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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 419717, 25 pages
Asymptotic Energy Estimates for Nonlinear Petrovsky Plate Model Subject to Viscoelastic Damping
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
Received 1 July 2012; Accepted 16 September 2012
Academic Editor: Liming Ge
Copyright © 2012 Xiuli Lin and Fushan Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the nonlinear Petrovsky plate model under the presence of long-time memory. Under suitable conditions, we show that the energy functional associated with the equation decays exponentially or polynomially to zero as time goes to infinity.
Let be a bounded domain of with a sufficiently smooth boundary . We will assume that and have positive measures. The vector is the unit exterior normal and represents the tangential direction to . Here, the variable represents displacement of a plate occupying the domain . The governing equation is given by where the notations . The positive constant is the proportional to the thickness of the plate, is relaxation function, div stands for scalar divergence of a vector field, the stress resultant is given by and the nonlinear function is defined as for all . is a linear operator defined on with value on , the space of symmetric matrices, and defined as for any , where is the identity matrix and denotes the trace of . Moreover, is the density of the shell, denotes the Young modulus, and is the Poisson’s ratio.
With (1.1), we associate the boundary conditions on the portion of the boundary , and the boundary conditions on the remaining portion of the boundary , where and the boundary operators and are defined by
With (1.1), we also associate the initial conditions where the symbol denotes the subspace of such that there exists a constant such that the functions vanish as .
This problem has its origin in the mathematical description of viscoelastic materials. It is well known that viscoelastic materials exhibit natural damping, which is due to the special property of these materials, to retain a memory of their past history. From the mathematical point of view, these damping effects are modeled by integro-differential operators. Therefore, the dynamics of viscoelastic materials are of great importance and interest as they have wide applications in natural sciences. From the physical point of view, the problem (1.1) describes the position of the material particle at time , which is clamped in the portion of its boundary.
Models of Petrovsky type are of interest in applications in various areas in mathematical physics, as well as in geophysics and ocean acoustics [1, 2]. The Petrovsky type models without memory were discussed in [3, 4]. Messaoudi  considered the initial-boundary value problem established an existence result for (1.10), and showed that the solution continues to exist globally if , however if and the initial energy is negative, the solution blows up in finite time. Chen and Zhou  proved that the solution of (1.10) blows up with positive initial energy. Moreover, she claimed that the solution blows up in finite time for vanishing initial energy under the condition by different method.
The Petrovsky type equations with memory arouse the attention of mathematicians to study them. Alabau-Boussouira et al.  discussed the initial-boundary value problem of linear Petrovsky equation related to a plate model with memory, and showed that the solution decays exponentially or polynomially as if the initial data is sufficient small. Yang  considered the problem in -dimensional space, and proved that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as in the states of large initial data and small initial energy. In particular, in the case of space dimension , the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. Muñoz Rivera et al.  considered the initial-boundary problem for viscoelastic plate equation, and proved that the first and second order energies associated with its solution decay exponentially provided the kernel of the convolution also decays exponentially. When the kernel decays polynomially then the energy also decays polynomially. More precisely if the kernel satisfies then the energy decays as . On the recently related papers concerning the Petrovsky type models, the readers can see references [8–12].
In [13–15], Li et al. proved the existence uniqueness, uniform rates of decay, and limit behavior of the solution to nonlinear viscoelastic Marguerre-von Karman shallow shells system, respectively. To our best knowledge, we do not find the research report on the problem (1.1) which is considered in this paper.
Motivated by the above work, we obtain the energy functional associated with the equation decays exponentially or polynomially to zero as time goes to infinity. The main contribution of this paper are as follows. (a) The problem considered in this paper is nonlinear equation with integral dissipation, to our knowledge this model has not been considered; (b) the hypothesis on and initial data are weaker; (c) we naturally define the energy by simple computation and only define simple auxiliary functionals to prove our result by precise priori estimates.
2. Notations and Preliminaries
In this section, we will prepare some material needed in the proof of our main results. We use standard Lebesgue space and Sobolev space and adopt the following notations We denote for any pair of tensors and . We introduce the following space Define the bilinear form as follows For simplicity, we denote by . For the relaxation function , we assume that is a function satisfying Both and exist, and Hypotheses () assure that the viscoelastic energy (defined below) is nonincreasing and the assumption () means that the material behaves like an viscoelastic solid at (cf. ).
