Global Existence and Asymptotic Behavior of Solutions for Quasi-linear Dissipative Plate Equation

In this paper we focus on the initial value problem for quasi-linear dissipative plate equation in multi-dimensional space $(n\geq2)$. This equation verifies the decay property of the regularity-loss type, which causes the difficulty in deriving the global a priori estimates of solutions. We overcome this difficulty by employing a time-weighted $L^2$ energy method which makes use of the integrability of $\|(\p^2_xu_t,\p^3_xu)(t)\|_{L^{\infty}}$. This $L^\infty$ norm can be controlled by showing the optimal $L^2$ decay estimates for lower-order derivatives of solutions. Thus we obtain the desired a priori estimate which enables us to prove the global existence and asymptotic decay of solutions under smallness and enough regularity assumptions on the initial data. Moreover, we show that the solution can be approximated by a simple-looking function, which is given explicitly in terms of the fundamental solution of a fourth-order linear parabolic equation.


Introduction
In this paper we consider the initial value problem of the following quasilinear dissipative plate equation in multi-dimensional space R n with n ≥ 2: The initial data are given as u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x). (1.2) Here u = u(x, t) is the unknown function of x = (x 1 , · · · , x n ) ∈ R n and t > 0, and represents the transversal displacement of the plate at the point x and the time t. The term u t represents a frictional dissipation in the plate, and the term ∆u tt corresponds to the rotational inertia effects. Also, b ij = b ij (V ) are smooth functions of V = (V ij ) ∈ S n satisfying the following structural conditions, where S n denotes the totality of n × n real symmetric matrices: [A1] There is a smooth potential function φ = φ(V ) such that b ij (V ) = (∂φ/∂V ij )(V ) for V = (V ij ) ∈ S n .
(1. 6) In this case we have b ij (V ) = αβ b ij αβ (O)V αβ +O(|V | 2 ), so that the linearized equation of (1.1) becomes to (1.7) Equations of the fourth-order appear in problems of solid mechanics and in the theory of thin plates and beams, and elliptic equations of the fourthorder appear in some formulations of problems related to the Navier-Stokes equations (see [24]). L. Sánchez [22] studied the existence and uniqueness of solutions of the full von Kármán system in an exterior domain and in the whole space, and proved that the model for thermoelastic plates is a singular limit of the von Kármán system under thermal effects. In [17] Perla Menzala and Zuazua also showed that the plate equation can be obtained as a singuler limit of the von Kármán system. Enomoto [6] studied a linear thermoelastic system associated with the plate equation in an exterior domain and proved a polynomial decay of the local energy for initial data with compact support. Buriol [2] considered the Timoshenko system of thermoelastic plates in R n and showed an exponential decay of the energy by considering two types of dissipation in the system. In [3], Charão et al. studied the asymptotic behavior of solutions of a dissipative plate equation in R n with periodic coefficients. They used the Bloch waves decomposition and a convenient Lyapunov function to derive a complete asymptotic expansion of solutions as t → ∞, and also proved that the solutions for the linear model behave as the homogenized heat kernel.
In [4], da-Luz and Charão studied a semi-linear dissipative plate equation whose linear part is given by (1.7) with b ij αβ (O) = δ ij δ αβ : u tt − ∆u tt + ∆ 2 u + u t = 0. (1.8) They proved the global existence of solutions and a polynomial decay of the energy by exploiting an energy method. However their result was restricted to the lower dimensional case 1 ≤ n ≤ 5. This restriction on the space dimension was removed by Sugitani and Kawashima [23] by making use of the sharp decay estimates for the linearized equation (1.8). One of the decay estimates obtained in [23] for (1.8) is given as follows: When u 0 ∈ H s+1 ∩ L 1 and u 1 ∈ H s ∩ L 1 , structure of the regularity-loss type and is characterized by the property where λ(ξ) denotes the eigenvalue of the equation obtained by taking Fourier transform of (1.8). See [23] for the details. A similar decay structure of the regularity-loss type was also observed for the dissipative Timoshenko system ( [9,21]) and a hyperbolic-elliptic system related to a radiating gas ( [8]). For more studies on various aspects of dissipation of plate equations, we refer to [1,5,10,13,15,19,20,26]. Also, as for the study of decay properties for dissipative hyperbolic-type equations, we refer to [11,14,18,25] for damped wave equations and [7,12,16] for wave equations of memory-type dissipation.
The main purpose of this paper is to study the global existence and largetime behavior of solutions to the initial value problem (1.1), (1.2). For this problem, we will observe that the regularity-loss also occurs in the dissipative part of the energy estimates, which makes it difficult to show the global a priori estimates of solutions to the nonlinear problem. To overcome the difficulty, we employ a time-weighted energy method, in which we make good use of the integrability of (∂ 2 x u t , ∂ 3 x u)(t) L ∞ and the optimal decay of the lower-order derivatives of solutions. Consequently, we obtain the global existence and the optimal decay estimates of solutions for small initial data with sufficient regularity. Moreover, we also show that for t → ∞, the global solution obtained is asymptotic to a simple-looking function which is given explicitly in terms of the the fundamental solution of the fourth-order linear parabolic equation (1.9) The contents of the paper are as follows. In Section 2 we give full statements of our main theorems. A time-weighted energy method is introduced in Section 3, and we explain why the usual energy method does not work well for our problem. In Section 4, we give the optimal decay estimates of solutions. Section 5 gives the proof of the first main theorem on the global existence and the optimal decay estimates of solutions, and the last section gives the proof of the second theorem on the asymptotic profile.
Before closing this section, we give some notations to be used below. Let F [f ] denote the Fourier transform of f defined by and we denote its inverse transform by F −1 .
Let s be a nonnegative integer. Then H s = H s (R n ) denotes the Sobolev space of L 2 functions, equipped with the norm Here, for a nonnegative integer k, ∂ k x denotes the totality of all the k-th order derivatives with respect to x ∈ R n . Also, C k (I; H s (R n )) denotes the space of k-times continuously differentiable functions on the interval I with values in the Sobolev space H s = H s (R n ).
Finally, in this paper, we denote every positive constant by the same symbol C or c without confusion.

