Maximal regularity and global existence of solutions to a quasilinear thermoelastic plate system

We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in R^n. Well-posedness of solutions reconstructing maximal parabolic regularity in nonlinear thermoelastic plates is established. In addition, exponential decay rates for strong solutions are also shown.


Introduction
In this paper we study the existence and exponential stability of solutions to a quasilinear system arising in the modeling of nonlinear thermoelastic plates. The mathematical analysis of thermoelastic systems has attracted a lot of attention over the years. An array of new and fundamental results in the area of wellposedness and stability of solutions to both linear and nonlinear thermoelasticity have been contributed to the field (see [10,11,23,15,16,48,49] and references therein).
The focus of this paper is on thermoelastic plates and associated uniform stability issues. This particular class of problems has received considerable attention in recent years, particularly in the context of some new developments in control theory. Questions such as exponential stability, controllability, observability, unique continuation have been asked and partially answered for both linear and nonlinear plates (see [28] and references therein). While there is at present a vast literature dealing with well-posedness and stability of linear and semilinear thermoelastic equations (see above), the treatment of quasilinear and fully nonlinear models defined on multidimensional domains is much more subtle and requires different mathematical approaches. This paper deals with global and smooth solutions defined for small initial data.
The equations we consider arise from a model that takes into account the coupling between elastic, magnetic and thermal fields in a nonlinear elastic plate model (see [1], [9], [22], [39], [19]). In non-dimensional form, the equations we consider are given below in (1)- (3). The nonlinearity arises from the nature of the magnetoelastic material, owing to a nonlinear dependence between the intensities of the deformation and stress. We also assume that the material nonlinearity is cubic, as in the original plate model [19]. However, the arguments provided depend neither on structure of nonlinearity nor on the order near the origin. We put this generalization in evidence by considering a more general system under the sole Assumption 1 (see below).
Let Ω be a bounded domain of R n , n ∈ N, with boundary ∂Ω ∈ C 2 . Consider the system We assume that the material constant a is positive.
In fact, in what follows we will be able to obtain results for a more general version of equation (1) where the cubic nonlinearity is replaced by a more general nonlinear function of superlinear growth. More specifically, we consider where the function φ satisfies:
(3) There exists ω > 0 and a constant C > 0 such that for |x(0)| Xp,µ ≤ ρ the following exponential estimate holds: (4) For all σ > 0 there exists ω > 0 (independent of σ) and a constant C(σ) > 0 such that for |x(0)| Xp,µ ≤ ρ the following exponential decay rate holds: By specializing φ to φ(s) = s 3 we obtain at once Remark 1. The result obtained in Theorem 2.1 uses weighted norms X p,µ . For µ = 1 one obtains 'classical' L p estimates. These norms account for singularity at the origin and provide trade-off between singularity and additional fractional regularity. Taking p → ∞ allows to obtain "almost" L ∞ -estimates. This is reminiscent to some of the framework introduced in [42,44] 2.3. Comments.
(1) It is interesting to contrast the result of Theorem 2.1 with the one of Theorem 1.3 and Theorem 1.5 of [32] obtained for the original model (1)-(3) within the framework of L 2 theory. More specifically, in [32] global existence and exponential decay rates are shown in the so called finite energy which is [L 2 (Ω)] 3 for the variable x(t). There is no uniqueness result obtained within this framework. This, of course, raises a familiar dilemma of discrepancy between uniqueness and globality of solutions. It is an interesting problem that is still open to the best knowledge of the authors. (2) Unique and "small" solutions for equations (1)- (3) have been also obtained in [32] within the framework of maximal regularity with the spaces C 1 (Ω). However, the above framework leads to the "loss"of incremental differentiability with respect to the initial data. This drawback is no longer present in Theorem 2.1 where the space X p,µ is invariant under the flow. (3) One can consider more general structure of linear matrix operator in (1) as long as it is associated with an exponentially stable semigroup. This is to say that the coefficients of matrix M introduced in (8) may be arbitrary as long as all eigenvalues of M have positive real parts. We shall next address the issue of higher regularity of solutions given by Theorem 2.1. Among other things it will be shown below that under the additional assumption that φ ∈ C ∞ , the solution x(t) is infinitely many times differentiable in time away from t = 0.

Proof of Theorem 2.1
The proof employs techniques developed in the context of abstract parabolic problems and related maximal regularity.
