Abstract

This paper deals essentially with a nonlinear degenerate evolution equation of the form supplemented with nonlinear boundary conditions of Neumann type given by . Under suitable conditions the existence and uniqueness of solutions are shown and that the boundary damping produces a uniform global stability of the corresponding solutions.

1. Introduction

Let be a smooth bounded open set of , with , and its boundary of class . Assume that is constituted by two disjoint closed parts and both with positive Lebesgue measure.

The main goal of this paper is to prove the existence and uniqueness as well as the uniform decay rates for the energy of the following nonlinear initial boundary value problem: where , , and are real functions, denotes the unit outward normal at , and is a constant.

The parabolic-hyperbolic equation when or ; this equation governs the motion of a nonlinear Kelvin solid. That is, a bar for and a plate for , subject to no nonlinear elastic constraints. The function represents the mass density of the solid.

The existence of solutions of the linear problem associated with   (, , and without the function and with ) was established by Komornik and Zuazua in [1], via semigroup theory and by Milla Miranda and Medeiros in [2], applying the Galerkin’s method, with a special basis. The advantage of this second method is to define the Sobolev space where is lying. In the same context, applying this second method for a wave equation with a nonlinear term, Araruna and Maciel [3], derive similar results. In Cavalcanti et al. [4] the existence of solution and an exponential decay rate is established supposing and being a particular function considered in our work; see also Cavalcanti et al. [5].

For the wave equation with and there is a vast literature on this problem. We cite the papers Cavalcanti et al. [6], Lasiecka and Tataru [7], and references contained therein for the reader.

Following the ideas delivered in Milla Miranda and Medeiros [2], but bringing more technical difficulties, Milla Miranda and San Gil Jutuca [8] applied the Galerkin’s method with a special basis to show the existence of solutions for Kirchhoff’s equation with a linear dissipation on the boundary. Applying a similar approach, Lourêdo and Miranda [9] obtained the existence of solutions for a coupled system of Kirchhoff’s equations with nonlinear boundary dissipation. For other models, but in the same context, we cite to the reader Lourêdo and Miranda [10], and Lourêdo et al. [11].

Park and Kang studied the existence, uniqueness, and uniform decay for the nonlinear degenerate equation with memory condition on the boundary in [12]. For the asymptotic behavior they also used the Nakao’s method. de Lima Santos and Junior [13] studied the equation with a boundary condition with memory for Kirchhoff plates equations. An abstract formulation with the coefficient satisfying the same conditions as in our paper was studied by Pereira in [14] and was established the existence, uniqueness, and asymptotic behavior for the solutions associated with a nonlinear beam equation.

In this paper we are interested in showing the global existence of solutions for Problem under very general conditions to be fixed in the next section.

In our approach, we apply the Galerkin’s method for a perturbed problem and a special basis; an appropriate Strauss’ Lipschitz-continuous approximation of ; the compactness method; and results on trace mapping of nonsmooth functions. Finally, the uniform stabilization of solutions is accomplished by using the Nakao’s method.

2. Notations and Main Results

In order to establish the main results of this paper we assume the following assumptions on the objects of problem :(H1)    (A1) and there exists a positive constant such that(A2)  , , where ;(H2)(H3) with strongly monotone; that is,  where . We use the notation ;(H4) with , , a.e. and there exists such that , a.e. ;(H5), for all ;(H6).

The scalar product and norm of are denoted, respectively, by and . By we represent the Hilbert space which is equipped with the scalar product and norm

The operator is defined by the triplet . Then its domain is given by From spectral theory it follows that is dense in ; see [15]. Moreover, it will be denoted

Theorem 1. Assume hypotheses (H1)–(H6); there exists at least a function in the class satisfying

In addition, if(H7)   and a.e. in ( a positive constant), holds true, we have the following result.

Corollary 2. Under the hypothesis of Theorem 1 and (H7), the solution of Problem (obtained in Theorem 1) is unique and has the following regularity:

Remark 3. If we replace the function in Theorem 1 by a continuous function such that and further Theorem 1 remains valid. Indeed, from (10), we obtain where is the Strauss’ approximations (see [16]) of the function .

Remark 4. Analogously if    and a.e. in ( a positive constant), we obtain for all (see [10]).

