DECAY PROPERTY FOR A PLATE EQUATION WITH MEMORY-TYPE DISSIPATION

In this paper we focus on the initial value problem of the semilinear plate equation with memory in multi-dimensions (n ≥ 1), the decay structure of which is of regularity-loss property. By using Fourier transform and Laplace transform, we obtain the fundamental solutions and thus the solution to the corresponding linear problem. Appealing to the point-wise estimate in the Fourier space of solutions to the linear problem, we get estimates and properties of solution operators, by exploiting which decay estimates of solutions to the linear problem are obtained. Also by introducing a set of time-weighted Sobolev spaces and using the contraction mapping theorem, we obtain the global in-time existence and the optimal decay estimates of solutions to the semi-linear problem under smallness assumption on the initial data.

1. Introduction. In this paper we consider the initial value problem of the following semi-linear plate equation with memory term in multi-dimensional space R n with n ≥ 1: u tt + ∆ 2 u + u + g * ∆u = f (u), (1.1) with the initial data 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.

YONGQIN LIU AND SHUICHI KAWASHIMA
where C i (i = 0, 1, 2, 3) are positive constants. Assumption [B]: Assume that f ∈ C ∞ (R \ {0}), and f (u) = O(|u| α ) as |u| → 0, here α > α n and α n := 1 + 2 n , n ≥ 1, and α is assumed to be an integer for n ≥ 3. In [10], we studied the inertial model of quasilinear dissipative plate equation, whose linear part in a simpler case is given by: here −∆u tt corresponds to the rotational inertia, and u t is the linear dissipative term. In that paper, we obtained the global existence and asymptotic behavior of solutions by employing the time-weighted energy method combined with a semigroup argument. In this paper, one point worth noticing is that , the dissipation given by the memory term g * ∆u is relatively weaker compared with the linear term u t . This weak dissipative mechanism could be reflected from the decay structure of solutions. Same as the inertial model of dissipative plate equation (1.3) in [10,17], the plate equation with memory (1.4) is also of regularity-loss property. The decay structure of the regularity-loss type is characterized by the property where ρ(ξ) is introduced in the point-wise estimate in the Fourier space (3.1) of solutions to the linear problem. It is not difficult to see that the decay structure is very weak in high frequency region since ρ(ξ) may tend to zero as |ξ| → ∞. A similar decay structure of the regularity-loss type was also observed for the dissipative Timoshenko system ( [9,15]) 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 [1,2,3,4,6,12,14,16,18]. Also, as for the study of decay properties for wave equations and hyperbolic systems of memory-type dissipation, we refer to [5,7,11,13].
The main purpose of this paper is to study decay estimates of solutions to the initial value problems (1.4), (1.2) and (1.1), (1.2). For our problem, it is difficult to obtain explicitly the solution operator or its Fourier transform due to the presence of memory term. However, by using Fourier transform and Laplace transform, we obtain the solutionū to the linear problem (1.4), (1.2) given by (2.5) and the solution operators G(t) * and H(t) * . Moreover by employing the energy method in the Fourier space, we obtain the point-wise estimate in the Fourier space of solutions to the corresponding linear problem (1.2) and Appealing to this point-wise estimate, the corresponding point-wise estimate of solution operators and their properties are obtained. Consequently, the decay estimates of solutions to (1.4) (1.2), and the global existence and optimal decay estimates of solutions to (1.1), (1.2) are achieved. As for the semi-linear problem, one point worthy to be mentioned is that we obtain the results for α > α n in the case n = 1, 2, while α n = 1 + 2 n is the well-known critical Fujita exponent in dealing with the global existence and blow up of solutions to some semi-linear parabolic differential equations.
The contents of the paper are as follows. Solution formula are obtained in section 2. In section 3, we obtain the estimates and properties of solutions operators, which is based on the point-wise estimate in the Fourier space of solutions to the corresponding linear problem. In section 4, we prove the decay estimates of solutions to the linear problem by virtue of the properties of solution operators. In the last section, the global existence and the optimal decay estimates of solutions to the initial value problem (1.1), (1.2) are obtained.
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 as F −1 .
Let L[f ] denote the Laplace transform of f defined by and we denote its inverse transform as L −1 .
In particular, we use W m, 2 = H m . Here, for a nonnegative integer k, ∂ k x denotes the totality or each of all the k-th order derivatives with respect to x ∈ R n . Also, C k (I; H m (R n )) denotes the space of k-times continuously differentiable functions on the interval I with values in the Sobolev space H m = H m (R n ).
Finally, in this paper, we denote every positive constant by the same symbol C or c without confusion. [·] is Gauss' symbol.
2. Solution formula. In this section we try to obtain the solution formula for the problems (1.4) (1.2) and (1.1) (1.2). Assume that G(x, t) and H(x, t) are solutions to the following problem, Apply Fourier transform and Laplace transform to (2.1) and (2.2), then we have formally thatĜ here C is a constant determined by the initial data in (2.1) and (2.2). The following lemma guarantees thatĜ(ξ, t) andĤ(ξ, t) are well defined.
Combining the two cases, we know that λ F (λ) is analytic in {λ ∈ C; Re(λ) > 0} if ξ = 0 and in {λ ∈ C; Re(λ) ≥ 0} if ξ = 0. Take λ = σ + iν, σ > max{Reλ s }, here {λ s } is the set of all the singular points of F (λ), then we have that and |L[g](λ)| ≤ C, then it is not difficult to prove that J 2 converges. The constant C in the expression ofĜ(ξ, t) is determined by the initial data of G(x, t). So far we complete the proof.
In view of Lemma 2.1 and Duhamel principle, the solution to the problem (1.1)(1.2) could be expressed as following: 3. Decay properties of solution operators. In this section our aim is to obtain the following decay estimates of the solution operators G(t) * and H(t) * appearing in the solution formula (2.4).
To prove the proposition, the key point is to obtain the point-wise estimates of the fundamental solutions in the Fourier space. In fact this could be achieved by using the following point-wise estimate of solutions to the linear problem (1.4) (1.2). Lemma 3.1 (point-wise estimate). Assume u is the solution of (1.4)(1.2), then it satisfies the following point-wise estimate in the Fourier space:

