ON SOME NONLINEAR PROBLEM FOR THE THERMOPLATE EQUATIONS

. In this paper, we prove the local and global well-posedness of some nonlinear thermoelastic plate equations with Dirichlet boundary conditions. The main tool for proving the local well-posedness is the maximal L p - L q regularity theorem for the linearized equations, and the main tool for proving the global well-posedness is the exponential stability of C 0 analytic semigroup associated with linear thermoelastic plate equations with Dirichlet boundary conditions.


Introduction.
1.1. Previous research. In this paper, we consider a nonlinear thermoelastic plate equation given by u tt + ∆ 2 u + ∆θ + b∆((∆u) 3 with positive constants b and T , subject to the initial condition u| t=0 = u 0 in Ω, u t | t=0 = u 1 in Ω, θ| t=0 = θ 0 in Ω, (2) and the Dirichlet boundary condition In equations (1), u = u(x, t) denotes a vertical displacement of the plate and θ = θ(x, t) describes the temperature relative to a constant reference temperature at time t ∈ (0, T ) and point x = (x 1 , · · · , x n ) ∈ Ω (see e.g. [14]). In (3), ν is the unit outer normal to Γ. Here, Ω is a domain in R N (N ≥ 2) whose boundary Γ is a C 4 -hypersurface and ∆ stands for the Laplace operator in Ω. The nonlinear term appearing in (1) represents the nature of magnetoelastic material due to a nonlinear dependence between the tension of deformation and stress. Numerous studies have attempted to prove the existence and uniqueness of solutions of the thermoelastic plate equations in the area of both linear and nonlinear. For linearized problem (b = 0), Lasiecka and Triggiani in [16] provided in-depth analysis on physically relevant boundary conditions. For a bounded domain Ω, exponential stability of the associated semigroup in L 2 have been studied by Kim [12], Munos Rivera and Racke [25], Liu and Zheng [24], Lasiecka and Triaggini [16,17,18], and Shibata [28]. The study on general von Karman evolution equations was carried out by Chuesov and Lacieska [2]. Moreover, the analyticity of the associated semigroup is the important aspect of the equation (1). Although the first equation in (1) is a simply dispersive equation (the product of two Schrödinger equations) with respect to u, the heat equations in θ has vigorous effect to have the analyticity of the whole system. This fact was addressed by Liu and Renardy [23], Liu and Liu [21], Liu and Young [22] in the L 2 -setting.
On the other hand, the L p -approach is appropriate to treat the equations with low regularity of the initial data. Therefore, it is essential to analyze the problem (1) to (3) in the L p -setting. Concerning L p framework, Denk and Racke [4] found the analyticity of the generated semigroup of the linearized problem in the whole-space. After that, Naito and Shibata [27] and Naito [26] studied the linearized equation (1) - (3) in the half-space case and proved the generation of C 0 -analytic semigroup. Additionally, Denk, Racke and Shibata [5] proved some decay estimates in bounded domains and exterior domains.
Recently, Lasiecka, Maad and Amol [20] and Lasiecka and Wilke [19] provided a rather complete analysis of the equation (1) with the boundary condition u = ∆u = 0. In [19], the maximal regularity of the system was proved by using an abstract operator-theoretic idea, since the operator ∆ 2 appearing in the first equation in (1) can be defined as a square of Dirichlet-Laplace operator due to the boundary condition: u = ∆u = 0. In the case of general domain, Denk and Shibata [8,9] proved the maximal L p -L q regularity of the linearized system with free boundary conditions: ∆u − (1 − β)∆ u + θ = g 1 on Γ × (0, T ), where ∆ is the Laplace-Beltrami operator on Γ. In that paper, some localization technique was used to handle the free boundary conditions because an abstract approach similar to that in [19] was seemly not available. Regarding to the free boundary conditions, analyticity properties of the associated semigroup in a Hilbert setting has been shown by Lasiecka and Triaggini in [15] and [17]. More recently, Inna [11] proved the maximal L p -L q regularity of the linearized system (1) to (3).
The generation of C 0 analytic semigroup was also shown in [11].

