Weak and Strong Solutions for a Strongly Damped Quasilinear Membrane Equation

Abstract

We consider a strongly damped quasilinear membrane equation with Dirichlet boundary condition. The goal is to prove the well-posedness of the equation in weak and strong senses. By setting suitable function spaces and making use of the properties of the quasilinear term in the equation, we have proved the fundamental results on existence, uniqueness, and continuous dependence on data including bilinear term of weak and strong solutions.

1. Introduction

Let be an open bounded set of with the smooth boundary . We set , for The nonlinear equation of the longitudinal motion of vibrating membrane surrounding with clamped boundary is described by the following Dirichlet boundary value problem:where is the height of a membrane, , is a forcing function, and denotes the Euclidean norm on . A brief physical background of (1) is given in our previous paper [1].

For damped linear or semilinear systems, there are many books and articles about the well-posedness with applications to various dynamic system’s topics (cf. [2–4], etc.) with semigroup or unified variational treatments. However, the quasilinear cases like (1) require more manipulations in the analysis of systems, because the systems like (1) are very much model-dependent due to the strong nonlinearity.

Equation (1) is proposed in Kobayashi et al. [5] and the well-posedness of strongly regular solutions is studied by using the resolvent estimates of linearized operators in a modified Banach space. Besides, the well-posedness of less regular solutions is proved in [1], called weak solutions in the framework of the variational method in Dautray and Lions [3]. Based on these results, we have treated the associated optimal control and identification problems in [6] and [7], respectively. Furthermore, in [8] we have extended the results in [1] to more general quasilinear nonautonomous wave equation with strong damping term.

In this paper, our concerned model is given by the following problem:where is a bilinear forcing term which is usually referred to as bilinear control variable acting as a multiplier of the displacement term.

Bilinear optimal control problems with the state equation being a linear first or second PDEs such as reaction diffusion equation or Kirchhoff plate equation are studied by some authors (see [9–12] and references therein).

For future work, we will study bilinear optimal control problem with state equation (2). Then we must be faced with many difficulties because of the quasilinear term in (2). However, more regular solution of (2) corresponding to more regular data than weak one enables us to overcome these difficulties. This is motivation of this paper.

As is recognized, existence of regular solution of a quasilinear PDE is also quite model-dependent due to the strong nonlinearity.

We briefly summarize this paper as follows. At first, referring to the results in [1], we shall prove the well-posedness of weak solution of (2). Secondly, we shall analyze (2) in higher regularity than in [1] by employing newly constructed energy equality for (2). Finally, we shall prove by exploiting the well-posedness of weak solution of (2) that the regular solution of (2) is continuous with respect to regular data.

The most difficult part of the existence proof is to show the strong convergence of nonlinear terms, and the part is completed by using the argument in [3, p. 569]. This is another novelty of the paper.

2. Notations and Main Results

If is a Banach space, we denote by its topological dual and by the duality pairing between and . We introduce the following abbreviations: with And mean the completions of in for . Let . If we denote the scalar product on by , then the scalar products on and are given as follows: Then obviously, The duality pairing between and is denoted by .

It is clear that and each space is dense in the following one and the injections are continuous.

Related to the nonlinear term in (2), we define the function by . Then it is easily verified thatThe nonlinear operator is defined byBy the definition of in (8), we have the following useful property on :The solution space for weak solutions of (2) is defined by endowed with a norm where and denote the first- and second-order distributive derivatives of . We remark that is continuously embedded in (cf. Dautray and Lions [3, p. 555]).

Definition 1. A function is said to be a weak solution of (2) if and satisfies

The following theorem gives the fundamental results on existence, uniqueness, and regularity of weak solutions of (2).

Theorem 2. Assume that , and Then problem (2) has a unique weak solution . Moreover, the solution mapping of into is locally Lipschitz continuous.

Indeed, let and We prove Theorem 2 by showing the inequalitywhere is a constant depending on data.

Next we introduce the solution space for strong solutions of (2) defined by endowed with a norm where and denote the first- and second-order distributive derivatives of . We remark also from Dautray and Lions [3, p. 555] that is continuously embedded in

Definition 3. A function is said to be a strong solution of (2) if and satisfies

The next theorem gives a well-posedness result for strong solutions of (2).

Theorem 4. Assume that , and Then (2) has a unique strong solution and it satisfieswhere is a constant depending on data.

Now we give the result on the continuous dependence of strong solutions of (2) on . Let be a product space defined byendowed with a norm For each we have a strong solution of (2) by Theorem 4. Thus, we can define the solution mapping of into .

