different at different occurrences.
In view of the generalized Hölder inequality ( ρ 2(ρ+1) + 1 2(ρ+1) + 1 2 = 1), Sobolev imbedding V → L 2(ρ+1) (Ω) and Lemma 2.1, we conclude that where the constant C are positive constants independent of m and t. By (3.9) and (3.10), it is inferred that Integrating the above inequality over (0, t), and taking (3.8) into account, we get that The Gronwall Lemma guarantees that From the inequality (3.13) and Trace Theorem [10], we also obtain the following estimate where the constant C > 0 is independent of m and t. Furthermore, taking assumption (A2) into account, we know that if |u m t (t)| > 1, then If |u m t (t)| ≤ 1, we obtain from the continuity of the function f 2 that |f 2 (u m t (t))| ≤ C. Thereby, we obtain that Using analogous arguments, from the assumption (A2) and (3.14), we also obtain that Step 4. Global existence.
From the above estimates, we can show that there exists a subsequences of {u m } which from now on will be also denoted by {u m } and function u : Consequently, making use of Lion's Lemma [38, Lemma 1.3, Chapter 1], it follows that In addition, we also obtain Therefore, (3.19)-(3.27) permit us to pass to the limit in equation (3.2). Since {w j } is a basis of V , then for all T > 0, for all d(t) ∈ D(0, T ) and for all w ∈ V , we have Taking into account w ∈ D(Ω) and (3.28), we deduce that Utilizing the convergences of (3.19) and (3.24), there appear the relations that u tt ∈ L ∞ (0, T ; L 2 (Ω)) and |u| ρ u ∈ L ∞ (0, T ; L 2 (Ω)). Hence, we deduce that 2 u ∈ L ∞ (0, T ; L 2 (Ω)) and Combining (3.19) and (3.26), it is easy to see that the approximate solutions {u m } possess the following property for all w ∈ V , which implies that Taking (3.28)-(3.30) into account, and making use of generalized Green formula, we discover that Next, we need to prove that In deed, replacing w j by u m in equation (3.2), and integrating the obtained expression over (0, T ), it is inferred that In view of the first and second estimates, Sobolev imbedding, Poincaré inequality, and Trace Theorem [10], it follows that Making use of the Aubin-Lions Theorem [23, Chapter 1] again, we have that Combining (3.29), (3.32), (3.39) and the generalized Green formula, it is found that which implies that Now, in view of (3.26), (3.27), (3.38), and using the standard Lebesgue controlconvergent Theorem, we obtain that Utilizing the non-decreasing monotonicity of functions f i (i = 1, 2), it follows that for all ψ ∈ L 2 (Γ 1 ). Then, from the inequalities (3.43), (3.44), we discover that and then passing to the limit as m → ∞, In order to prove (3.33) from (3.47) and (3.48), we use the semi-continuous [23,Chapter 2]. Let ψ = u νt − λϕ, ∀ ϕ ∈ L 2 (Γ 1 ) and λ ≥ 0, then we have Pass to the limit as λ → 0 gives that In a similar way, let ψ = u νt − λϕ, λ ≤ 0 and ∀ ϕ ∈ L 2 (Γ 1 ), we obtain From (3.50) and (3.51), we see that Using the analogous arguments, taking ψ = u νt − λϕ, and ∀ ϕ ∈ L 2 (Γ 1 ), we also get from (3.48) that which implies that Thus, we obtain that u is a global regular solutions of problem (1.1).
Let u, u be two solutions of problem (1.1). Then, y = u − u satisfies for all w ∈ V . Replacing w by y t in the above identity, and noting that f i (i = 1, 2) are monotone functions, it follows that Using the Hölder inequality, Sobolev imbedding V → L 2(ρ+1) (Ω) and taking the first estimate into account, we thereby deduce that Then, apply the Gronwall Lemma yields that y t (t) 2 2 = y(t) 2 2 = 0. This completes the proof of Theorem 2.1.

Existence, uniqueness of weak solutions
Our attention in this section is turned to the existence, uniqueness of weak solutions for problem (1.1). Applying the standard density argument, we extend the existence, uniqueness results of regular solutions to the weak solutions.