1)
here ρ(ξ) = |ξ| 2 1+|ξ| 4 . To prove Lemma 3.1 we need some notations. For any real or complex-valued function f (t), we define By direct calculation we have the following lemma, which is useful in obtaining our point-wise estimate of solutions in the Fourier space. 6 YONGQIN LIU AND SHUICHI KAWASHIMA Lemma 3.2. For any function k ∈ C(R), and any φ ∈ W 1,2 (0, T ), it holds that Now we will come to get the point-wise estimates in the Fourier space.
Proof of Lemma 3.1.
Step 1: Apply Fourier transform to (1.4) we have that By multiplying (3.2) byū t and taking the real part, we have that Apply Lemma 3.2 2) to the term Re{g * ûū t } in (3.3), and denote Step 2: By multiplying (3.2) by {−(g * ū) t } and taking the real part, we have that Since (g * ū) t = g(0)ū + g * ū, the second term in (3.5) yields that, Step 3: By multiplying (3.2) byū and taking the real part, we have that In view of Lemma 3.

It yields 3) and 4).
Now we use Lemma 3.3 to prove Proposition 1.
Proof of Proposition 1. In view of Lemma 3.3 1), we have that Assume that p satisfies 1 p + 1 p = 1, then Next we prove 3) and 4). It follows from Lemma 3.3 3) that Theorem 4.1 ( energy estimate for linear problem). Let s ≥ 1 be an integer. Assume that u 0 ∈ H s+1 (R n ) and u 1 ∈ H s (R n ), and put Then the solutionū to the problem (1.4), (1.2) given by (2.5) satisfies and the following energy estimate: Proof. We have obtained the solutionū of (1.4)(1.2) given by (2.5) and proved that it satisfies the point-wise estimate in the Fourier space (3.1). From (3.14) and (3.15) we have that ∂ ∂t E(ξ, t) + Cρ(ξ)E(ξ, t) ≤ 0.
Integrate the previous inequality with respect to t and appeal to (3.11), then we obtain Multiply (4.1) by (1 + |ξ| 2 ) s−1 and integrate the resulting inequality with respect to ξ ∈ R n , then we have that (4.2) guarantees the regularity of the solution (2.5). So far we complete the proof of Theorem 4.1.
By using Proposition 1 with p = 2, we obtain the following decay estimates ofū given by (2.5), if initial data u 0 ∈ H s+1 (R n ) and u 1 ∈ H s (R n ).
Remark 1. Under the same assumptions as in Theorem 4.1,ū given by (2.5) also satisfies the following decay estimate, which we think is not optimal, for 0 Denote σ(k, n) = 2k + n + 1 2 , n ≥ 1, (4.5) then the theorem can be stated as follows.
Since proof of Theorem 4.2 and 4.3 are similar, here we only prove Theorem 4.3.
Take sum of (4.10) with 0 ≤ m ≤ s + 1 − σ(k, n) we obtain that Thus (4.7) is proved. 5. Global existence and decay for semi-linear problem. In this section we will first introduce a set of time-weighted Sobolev spaces and employ the contraction mapping theorem to prove the global existence and optimal decay estimates of solutions to the semi-linear problem, then obtain the decay estimate of u t by using the decay estimate for u and the semi-group method. First we give some useful lemmas.

YONGQIN LIU AND SHUICHI KAWASHIMA
Recall Assumption [B], we know that f ∈ C ∞ (R \ {0}), and f (u) = O(|u| α ) as |u| → 0, here α > α n and α n := 1 + 2 n , n ≥ 1, and α is assumed to be an integer for n ≥ 3. By using the decay properties of solution operators and the above lemmas, we obtain the following result about the global existence and optimal decay estimates of solutions to the semi-linear problems (1.1)(1.2). Theorem 5.3 (existence and decay estimates for semi-linear problem). Let s be an integer, s ≥ 2 for n = 1 and s ≥ [ n 2 ] for n ≥ 2 . Also assume that s + 1 ≤ α for n = 1, 2. Let u 0 ∈ H s+1 (R n ) ∩ L 1 (R n ) and u 1 ∈ H s (R n ) ∩ L 1 (R n ), and put I 1 := u 0 H s+1 + u 1 H s + (u 0 , u 1 ) L 1 .