Theorem 5. The nonlinear solution mapping of into of (2) is continuous.

Throughout this paper, we will use as a generic constant and omit writing the integral variables in any definite integrals without confusion.

3. Proof of Main Results

Proof of Theorem 2. Since , by the results in [1], we can deduce that the weak solutions of (2) corresponding to exist in such thatWe denote by . Then, we can get from (2) that satisfies the following equation in weak sense:where and
We multiply (21) by to haveBy integrating (22) over , we obtainLet be an arbitrary real number. Then, by (9), (20), and the Schwartz inequality we can obtain the following:We also note that Therefore, from (23) and (24), we can obtain the following inequality:If we choose , then by Bellman-Gronwall’s inequality it follows thatBy (21) and (26) we haveFinally, by combining (26) and (27) we obtain (13).
This completes the proof.

Lemma 6. Let , and be Banach spaces such that the imbeddings are continuous and the imbedding is compact. Then a bounded set of is relatively compact in .

Proof. See Simon [13].

Proof of Theorem 4. We divide the proof into three steps.
Step 1 (approximate solutions and a priori estimates). We construct approximate solutions of (2) by a Faedo–Galerkin’s procedure. Since is separable, there exists a complete orthonormal system in such that is free and total in . For each we can define an approximate solution of (2) by where satisfies (2). Then (2) can be written as vector differential equationswith initial values Notations of (29) can be explained as follows: where denotes the transpose of . Since is Lipschitz continuous and , we can deduce by Carathéodory type existence theorem that the nonlinear vector differential equation (29) admits a unique solution on . Hence, we can construct the approximate solution of (2). Next we shall derive a priori estimates of .
By analogy with (22), we take product of the equations for approximate solutions with to haveBy integrating (32) over , we obtainHere we note from the elliptic regularity theory thatThus, by (34) we can deduce for For other estimations of the terms in the RHS of (33) other than (35), we can follow the analogous process in the proof of Theorem 2 to getTherefore, it is shown by using Bellman-Gronwall’s inequality thatAnd also from Theorem 2, (2), (34), and (37), we havewhere
Step 2 (passage to the limits). Equations (37) and (38) imply thatAnd the nonlinear term is bounded in . Hence, by the extraction theorem of Rellich, we can extract a subsequence of and find and such thatas . Since is compact, we can apply Lemma 6 with and Aubin-Lions-Temam’s compact imbedding theorem (cf. Temam [14, p. 274]) to (39) to verify that and are precompact in and , respectively. Hence, we can find a subsequence , if necessary, such thatas
By the standard argument of Dautray and Lions [3, pp. 564–566], it can be verified that the limit of is a strong solution of the linear problemStep 3 (strong convergence of approximate solutions). In order to prove that is a strong solution of (2), it is sufficient to prove . For this, we shall show strongly in for all . To prove the strong convergence, we use the modified arguments in Dautray and Lions [3, pp. 579–581] and the classical compact imbedding theorem.
First as in (33), we take product equation (43) with and integrate it over to haveBy making use of the following trivial equalities: we add (33) to (44) and denote by to getwhere For simplicity we setIt is verified by direct computations thatwhereBy (49)–(51), (46) can be rewritten byThe term can be estimated asThen by routine calculations in (52) together with (53), we can derive the following inequality:By applying the extended Bellman-Gronwall’s inequality to (54), we deduceBy virtue of the strong convergence of the initial values and (40)–(42), we can extract a subsequence of such thatas . Therefore, in view of (44) we can deduce by the sum of limits in (56) thatAlso from (41), we can easily verify that there exists a subsequence of such thatfor all . Then it follows from (58) thatfor all . Hence, we haveSincewe see from (60), (61), and the Lebesgue dominated convergence theorem thatBy applying (57) and (62) to (55) with , we haveas From (49), (53), (62), and (63), it follows thatas Thus, we readily haveTherefore, we have proved the existence of a strong solution of (2).
By similar estimations as in (37) and (38), we can show (17).
The uniqueness of strong solutions is evident from the uniqueness of weak solutions.

Proof of Theorem 5. Let and be the strong solutions of (2) corresponding to and , respectively. Set Then satisfiesin the weak sense. Here we can repeat the same estimations in the proof of Theorem 4 mainly by performing -product of (66) with to obtainwhere is given by (51) in which and are replaced by and , respectively. By Theorem 2, we know that Equation (68) immediately impliesWhence from (58)–(62) we know thatFinally by (66), (67), and (70), we can conclude thatThis completes the proof.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This research was supported by the Daegu University Research Grant 2014.