Main theorems
Our first theorem is on the global existence and optimal decay of solutions to the problem (1.1), (1.2). To state the result, we need to introduce several special notations. Let k be a nonnegative integer and n be the space dimension. Let and put σ(k, n) = max{σ 0 (k), σ 1 (k, n)}, which indicates the loss of regularity. Since σ 1 (k, 3) = σ 0 (k) and σ 1 (k, n) is an increasing function of n, we have Also, for n ≥ 2, we define s(n) by which indicates the regularity of the initial data. Now we can state our first theorem as follows.
Theorem 2.1 (Global existence and optimal decay). Suppose that the conditions [A1] and [A2] are satisfied. Let n ≥ 2 and s ≥ s(n). Assume that u 0 ∈ H s+1 (R n ) ∩ L 1 (R n ) and u 1 ∈ H s (R n ) ∩ L 1 (R n ), and put Then there is a positive constant δ 0 such that if E 1 ≤ δ 0 , then the problem (1.1), (1.2) has a unique global solution u(x, t) with Moreover, the solution satisfies the following optimal decay estimates: Remark. The regularity assumption s ≥ s(n) might be technical but it seems necessary in our proof.
This global existence and optimal decay result is based on the following local existence result and the corresponding a priori estimates stated in Proposition 2.3 below. Assume that u 0 ∈ H s+1 (R n ) and u 1 ∈ H s (R n ), and put E 0 := u 0 H s+1 + u 1 H s . Then there is a positive constant T 0 depending on E 0 such that the problem (1.1), (1.2) has a unique solution u(x, t) with The solution verifies the following estimate for t ∈ [0, T 0 ]: This local existence result can be proved by the standard method based on the successive approximation sequence, so that the details are omitted.
To state the result on our a priori estimates, we introduce several timeweighted norms: where s ≥ 2, and where σ(k, n) is defined in (2. respectively. E(T ) is a time-weighted energy norm and D(T ) is the associated dissipation norm, while M 0 (T ) and M 1 (T ) are corresponding to the optimal decay estimates for u and u t , respectively. Now, the result on our a priori estimates is stated as follows.