Notations and main results.
To state our results, we first introduce several symbols used throughout the paper. Let N, R and C be the sets of all natural numbers, real numbers and complex numbers, respectively. Let N 0 = N ∪ {0}. For any scalar function f and vector-valued function g where ∂ α x = ∂ α1 1 . . . ∂ α N N , ∂ i = ∂/∂x i , and |α| = α 1 + . . . + α N for any multi-index α = (α 1 , . . . , α N ). For any domain G, we denote the Lebesgue space, Sobolev space and Besov space by L q (G), H m q (G) and B s q,p (G), respectively, and their norms are written by · Lq(G) , · H m q (G) and · B s q,p (G) , respectively. Here, H 0 q (G) is L q (G). In particular, we use the following symbols: where ∂G is the boundary of G and ν G the unit outer normal to ∂G. Let . Let X and Y be Banach spaces with the norms · X and · Y , and let L(X, Y ) be the space of all bounded linear operators from X to Y and L(X, X) is written simply by L(X). Let For scalar functions f and g, we set (f, g) For time interval J = (0, T ), we denote the X-valued Lebesgue space and the X-valued Sobolev space by L p (J, X) and H m q (J, X) (m ∈ N) and their norms by · Lp(J,X) and · H m q (J,X) , respectively. For any domain V in C, we denote the set of all X-valued functions f = f (λ), defined for λ = η + iτ ∈ V , that are continuously differentiable with respect to τ when λ ∈ V by Hol(V, X). Let Σ ϑ = {λ ∈ C \ {0} | | arg λ| < ϑ} for ϑ ∈ (0, π) and let Σ ϑ,λ0 = {λ ∈ Σ ϑ | |λ| ≥ λ 0 } for λ 0 ≥ 0. For any complex number λ, τ denotes the imaginary part of λ, that is λ = η + iτ ∈ C. The letter C denotes generic constants. C a,b,··· denotes a constant depending on the quantities a, b, · · · . The values of C and C a,b,··· may change from line to line.
Secondly, we make a definition.
Definition 1.1. A domain Ω is called a uniformly C 4 -domain if there exist positive constants K, L 1 , L 2 such that for any x 0 ∈ Γ there exists a coordinate number j and We now state the main results of this paper. Theorem 1.2. Let N < q < ∞, 2 < p < ∞, T 1 > 0 and R > 0. Assume that Ω is a uniform C 4 -domain in R N . If the initial data U 0 = (u 0 , v 0 , θ 0 ) and functions f 1 f 2 ∈ L p ((0, T 1 ), L q (Ω)) of the right side of (4) satisfy and satisfy compatibility conditions: NONLINEAR PROBLEM FOR THE THERMOPLATE EQUATIONS   759 and a positive condition: for some positive numbers A 1 and A 2 , then there exists a time T ∈ (0, T 1 ) depending on R, b, A 1 and A 2 such that problem (5) admits a unique solution U with ) possessing the estimate: Assume that Ω is a bounded domain whose boundary Γ is a compact C 4 hypersurface. Then, there exist small positive numbers and η such that if the initial data U 0 = (u 0 , v 0 , θ 0 ) and functions f 1 f 2 ∈ L p (R + , L q (Ω)) of the right side of (4) satisfy and satisfy compatibility conditions then the problem (6) with T = ∞ admits a unique solution U with ) possessing the estimate: e ηt U Lp(R+,H 2 q (Ω)) + e ηt ∂ t U Lp(R+,H 0 q (Ω)) ≤ C . To prove Theorem 1.2, the main step is to prove the L p -L q maximal regularity for the following linearized problem: with where α is a uniformly continuous function with respect to x ∈Ω and satisfies for positive constants A 1 and A 2 . Namely, we prove where (·, ·) θ,p denotes the real interpolation functor. Then, there exists a positive number λ 1 such that for any initial data U 0 = (u 0 , v 0 , θ 0 ) ∈ D q,p (Ω) and F = (f, g, h) ∈ L p ((0, T ), H 0 q (Ω)), Eq. (11) admits a unique solution U with U ∈ L p ((0, T ), H 2 q (Ω)) ∩ H 1 p ((0, T ), H 0 q (Ω)) possessing the estimate: (Ω)×B (Ω) 2 + F Lp((0,T ),H 2 q (Ω)×Lq(Ω) 2 ) ).
To prove Theorem 1.3, the main step is to prove the exponential stability of solutions of the following linearized equations: Theorem 1.5. Let 1 < p, q < ∞ Assume that Ω is a bounded domain whose boundary Γis a compact C 4 hypersurface. Let D q,p (Ω) be the same set as in Theorem 1.4. Then, there exists a positive number η such that for any initial data ) possessing the estimate: (Ω)×B (Ω) 2 + e ηt G Lp(R+,H 2 q (Ω)×Lq(Ω) 2 ) ). Here and in the following, we write