3.1. Abstract parabolic problems and maximal regularity. Let X be a given Banach space and J = [0, T ] or J = [0, ∞) and let A : D(A) ⊂ X → X be a closed operator that is also densely defined . Consider an abstract Cauchy problem and (·, ·) θ,p denotes the real interpolation method.
Definition 3.2. We say that the abstract inhomogeneous Cauchy problem admits maximal L p regularity, if the solution map In particular, the following estimate holds for operators A with maximal L p regularity: 3.2. Setting up (2)-(4) as an abstract parabolic problem. We define [41,37] U := W t , Z := ∆W and set x := (Z, U, Θ).
The differential operator ∆, equipped with zero Dirichlet boundary conditions, generates an analytic semigroup on L p (Ω). With the above notation, the original system can be written in the following operator form: where M is 3 × 3 nonsingular matrix with eigenvalues having positive real parts. It is easily seen that A is the generator of an exponentially stable analytic semigroup e At on X 0 := L p (Ω) × L p (Ω) × L p (Ω) and (7) can be rewritten as (9) x Equation (9) is a nonlinear abstract parabolic system defined on X 0 . The nonlinearity enters via the generator A, and so solvability of the system must depend on "maximal regularity" properties [12,42,47]. Since maximal regularity does not hold within the context of the L ∞ ([0, T ]; X 0 )-topology [42], one should consider the problem within the framework of L p -spaces.
3.3. Representation as a quasilinear abstract parabolic system. Rewriting we obtain from (9) that leads to the consideration of a quasilinear system of the form: (11) x Equation (11) is a quasilinear abstract parabolic system. Since A = M ∆ where M is a real valued 3 × 3 matrix with eigenvalues possessing positive real parts, the operator A has maximal parabolic regularity when considered on the space L p (J, X 0 ) (see e.g. [14]). The interval J can be extended to the positive real axis due to exponential stability of e At . This of course implies that A(0) = A enjoys maximal parabolic regularity on J = (0, ∞) = R + . By [47] the operator A has the property of maximal parabolic regularity in the weighted L p -spaces where µ ∈ (1/p, 1]. In particular, in [47] the authors have shown that the problem if and only if f ∈ L p,µ (J; X 0 ) =: E 0,µ (J) and Moreover the estimate |v| E1,µ(J) ≤ C(|f | E0,µ(J) + |v 0 | Xp,µ ) holds for some constant C > 0.
Let s(A) < 0 be the spectral bound of A and let f ∈ e −ω L p,µ (J; X 0 ) as well as v 0 ∈ X p,µ be given. Consider the problem (12) v in e −ω L p,µ (J; X 0 ). The scaled function u(t) = e ωt v(t) then solves the problem ). Since by assumption e ωt f ∈ L p,µ (J; X 0 ) and v 0 ∈ X p,µ it follows that there exists a unique solution u ∈ E 1,µ (J) of (13). But this in turn implies that there exists a unique solution v ∈ e −ω E 1,µ (J) of problem (12) satisfying the estimate In other words we have shown that the operator A has maximal parabolic regularity in the weighted spaces e −ω L p,µ (J; X 0 ) as long as ω ∈ [0, −s(A)) and µ ∈ (1/p, 1]. The above allows to consider system (11) within this maximal regularity framework. In order to be able to use maximal regularity theory we need to verify several assumptions regarding the operator A(x) and the forcing term f (x). This is done below.
3.4. Supporting estimates. We shall present several estimates which will be used later for the proof of main theorems.
Proof. We first solve the problem v t = Av, v(0) = x 0 in e −ω E 0,µ (J). This yields a solution v = e At x 0 ∈ e −ω E 1,µ (J) satisfying the estimate (20) |v| e −ω E1,µ ≤ C|x 0 | Xp,µ Our next step is to homogenize the equation with respect to the initial data. For this we introduce change of variable w := x − v , so that w| t=0 = 0. Then the sought after solution x can be expressed as being a given function. The regularity g ∈ e −ω E 0,µ (J) follows directly from Lemma 3.3. Thus, our goal is reduced to establishing well-posedness of (21) . Writing A(V ) = A + B(V ), where A has maximal parabolic regularity in e −ω E 0,µ (J), we may rewrite linear equation in w given in (21) as in the space e −ω 0 E 1,µ (J). By Lemma 3.3 and maximal parabolic regularity of A, which then implies invertibility of (∂ t − A) −1 from e −ω E 0,µ into e −ω E 1,µ , we obtain the estimate where s > 0 by the assumption imposed on φ. Therefore, if ρ ∈ (0, ρ 0 ) and ρ 0 > 0 is sufficiently small, a Neumann series argument yields the statement. Recall that equation for w is linear ( w → f (V, w) is also linear). The above and maximal regularity imply the estimate for w The above estimate along with the estimate in (20) leads to the final conclusion in the Lemma.