Proposition 2.3 (A priori estimates). Suppose that the conditions [A1]
and [A2] are satisfied. Let n ≥ 2 and s ≥ s(n), and assume that u 0 ∈ H s+1 (R n ) ∩ L 1 (R n ) and u 1 ∈ H s (R n ) ∩ L 1 (R n ). Let T > 0 and let u(x, t) be the corresponding solution to the problem (1.1), (1.2) satisfying and the a priori bound (3.3) below. Then there is a positive constant δ 1 independent of T such that if E 1 ≤ δ 1 , then the solution verifies the timeweighted energy estimate and the following optimal decay estimate: Here E 1 is given in Theorem 2.1.
Remark. In order to obtain the above a priori estimates, we employ a timeweighted energy method. To close our energy estimates, we make use of the following decay estimates for the L ∞ norm of the derivatives ∂ 2 x u t , ∂ 3 x u and ∂ 2 x u:

For the details, see Sections 3, 4 and 5.
Our next result is concerning the asymptotic profile of the global solution obtained in Theorem 2.1. First we show that the solution to the problem (1.1), (1.2) can be approximated by the solution to the corresponding linear problem (1.7), (1.2). Then we prove that the solution to this linear problem can be further approximated by the profile MG 0 (x, t + 1), where M = R n (u 0 + u 1 )(x)dx denotes the "mass" and is the fundamental solution to the fourth-order linear parabolic equation Thus we conclude that MG 0 (x, t + 1) is an asymptotic profile of the solution to our problem (1.1), (1.2). This result on the asymptotic profile is stated as follows.
Then the global solution u(x, t) to the problem (1.1), (1.2), which is constructed in Theorem 2.1, is asymptotic to the profile MG 0 (x, t + 1) as t → ∞ in the following sense: Here M is a constant given by M = R n (u 0 + u 1 )(x)dx and G 0 (x, t) is the fundamental solution of (1.9) given in (2.10).