R-bounded solution operators.
To prove the maximal L p -L q -regularity of problem (11), we show the existence of an R-bounded solution operator of the associated resolvent problem: with α satisfying (12). Let ϑ 0 (π/2 < ϑ 0 < π) be a constant given in (20) below. In this section, we shall prove the following theorem.
Our analysis is based on the results due to Inna [11] (cf. Naito and Shibata [27] and Denk and Shibata [8]). Before proving Theorem 2.1, we introduce the definition and some fundamental properties of R-bounded operators and Bourgain's results concerning Fourier multiplier theorems with scalar multiplier. .
Here the Rademacher function r k , k ∈ N, are given by r k : [0, 1] → {−1, 1}, t → sign(sin(2 k πt)). The smallest such C is called the R-bound of T on L(X, Y ) which is written by R L(X,Y ) (T ) in what follows.
(a) Let X and Y be Banach spaces, and let T and S be R-bounded families in L(X, Y ). Then, T + S = {T + S | T ∈ T , S ∈ S} is also an R-bounded family in L(X, Y ) and Let X, Y and Z be Banach spaces, and let T and S be R-bounded families in L(X, Y ) and L(Y, Z), respectively. Then, ST = {ST | T ∈ T , S ∈ S} also an R-bounded family in L(X, Z) and Let T n be the scalar-valued Fourier multiplier defined by ). Then, T n is extended to a bounded linear operator from L p (R, L q (D)) into itself. Moreover, denoting this extension also by T n , we have T n L(Lp(R,Lq(D))) ≤ C p,q,D γ. Here, D(R, L q (D)) denotes the set of all L q (D)-valued C ∞ -functions on R with compact support.
2.1. Analysis in the whole space. Let A 1 and A 2 be two positive numbers given in (12). In this section, we assume that α is a positive number such that A 1 ≤ α ≤ A 2 . We consider the resolvent problem in the whole space: and then by using the Fourier transform, we transform Eq. (16) to the following linear equation: Here, the Fourier transform of a function g is defined bŷ The determinant of λI −Â α (ξ) is given by where γ 1 , γ 2 and γ 3 are numbers such that Let γ 1 be a real number, and then γ 1 ∈ (0, 1). Moreover, since f (s) > 0, γ 2 and γ 3 are complex numbers such that γ 3 = γ 2 and Re γ 2 = Re γ 3 . Since γ i is a continuous function with respect to α, because Re γ 2 = Re γ 3 ∈ (0, 1/2). Let ϑ 0 be a number such that π/2 < ϑ 0 < π and ϑ 0 + γ max = ϑ 1 < π. Then, we have Employing the same argumentation as in Naito and Shibata [27] in R N (cf. also Denk and Shibata [8], Inna [11]), we have the following theorem.
for s = 0, 1 and (κ, k) ∈ N N +1 0 with |κ| + k = 2 and k = 0, 1, 2. Here, γ α,λ0 is a constant depending on α and λ 0 . Moreover, we have set We next show that there exists a γ 0 depending on λ 0 such that Let α and β be two positive numbers in [A 1 , A 2 ], and we consider the equation: Then, we have Note that Let U = A α (λ)F , and then

ON SOME NONLINEAR PROBLEM FOR THE THERMOPLATE EQUATIONS 763
From the estimate: for s = 0, 1 and j = 0, 1, 2. Moreover, for any λ ∈ Σ ϑ,λ0 and F ∈ H 0 q (R N ), U = A β (λ)F is a unique solution of Eq. (22). Summing up, we have proved the following theorem.