Our results are based on the following existence theorem.
Theorem 2.1. One assumes that satisfies conditions ()-(). For any initial data subject to the compatibility conditions satisfied on the boundary , Then for any , there exists a unique global solution to (1.1)–(1.10) satisfying
At First, the energy of the system must be properly defined. The total energy can be expected to consist of two parts. One part involves the current kinetic and strain energies, and the other will involve the past history of strains. To obtain the appropriate expression of energy functional, using assumption (), we rewrite system (1.1) and (1.5) in the form
In order to define the energy functional, we give the following lemma.
Lemma 2.2. Let and be the functions in . Then, one has
In order to define the energy function of the problem (1.1)–(1.8), we give the following computations. Multiplying equation (2.10) by , integrating the result over and adding Green’s formula, we get from Lemma 2.2 and (1.4) that Clearly By applying Lemma 2.2, the first term on the right-hand side of (2.21) equals In the same way, the second term on the right-hand side of (2.21) equals Therefore, from (2.21)–(2.23), we have Note and (by ) Consequently, we conclude form (2.24)–(2.26) that Summing (2.20) and (2.27), using the boundary conditions (2.11) together with the symmetric of yields Note the relation In fact with . Denote . Hence which implies that From (2.30) and (2.32), we conclude that (2.29) holds. Combining (2.28) and (2.29), we conclude that The relation (2.33) inspires us to define energy functional as the following
Lemma 2.3. Under the above notations and assumptions () and (), one has that
3. Main Results
In this section, , and denote some positive constants. Here, we need to point out that denotes the small enough different positive constant and denotes the different positive constant depending on in a different place, respectively. The main goal of this paper is to show, respectively, that the solution decays exponentially or polynomially to zero as the time goes to infinity under suitable conditions. Our main results are formulated below.
To prove our main result, we will give some important preliminaries.
Lemma 3.2. Assume , functional and tensor are defined as above. Then(1)(2)(3)(4).
Proof. In fact
Combining (3.2)–(3.4) and noting , we finish the desire conclusion (1).
By , the definition of , and Poincaré inquality, we know
Using Poincaré inequality, we get which with conclusion (2) shows the desire result.
Denote . Hence with , and which imply From (3.9) and noting , we obtain the desire conclusion (4).
The proof is completed.
Lemma 3.3. Suppose and is defined as above, then
Lemma 3.4. Define the functional then
Proof. (1) By the formula of integral by part and (), we deduce
(2) Using () and (3.1), we infer the conclusion.
(3) By and the formula of integral by part, we get the conclusion (3).
The proof is completed.
Lemma 3.6. Define the functional then for some small enough, there exist and as the following inequality holds
Proof. Multiplying (1.1) with and integrating the result over , we have
which with Lemma 3.5 gives
By Green formula, we have
Inserting (3.22) and (3.23) into (3.21) yields
Applying Lemma 2.2, Lemma 3.5 and noting , we have
Inserting (3.25) into (3.24) and owing to (1.5) yield
Now, we estimate some terms on the right-hand side of (3.26). The Lemma 3.3 and (3.1) imply
Using Cauchy inequality, Hölder inequality, and Lemma 3.2 and noting (3.1), we obtain
Applying Hölder inequality, Lemma 3.2, and Lemma 2.3, we conclude
Combining (3.26)–(3.29), we conclude that
The proof is completed.
proof of Theorem 3.1. Let By Lemma 2.3, Lemma 3.4, and Lemma 3.6 and noting (3.1), we have with Fixing , we obtain . Under the conditions, we obtain from (2.34) and (3.32) that there exists a positive such that Fixing a large enough positive constant , we prove and are equivalent, that is, with . Indeed, using Cauchy inequality, Lemma 3.2, and (2.34), we have Using the definition of and Green’s formula, we infer Applying Cauchy inequality, Lemma 3.2 and noting (3.1), we conclude that