Time-weighted energy method
In this section, we introduce a time-weighted energy method for our nonlinear problem (1.1), (1.2) and explain why the standard energy method does not work well for our problem. First we give a lemma which will be used in the next energy estimates.
Lemma 3.1. Let n ≥ 1, 1 ≤ p, q, r ≤ ∞ and 1 p = 1 q + 1 r . Then the following estimates hold: Proof. These estimates can be found in a literature but we give here a proof. To prove (3.1), it is enough to show that, for k 1 ≥ 1, k 2 ≥ 1 and k 1 + k 2 = k, the following estimate holds: Since θ 1 + θ 2 = 1, we have 1 p = 1 p 1 + 1 p 2 . By using the Hölder inequality and the Gagliardo-Nirenberg inequality, we have In the last inequality, we have used the Young inequality. Thus (3.1) is proved.
which gives (3.2). This completes the proof of Lemma 3.1.
Energy estimates: Now, let T > 0 and consider solutions to the problem (1.1), (1.2), which are defined on the time interval [0, T ] and verify the regularity mentioned in Proposition 2.3. We derive energy estimates for the solutions under the following a priori assumption: whereδ > 0 is a given small number not depending on T . First, we multiply the equation (1.1) by u t . After straightforward computations, we have the energy equality . We integrate this equality in x ∈ R n , obtaining Here the Taylor expansion, using (1.6), shows that Next we derive a similar energy equality for derivatives. Notice that We multiply this equation by ∂ l x u t . After direct computations, we obtain We integrate the above equality in x ∈ R n , obtaining where we put R (l) = R n |r (l) | dx and Here, using (1.5) and (3.3), we see that Therefore the term R (l) can be estimated as We can use (3.7) even for l = 0 instead of the energy equality (3.4).
Finally, we multiply (1.1) by u. This yields We integrate this equality in where we have used (1.5) and (3.3). To get a similar estimate for derivatives, we multiply (3.6) by ∂ l x u. After direct computations, we have Integrating the above equality in x ∈ R n and using (1.5) and (3.3), we obtain Notice that (3.10) coincides with (3.11) with l = 0 if we setR (0) = 0. We explain why the standard energy method does not work well for our problem. To this end, we integrate (3.7) with respect to t and add the resulting inequality for l with 0 ≤ l ≤ s − 1. This yields Also, we integrate (3.10) and (3.11) with respect to t and add the resulting inequalities for l with 1 ≤ l ≤ s − 2; we use (3.11) only for s ≥ 3. This gives where (l) (τ )dτ , in which we can regardR (0) = 0. Combining (3.13) and (3.14), we arrive at the final energy inequality where R(t) = R 1 (t) + R 2 (t). We note that To complete our energy method, we need to control the term R(t) which comes from the nonlinearity of our equation (1.1). Usually, this can be done by using the dissipative term, namely, the third term on the left hand side of (3.15). Following to this strategy, we estimate the term R(t) as In our case, however, the dissipative term does not contain the highestorder term t 0 ( ∂ s x u t (τ ) 2 L 2 + ∂ s+1 x u(τ ) 2 L 2 )dτ because of the loss of regularity, and therefore it could not control the nonlinearity R(t). Consequently, the standard energy method does not work well for our problem.
Time-weighted energy estimates: To resolve the above difficulty caused by the regularity-loss property, we try to estimate the nonlinearity R(t) as This requires the integrability of (∂ 2 x u t , ∂ 3 x u)(t) L ∞ over t ≥ 0. To ensure this integrability, we need to show a suitable decay estimate for (∂ 2 x u t , ∂ 3 x u)(t) L ∞ and this will be done by employing the time-weighted energy method combined with the optimal L 2 decay estimates for lower-order derivatives of solutions. Now we estimate the time-weighted energy norm E(T ) and the associated dissipation norm D(T ) defined in (2.6) by applying the time-weighted energy method mentioned above. We make use of the following integral norm L(T ): The result is stated as follows.
Proof. For the proof, it is enough to show the following estimates for any t ∈ [0, T ]: ]. We use (3.7) and (3.11). Let λ ≥ 0. We multiply (3.7) by (1 + t) λ and integrate with respect to t. This yields for 0 ≤ l ≤ s − 1. Here we can regard R (0) = 0 if we use (3.4) in place of (3.7) with l = 0. Also, we multiply (3.11) by (1 + t) λ and integrate with respect to t. This yields which corresponds to (3.13). Also, we put λ = 0 in (3.21) and add for l with 0 ≤ l ≤ s − 2. Since t 0 s−2 l=0R (l) (τ )dτ ≤ CL(T )E(T ) 2 , we obtain where we have used (3.22), (3.23) and the fact that the last integral involving R (l) can be estimated by  (3.19) for j = k + 1. To this end, we first put λ = k in (3.21) and add for l with 2k ≤ l ≤ s − k − 2. This yields Here the last integral can be estimated by Also, the term C t 0 (1 + τ ) k−1 ∂ 2k x u(τ ) 2 H s−3k dτ is estimated by using (3.18) with j = k, while the other terms on the right hand side of the above inequality are estimated by making use of (3.19) with j = k. Consequently, we obtain which implies the desired estimate (3.18) for j = k + 1. Finally, we show (3.19) for j = k + 1. Let λ = k + 1 in (3.20) and add for l with 2k ≤ l ≤ s − k − 2. This yields Here the last integral can be estimated by H s−3k−2 dτ can be estimated by using (3.18) with j = k + 1 which has already been proved. Thus we get which proves (3.19) for j = k + 1. This completes the proof of Proposition 4.1.