2.2.
Analysis of a perturbed problem in R N . Let α(x) be a real valued continuous function satisfying the assumption (12). Let x 0 be any point in Ω and let M 1 be any number in (0, 1). Let d 0 > 0 be a small positive number such that We may assume that In this subsection, we consider the resolvent problem: We shall prove the following theorem.
To prove the uniqueness, let U ∈ H 2 q (R N ) satisfy the homogeneous equation: (27) and (23) . Since γ 0 M 1 < 1/2, we have U = 0. This shows the uniqueness, and therefore the proof is completed.
2.3. Analysis in the half space. Let A 1 and A 2 be two positive numbers such that A 1 < A 2 . Let α be a positive number such that A 1 ≤ α ≤ A 2 . We consider the resolvent problem: The following theorem is the main result of this section.
In the following, we prove Theorem 2.6. Given k defined on R N + . let k 0 be the odd extension of k, which is defined by be the solution operator given in the Theorem 2.4, and let for any F ∈ H 0 q (R N + ). Let U 0 = A + (λ)F , and then Moreover, by Theorem 2.4, we have for s = 0, 1 and j = 0, 1, 2, because F o Thus, we will construct a solution U 1 = (u 1 , v 1 , θ 1 ) such that where Let w = u 1 , λw = v 1 , τ = θ 1 and W = ∂u 0 ∂x N , and then To solve (37), we apply the partial Fourier transform, defined bỹ to (37), where ξ = (ξ 1 , . . . , ξ N −1 ), and then we obtain a system of ordinary differential equations: with where γ i (i = 1, 2, 3) are the numbers given in (18), while and We have Employing the same argument as in Naito and Shibata [27] (cf. also Denk and Shibata [8], Inna [11]), we have the following theorem.
Employing the same argument as in Subsec 2.1 and using Theorem 2.7, we have Theorem 2.6.
Here, C K is a constant depending on the constants K, L 1 and L 2 appearing in Definition 1.1 but independent of M 1 . We choose M 1 small enough eventually, so that we may assume that 0 < M 1 ≤ 1 ≤ M 2 without loss of generality. Let Let ν + be the unit outer normal to Γ + and let ∂ ν+ = ν + · ∇. Let α be a real valued function satisfying the condition (12). Let y 0 be any point on Γ + and we assume in this subsection that there exists a positive number d 0 such that |α(y) − α(y 0 )| ≤ M 1 for any y ∈ B d0 (y 0 ) ∩ Ω + .
In this section, consider the following resolvent problem in a bent half space: for U = (u, v, θ) . We shall prove the following theorem.
(54), we have By (52) and (45), Here and in the following, C denotes generic constants independent of M 1 and M 2 and C M2 denotes generic constants depending on M 2 . Let λ 0 be any number ≥ 1.
The uniqueness can be shown by a priori estimates of the solutions of homogeneous equations. The argument is the same as in the proof of Theorem 2.5, and so we may omit the proof of the uniqueness. This completes the proof of Theorem 2.8.

Analysis in a general Domain.
To prove Theorem 2.1, we use several properties of uniform C 4 domain, which are stated in the following proposition.

Proposition 2.
Let Ω be a uniform C 4 -domain in R N with boundary Γ. Then, for any positive constant M 1 , there exist constants M 2 > 0, d 0 , d 1 ∈ (0, 1), at most countably many functions Φ j ∈ C 4 (R N ) and points x 0 j ∈ Γ and x 1 j ∈ Ω such that the following assertions hold: Here, c 0 is a constant which depends on M 2 and N , but is independent of j ∈ N.
Moreover, R i j and R i,− j satisfy the conditions: Here, C K is a constant depending on the constants K, L 1 and L 2 appearing in Definition 1.1 but independent of j ∈ N, M 1 and M 2 . In what follows, we write Ω 0 = R N , Ω 1 = Φ (R N + ) and Γ = Φ (R N 0 ) for ∈ N. Furthermore, we simply write V i = Ω i ∩ B i d (x i ) ( ∈ N, i = 0, 1). Moreover, ζ i j and ζ i j are functions given in Proposition 2, From Proposition 2 (5), we have for any f ∈ L q (Ω) and 1 ≤ q < ∞. Then by (61) we have the following lemma [11,Lemma 5.1].
3. Maximal L p -L q regularity and the local well-posedness.