For use later, note that where is the first eigenvalue of the spectral problem for all (see [15]).

In order to establish the uniform decay rate for energy, we assume(H8);(H9);(H10);(H11), , , where .

Remark 5. There are functions that satisfy hypothesis (H5) and (H8). In fact, the function with , , and satisfies such hypothesis, since and so

Theorem 6. Under the hypothesis of Theorem 1 and (H7)–(H11), with satisfying (9) and (10), the energy associated with the solution, , obtained in Corollary 2 is uniformly stable; that is, there exists a positive constant such that where and are positive constants.

For use later, we observe that hypothesis in on implies and . Thus, the Sobolev’s embedding gives Here, indicates that the subspace is continuously embedded in the space .

Next, following the ideas contained in Strauss [16], we approximate the function by Lipschitz-continuous ones .

3. Proof of Theorem 1

For our purposes we need the following previous results, whose proof can be seen in [10].

Lemma 7. Let be a function satisfying hypothesis (H3). Then there exists a sequence of functions in such that(i) a.e. in ;(ii), , and a.e. in ; ( positive constant);(iii)for any there exists a function in satisfying (iv) converges to uniformly on bounded sets on and a.e. in .

Lemma 8. Let be a real number. Consider the sequence of vectors in and vectors and such that(i) weak in ;(ii) in .
Then, .

Lemma 9. Let be a globally Lipschitz-continuous function with and let be the continuous trace of order zero. Consider then , , and a.e. .

Proof. We see that are continuous maps (see Brezis and Cazenave [17] and Marcus and Mizel [18]). Let . Consider a sequence of functions of such that Then by (21) and (22), we have in , and by (20) Also by (20) and (22), we deduce As , it follows from (23) and (24) that . This implies
Now, we consider the set . Then with (see Brezis and Cazenave, loc. cit). As then From this and since then . This and (25) furnish From (25) to (28) we have the results of this Lemma.

Proof of Theorem 1. We will use the Faedo-Galerkin’s method with a special basis of . Thus, let us consider the Strauss’ approximation of given by Lemma 7. Let us consider a sequence of vectors in such that Note that and on since . Thus, Now, we fix and construct the basis of such that , belong to the subspace spanned by and . Let be the subspace of spanned by . With this basis we determine the approximate solutions of Problem , where fixed.
Approximated Perturbed Problem. This consists to find the functions , solutions of the problem
The above finite-dimensional system has a solution, , defined on . The following estimate allows us to extend this solution to the whole interval .

3.1. Estimates
3.1.1. First Estimate

Considering in 1, integrating from to   , using the fact that (see Part (ii) of Lemma 7), assumptions (H4) and (H5), and since , we obtain Note that In fact, by the Gauss’s formula we have where is the th entry of the normal vector . Hence, by we obtain Using the hypothesis , choosing , and plugging (34) in (31), we find where for all , and . Moreover, for all . Therefore by the Gronwall’s inequality and (35)

3.1.2. Second Estimate

Differentiating with respect to the approximate equation 1 and taking , we obtain Note that and that where the last inequality becomes from Young’s inequality. Therefore, if then . Combining this inequality and (38) with (37), after that using that , and integrating from to , we obtain We also have used the above hypotheses (H4) and (H5). Now, by Hölder’s inequality for , embedding (18) and first estimate (36), we find where the constant is independent of , , and , and is a positive constant that depends on . Substituting this inequality in (41) yields Choosing and , small such that , we find Now we need to derive an estimate for . Thus, taking in approximate Problem 1 and choosing , one has Applying Green’s formula Thanks to (30), the integrals on in (46) are null. Using convergence (29), embedding (17), and the hypothesis (H4) give From this boundedness and (46), we get where is constant independent of , , and . Applying Gronwall’s inequality in (48) and using estimate (47), we have From estimates (36), (49), induction, and diagonal process, we obtain a subsequence of , which is still denoted by , and a function , such that From the convergence (49) we obtain

3.1.3. Analysis of the Nonlinear Terms

By estimates (36), (36), compactness method (cf. Lions [19] or Simon [20]), embedding (17), induction, and diagonal process, we obtain a subsequence of , which also is denoted by , such that From (48), we have that is bounded in . Thus, estimate (49) and the compactness embedding of in , give in . From this, property (iii) and Lemma 7 yield

3.1.4. Passage to the Limit as

Convergences (49), (51), and (53) permit us to pass to the limits in approximate equations , as . Thus, this fact and the density of in , give Now, if and we obtain using the regularity of (given by (49)), that From (49) and (55) we have and , respectively. Thus, . (compare to Lions [19] and Medeiros and Milla Miranda [21]). Multiplying both sides of (55) by , with and , and integrating over , then the preceding regularity, , gives where denotes the duality paring between and .