Proof. The main idea of this proof is the density method. We will divided it into four steps.
We start to approximate the initial data u 0 and u 1 with more regular data u 0 µ and u 1 µ , respectively. Indeed, let us assume that Moreover, using the continuity of functionals where E µ (0) = E(u 0 µ ). Therefore, for sufficiently large µ ≥ µ 0 , we get u 0 µ 2 < λ 1 and E µ (0) < d. Thus, for each µ ≥ µ 0 , let u µ be the solutions of problem (1.1) with the initial date {u 0 µ , u 1 µ }, which satisfies all the conditions of Theorem 2.1, so we obtain (4.7) Step 2. Energy estimates and global existence.
Applying the analogous arguments used to prove the first estimate of the above section, we deduce that there exist constants C (various positive constants C may be different at different occurrences) which are independent of µ and t ∈ [0, T ], such that Let us define y µ,σ (t) = u µ (t) − u σ (t), µ, σ ∈ N . From the monotonicity of functions f i , (i = 1, 2), it follows that which together with the Hölder inequality, Sobelev imbedding from V → L 2(ρ+1) (Ω) and (4.8) gives that Then, the Gronwall Lemma reveals that where the constant C > 0 is independent of µ, σ ∈ N .

Uniform decay rates of solutions
The focus of the development in this section is the decay rate estimates of the energy to problem (1.1). The proofs are based on the construction of a special weight function φ, nonlinear integral inequality and the Multiplier method.
First, by the virtue of Theorem 2.1, it is known that the solution u of problem (1.1) possesses the some properties listed in Theorem 2.1 and Theorem 2.2. Thus, we can apply the following energy identity Taking into account that f i (s)s > 0 if s = 0, we see that E(t) is a non-increasing function. Moreover, the weight function φ appeared in Lemma 2.3 (construction method of φ will be presented in the sequel) will play key role in the proof of energy decay rate estimates. Now, let us multiply the equation in (1.1) by Eφ M u, where the function M u is defined by Then, considering 0 ≤ S < T < +∞ and applying the generalized Green formula, we deduce that Applying integration by parts and Gauss Theorem, it follows that The application of Gauss Theorem gives that Estimate of I 3 = (n − 1) T S Eφ Ω u tt udxdt. By the integration by parts again, we also obtain that Inserting (5.4)-(5.6) into (5.3), noting that u νν = −f 1 (u νt ), u ννν = f 2 (u t ) on Γ 1 and ∇u = u ν · ν on Γ 0 , it follows that Using the definition of energy E(t) and the identity (5.7), we obtain that Next, we shall estimate the last two terms of the right hand side of the above identity (5.8). 1], then by the interpolation inequality of L p (Ω) spaces, s p ≤ s α 2 s 1−α q with p = ρ + 2, q = 2(ρ + 1) and α = 1 ρ+2 , we deduce that Setting h = n − 1 + 2 ρ+2 , by Poincaré inequality, Sobolev embedding from V into L 2(ρ+1) (Ω) and Young inequality, we obtain that for all ε > 0 and B 2 = 2(ρ+2) ρ ρ+1 E(0) ρ . Combining (2.8) and (2.14), a direct computation gives that Furthermore, replace (5.12) in (5.10) gives that From (5.13), we obtain that Estimate of D 2 = 2 T S Eφ Ω |u| ρ u(m · ∇u)dxdt. By the Hölder inequality and Poincaré inequality, we have that Taking into account that 0 < ρ < 4 n−4 , if n > 4, and 0 < s < 2n n−4 − 2(ρ + 1), and considering the interpolation inequality s p ≤ s α 2 s 1−α q with p = 2(ρ + 1), q = 2(ρ + 1) + s, we discover that Applying Poincaré inequality and Sobolev embedding from V → L 2(ρ+1)+s (Ω) 2(ρ + 1) + s < 2n n−4 , then we have Combining (5.15) and (5.18), we conclude that From the Young inequality, .

Combining (5.19) and (5.21), we have that
Therefore, in view of (5.8), (5.14) and (5.22), when m · ν ≤ 0 on Γ 0 and ε small enough, we can conclude that there exists δ 1 , δ 2 > 0 such that In order to estimate the last term of (5.23), let us give the following lemma.  1). Then for T > T 0 , where T 0 is sufficiently large, we have for all 0 ≤ S < T < +∞.