3.1.
A proof of Theorem 1.4. In this section, we shall prove the Theorem 1.4. First of all, we consider the Cauchy problem: Let A be an operator defined by AU = A α (D)U for U ∈ H 2 q (Ω), then by Theorem 2.1 we have Σ ϑ0,λ0 is contained in the resolvent set of A and (λI−A) −1 F = S α (λ)F for F ∈ H 0 q (Ω). Moreover, by the definition of R-boundedness with m = 1 in Definition 2.2, for any λ ∈ Σ ϑ0,λ0 and F ∈ H 0 q (Ω) with some constant C α > 0. Thus, A generates a C 0 analytic semigroup {T (t)} t≥0 on H 0 q (Ω) satisfying the estimate: with η 0 ≥ λ 0 ≥ 1. Employing the same argument as that in Shibata and Shimizu [30, Proof of Theorem 3.9], we see that for any U 0 = (u 0 , v 0 , θ 0 ) ∈ D q,p (Ω), Eq.
(71) admits a unique solution V with ) possessing the estimate: (Ω) 2 , with some positive constants C and c.
The next step is to homogenize the equation with respect to the initial data. For this purpose we introduce the unknown variable W = U − V so that W | t=0 = 0. Then the desirable solution U can be stated as U = V + W where W is the solution of wit F = (f 0 , f 1 , f 2 ) ∈ L q (Ω) 3 .

3.2.
A proof of the Theorem 1.2. Let T ∈ (0, T 1 ) and L > 0 be numbers determined later, and set Where, we have set for the notational simplicity. Given U = (u, v, θ) ∈ I T , let W W ∈ L p ((0, T ), H 2 q (Ω)) ∩ H 1 p ((0, T ), H 0 q (Ω)) be a unique solution of the equations: where α = 1 + bφ (∆u 0 ), and By real interpolation theorem and (77), we have where C is a constant independent of T > 0 and R is the number given in Theorem 1.2. Since p > 2 and q > N , we choose > 0 so small that N/q + < 1, and then by Sobolev's imbedding theorem In particular, by (80) we have Since (N (1/q − 1/(2q)) = N/(2q) < 1, we have Since N/q < 1, by Hölder's inequality we have We next consider (φ (∆u 0 ) − φ (∆u))∆ 2 u. By Sobolev's imbedding theorem and (79), Applying Theorem 1.4 to Eq. (78) and using (84) gives Choosing L > 0 so large that 2CeR = L and choosing T ∈ (0, T 1 ) so small that by (85) we have W E p,q,T ≤ L (86) with L = 2CeR. Let Φ be a map defined by ΦU = W , and then by (86), Φ maps Employing the same argument as that in proving (86), we have with some constants s > 0 and C L,R . Here, C L,R is a constant depending on L and R. Choosing T > 0 so small that C L,R T s ≤ 1/2, by (87), we see that Φ is a contraction map on I T , and so by Banach's fixed point theorem, there exists a unique U ∈ I T such that ΦU = U , which is a required unique solution of Eq. (5). This completes the proof of Theorem 1.2.
4. Exponential stability and the global well-posedness.
Since Ω is bounded, to prove the unique existence of solutions of Eq. (88), in view of the Riesz-Schauder theorem, the Fredholm alternative principle, it suffices to prove the uniqueness of Eq. (88). Let U ∈ H 2 q (Ω) be a solution of the homogeneous equation: λU − A(D)U = 0 in Ω, B(D)U | Γ = 0.
(91) Notice that Eq. (91) is written componentwise as We first consider the case where 2 ≤ q < ∞.
Summing up, we have proved the unique existence theorem of Eq. (88) for each λ ∈ Λ λ0 . Let A be an operator defined by letting Then, we see that for any λ ∈ Λ λ0 , (λI − A) −1 exists. By the Banach fixed point theorem, there exists a constant C λ for which . And then, employing the same argument as in the proof of Theorem 2.4, we see that there exists a constant C independent of λ ∈ Λ λ0 for which . This completes the proof of Proposition 3.
Let A be the operator defined in (95). Combining (89) and Proposition 3, we see that {T (t)} t≥0 is exponentially stable, that is there exist positive constants C and η for which for any t > 0 and F ∈ H 0 q (Ω). We now consider the evolution equations (14). Replacing α by 1, we know that Theorem 1.4 and Theorem 2.1 holds for the operator A(D). In particular, we know the existence of solution U of Eq. (14) with U ∈ (L p,loc ((0, T ), H 2 q (Ω)) ∩ H 1 p,loc ((0, T ), H 0 q (Ω))) for any initial data U 0 ∈ D q,p (Ω) and F ∈ L p (R, H 0 q (Ω)). By Duhamel's principle, the solution U = U (t) of Eq. (14) is written as and so for any η 1 ∈ (0, η), by (96) This completes the proof of Theorem 1.5.