Comparing (54) and (56) and using the Lipschitz property of , we obtain

3.1.5. Passage to the Limit in and

As estimates (36) and (49) are independent of , , and we obtain a subsequence of , which still denoted by , and a function such that all convergences (49) and (52) are valid. These convergences will be denoted by (49), (51), and (52), respectively. These results imply that there exists a function belonging to class (49) and it is a solution of equation Denoting these convergence in by (49), (51), and (52), respectively, then the convergence (49) gives us weak star in . From this and (55), weak star in . Then Moreover, convergence (49) furnishes . Now, we fix . The preceding convergence implies Fixing , then by (61) the set is bounded. Part (iv) of Lemma 7 says that converges to uniformly in bounded sets of , a.e. in . These two results and (61) give

On the other hand, by (58) and (59), we obtain By familiar inequalities, and from embedding (17), where is a constant independent of and .

As integrating the inequality above from to , and using the hypothesis (H5) and (H6), we find

Note that , and that and are bounded in and in , respectively. Taking into account the preceding considerations, estimates (49) and convergence (36) in (63), we obtain where is a constant independent of and . Note that .

From (62), (68), Strauss’ approximations, Lemma 9 and from a diagonal process, we get

Convergences (60) and (69) imply in and in . Now we take the limit in (59). Moreover the last two convergences and the regularity of imply in , which shows that satisfies Hence, the result is done as in (58).

The verification of the initial conditions follows by convergence (49).

Remark 10. If , then the sequence , which converges to , satisfies (see Lourêdo and Miranda [10]). In these conditions, the solution is unique and . Consequently, .
The proof of Corollary 2 follows from Remark 10 and from regularity of elliptic problems (see Lions and Magenes [22]).

4. Asymptotic Behavior

In this section, by applying Nakao’s method (see [23]), we will prove the uniform stabilization of the energy associated with the solution of the Problem .

Proof of Theorem 6. First, we prove the inequality (15) for the approximate energy given by and Theorem 6 will follow by taking the in .
Taking the scalar product of in both sides of with , we find Using (73) and the hypotheses (A2), (H3), (H8), and (H2), we obtain Note that is decreasing. From (73), the hypotheses (H1), (H2), and Remark 10, we have that Thus, where and . Integrating (76) from to , we obtain Furthermore, taking the scalar product in both sides of (72) with , we find As then Now note that and by Gauss’ formula, we have Therefore, Hence, Substituting (84) in (80), we obtain Integrating (85) from to yields Using the embedding of in and in (see (12)), it follows that(i);(ii);(iii).(iv) ;(v);(vi).
Note that since is decreasing and , . Analogously, we obtain Using (87) and (88) in (vi) we find Substituting (i)–(vi) and (89) in (86), we obtain It follows from (74) that where and . Hence where .
From (92), we obtain and from (93), we get By the Mean Value Theorem, there are and , such that From (95), we can write Now from (93) we have where .
Analogously we obtain where .
Substituting (96), (97), and (98) in (90) and considering we get As then , and substituting this inequality in (100), we obtain Replacing and in (93), we have Adding (101) and (102), we obtain and this implies where The hypothesis (H11) yields and as then Since , and , where .
From (106), we obtain
Note that by hypothesis (H11), , thus from (107), we find Again, by the Mean Value Theorem there exists , such that Integrating (76) from to and using (77) and (93), it follows that Substituting (109) in (110), we get Now, substituting (108) in (111), we obtain As is decreasing, the inequality (113) provides that where Thus, it follows using (114) and from Nakao’s Lemma, (see [23]), that Taking the as in (116), we obtain where is positive constant.

Conflict of Interests

The authors report that there is no conflict of interests in the publication of this paper.