Optimal decay estimates
This section is devoted to the proof of the optimal decay estimates for solutions to the problem (1.1), (1.2), which are defined on the time interval [0, T ] and verify (3.3).
Consequently, by the Duhamel principle, we can express the solutions to the problem (1.1), (1.2) as where G(x, t) and H(x, t) are the fundamental solutions to the linearized equation (1.7). Here and in the following, we use the abbreviation ∂ 2 The fundamental solutions G(x, t) and H(x, t) are given by the formulas where λ ± (ξ) are the eigenvalues given explicitly by with γ(ω) being defined in [A2]. We decompose the solution formula (4.2) in the form u(t) =ū(t) − F (u)(t), whereū(x, t) and F (u)(x, t) denote the linear and nonlinear parts, respectively: We review the basic decay property for the equation (1.8) which was studied in [23]. Notice that (1.8) is a special case of our linearized equation (1.7) with b ij αβ (O) = δ ij δ αβ , in which we have γ(ω) = 1. Therefore the corresponding fundamental solutions to (1.8) are given by the above formula with γ(ω) = 1. In particular, we havẽ The following decay result was proved in [23]. 23]). Let n ≥ 1 and s ≥ 0, and assume that φ ∈ H s (R n ) ∩ L 1 (R n ). Then the following decay estimates hold: for integers k and l with k ≥ 0, l + 1 ≥ 0 and 0 ≤ k + l ≤ s, and for integers k and l with k ≥ 0, l ≥ 0 and k + l ≤ s.
In view of the structural assumption [A2], the quantity γ(ω) has a positive minimum over S n−1 . By virtue of this fact, we find that the proof of Lemma 4.1 in [23] is valid also for our fundamental solution G(x, t). Thus we conclude that our G(x, t) satisfies the same decay estimates in Lemma 4.1. Moreover, as a simple modification of this lemma, we also have: for k ≥ 0, l + 1 ≥ 0 and 2 ≤ k + l ≤ s + 2, and for k ≥ 0, l ≥ 0 and 2 ≤ k + l ≤ s + 2.
As for the solution to our linearized problem, by similar calculations as in [23], we obtain the following decay result.
Second, we prove (4.10). We apply ∂ t to F (u) in (4.3) to get Moreover, applying ∂ k+h x and taking the L 2 norm, we obtain (4.22) We estimate the term I 3 by applying (4.5) with k replaced by k + h + 2 and with φ = g(∂ 2 x u) as (4.23) Here the term I 31 is estimated just in the same way as I 11 in (4.13) and we have provided that s ≥ [ n−1 4 ] + 4. Also, similarly to I 12 in (4.14), we get We choose l as the smallest integer satisfying A similar observation as in (4.15) shows that the desired choice is l = σ(k, n) − k + 3. For this choice of l, we obtain for h satisfying 0 ≤ h ≤ s − 4 − σ(k, n); here we need to assume that s ≥ σ(0, n) + 4 = [ n−1 4 ] + 4. For the term I 4 , we apply (4.5) with k = h, φ = ∂ k+2 x g(∂ 2 x u) and l = 2. This gives (4.28) Here we have ∂ k+2 where we have used the requirement d > d(n) = n 8 + 1 2 . Substituting all these estimates into (4.22) and adding for h with 0 ≤ h ≤ s − 4 − σ(k, n), we arrive at where we have assumed that s ≥ [ n−1 4 ] + 4. This estimate together with (4.7) gives the desired estimate (4.10). Therefore the proof of Proposition 4.3 is complete.

Proof of Theorem 2.1
The aim of this section is to prove Theorem 2.1. Since a local existence result is obtained in Theorem 2.2, we only need to show the a priori estimates stated in Proposition 2.3.
By virtue of the a priori estimate (2.8), we can continue a unique local solution obtained in Theorem 2.2 globally in time, provided that E 1 is suitably small, say, E 1 ≤ δ 0 . The global solution thus obtained satisfies (2.8) and (2.9) for any T > 0. In particular, we have the decay estimates (2.4) and (2.5) from (2.9). This completes the proof of Theorem 2.1.

Asymptotic profile
The aim of this section is to prove Theorem 2.4 on the asymptotic profile. First we prove that the solution to the problem (1.1), (1.2) can be approximated by the solution to the corresponding linear problem (1.7), (1.2). provided that s ≥ σ(k, n) + 6.