Note that T is well-defined by Lemma 3.4 and we have the estimate From now on we assume that |x 0 | Xp,µ ≤ δ. This yields Here M ≥ 1 denotes the embedding constant from e −ω E 1,µ ֒→ e −ω BU C(J; X p,µ ). Therefore, if δ = ρ/(M (C(ρ 0 ) + c)), it follows that T (W) ⊂ W.
We shall now show that T is a contraction on W. Let W,W ∈ W and x = T (W ), x = T (W ). By the proof of Lemma 3.4 we have For the first term on the right side we estimate as follows. for some s > 0, since |V | e −ω BUC(J;Xp,µ) , |V | e −ω BUC(J;Xp,µ) , |x| e −ω E1,µ ≤ ρ and X p,µ ֒→ L ∞ (Ω). In a similar way we obtain for some s > 0, by Assumption 1. Since it follows that T is a strict contraction on W provided that ρ > 0 is sufficiently small. The contraction mapping principle yields a unique fixed point x * ∈ W of T , i.e. T (x * ) = x * . By construction of T , the fixed point x * is the unique solution of (11) in e −ω E 1,µ . Moreover x * satisfies as well as where J σ = [σ, ∞) for some σ > 0 and J = J 0 . Here the constant M 1 > 0 comes from the embedding (see [47]) and the constant M 2 > 0 does not depend on σ > 0. This can be seen as follows.
where [T (σ)f ](τ ) := f (τ + σ), τ, σ ≥ 0, is the semigroup of left-translations and M 2 > 0 denotes the embedding constant associated to This yields the estimates valid for all |x 0 | Xp,µ ≤ δ. It follows that x * (t) → 0 in X p as t → ∞ at an exponential rate and the trivial equilibrium of (11) is exponentially stable in X p,µ for µ ∈ ( The proof of Theorem 2.2 follows from Theorem 2.1 and a suitable application of the implicit function theorem (see [13]), which gives both differentiability and analyticity of the nonlinear flow. The parameter trick which will be applied below goes back to [3] and in the context of maximal regularity it has been applied e.g. in [46]. 4.1. Differentiability of solutions. We will show that where A(x) = A + B(Z) and f (x) = −a[0, φ ′′ (Z)|∆Z| 2 , 0] T To this end, let φ satisfy Assumption 1 and, in addition, assume that φ ∈ C 3 (R). The natural candidate for f ′ (x * )x is We have f 3 (Z, Z * ) := φ ′′ (Z * + Z)|∇Z| 2 Since φ ∈ C 3 (R), it is easy to check the desired C 1 -property for [x → f (x)] with the help of Lemma 3.3. In the same way (which is actually easier) one can show be the solution according to Theorem 2.1. We introduce a new function x λ (t) := x * (λt) for λ ∈ (1 − ǫ, 1 + ǫ) and t ∈ J. It follows that Define a mapping H : and by the differentiability properties of A and f . This yields where as before A(x) = A + B(x). We already know that |x * | e −ω E1,µ(J) ≤ C|x * (0)| Xp,µ by (22). This yields if |x * (0)| Xp,µ ≤ δ. Similarly one can show that |B(x * )x − f ′ (x * )x| e −ω E0,µ(J) ≤ Cδ s |x| e −ω E1,µ(J) .

4.3.
Analyticity. If φ is even real analytic, then A and f are real analytic and then so is H. The real analytic version of the implicit function theorem (see e.g. [13,Theorem 15.3]) yields that Φ is real analytic, hence [λ → x λ ] is real analytic. Let t 0 > 0 be fixed and define e(x) := x(t 0 ). It is easy to see that e ∈ L(e −ω E 1 (J σ ); X p ), hence [λ → x λ (t 0 ) = x * (λt 0 )] is real analytic. But since this is true for any t 0 > 0, we obtain that x * is real analytic for all t > 0 with values in X p .