Proof. We shall argue by contradiction. Suppose that (5.24) is not verified. Let u k be a sequence of solutions to problem (1.1) such that (5.25) while the total energy E k (0) with initial data {u k (0), u k t (0)} remains uniformly bounded in k, that is , there exists M > 0 such that E k (0) < M .
Since E k (0) < M , by the non-increasing property of E k (t), we have E k (t) < M . Hence, there exists a subsequence of the sequence {u k }, still denoted by {u k }, which satisfies Applying the similar methods used to prove (3.18) and (3.25), we have that Notice that the Aubin-Lions type compactness gives us In what follows, we will apply the ideas contained in Lasiecka and Tataru [20] or Cavalcanti et al [4] to our context. Case (i). Let us consider that u = 0. By (5.30), it follows that |u k | ρ u k −→ |u| ρ u , a.e. in Q T = Ω × (0, T ), k −→ ∞. (5.32) Since the sequence {|u k | ρ u k } is bounded in L ∞ (0, T ; L 2 (Ω)), together with (5.32) and Lion's Lemma [23, Lemma 1.3, Chapter 1], we have |u k | ρ u k −→ |u| ρ u in L ∞ (0, T ; L 2 (Ω)) weakly star, k −→ ∞. (5.33) Taking into account that the Poincaré inequality and the boundedness of E k (t), it is found that where C is a positive constant independent of k and t. Thus, we can deduce that the term T S φ Ω |∇u k | 2 dxdt is bounded. Therefore, we have from (5.25) that (f 1 (u k νt )) 2 dΓdt = 0.
Therefore, we conclude In a similar way, we also conclude that Taking k → +∞ in the equation, we get for u (5.42) and for u t = v,
Hence, the equation (5.42) reduce to the elliptic equation for all u = 0. This is a contradiction.
Case (ii). Let us assume that u ≡ 0. Setting By the similar argument as (5.46), we deduce that which along with (5.49) yields that Furthermore, when u ≡ 0, we deduce that c k → 0 as k → +∞.
On the other hand, considering the energy identity, (5.53) and multiplying this identity by E k (t), then we obtain Integrating (5.54) with respect to t from S to T , we discover that In view of (5.54) and (5.55), we deduce that Replacing M u k = 2(m · ∇u k ) + (n − 1)u k in inequality (5.23), we obtain that Using Young inequality and a direct calculation gives that Applying integration by parts and the Young inequality, it follows that Considering Young inequality and Poincaré inequality, we obtain from the definition of E k (t) that where L is a positive constant which verifies |φ (t)| ≤ |φ (0)| = L, ∀ t ≥ 0. Therefore, we have Analogously, considering the same procedure used to prove (5.61), we also get that
Hence, there exists a subsequence of the sequence { u k }, still denoted by { u k }, which satisfies In addition, u k also satisfies (5.81) From (5.74), we see that Making use of the same procedure used to prove (5.85), we deduce that Considering that |y| ρ is a continuous in R, so we define M ε = sup |y|≤ε |y| ρ . Therefore, we obtain Combining (5.77) and hypotheses (A1), we deduce from the (5.88) that Then, taking ε → 0 and k → +∞, we get that |u k | ρ u k → 0 in L 2 (0, T ; L 2 (Ω)). (5.90) From what has been discussed above, passing to the limit in (5.81) as k → +∞, we have (5.91) Differentiating (5.91) with respect to t and taking v = u t , we conclude that Applying the standard uniqueness results of [16](see Chapter 6) or the uniqueness results of [29] to our context again, it comes that v = 0, that is u t = 0. Returning to the equation (5.91), we obtain (5.93) Multiplying the above problem by u, we see that which implies that u = 0. But from the (5.48) and (5.80), we conclude that u = 0. This is a contraction. This completes the proof of Lemma 5.1.
On the basis of Lemma 5.1, we are now in positive to give the straightforward proof of Theorem 2.3.
By a straightforward adaptation of the above result (5.101), we also obtain that where L 3 , L 4 are positive constants.
Hence, we have that .