\documentclass[reqno]{amsart} \usepackage{graphicx,amssymb,mathrsfs} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 48, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/48\hfil $L^p$-resolvent estimates] {$L^p$-resolvent estimates and time decay for generalized thermoelastic plate equations} \author[R. Denk, R. Racke\hfil EJDE-2006/48\hfilneg] {Robert Denk, Reinhard Racke} \address[Robert Denk]{Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany} \email{robert.denk@uni-konstanz.de} \address[Reinhard Racke]{ Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany} \email{reinhard.racke@uni-konstanz.de} \date{} \thanks{Submitted September 13, 2005. Published April 11, 2006.} \subjclass[2000]{35M20, 35B40, 35Q72, 47D06, 74F05} \keywords{Analytic semigroup in $L^p$; polynomial decay rates; Cauchy problem} \begin{abstract} We consider the Cauchy problem for a coupled system generalizing the thermoelastic plate equations. First we prove resolvent estimates for the stationary operator and conclude the analyticity of the associated semigroup in $L^p$-spaces, $1 0$, and $\alpha, \beta \in [0, 1]$ are parameters of the ``$\alpha$-$\beta$-system" (\ref{eq1.1}) (\ref{eq1.2}). The constants $a, b, d, g$ are positive and assumed to be equal to one in the sequel w.l.o.g. For $\eta = 2$ and $\alpha = \beta = 1/2$ we have the thermoelastic plate equations (\cite{La89}), \begin{gather} u_{tt}+ a \Delta^2 u + b \Delta \theta 0, \label{eq1.4}\\ \theta_{t} - g \Delta \theta - b \Delta u_{t} = 0 \label{eq1.4a} \end{gather} which has been widely discussed in particular for bounded reference configurations $\Omega \ni x$, see the work of Kim \cite{Ki92}, Mu\~noz Rivera \& Racke \cite{MR95}, Liu \& Zheng \cite{LZ97}, Avalos \& Lasiecka \cite{AL97}, Lasiecka \& Triggiani \cite{LTa, LTb, LTc, LTd} for the question of exponential stability of the associated semigroup (for various boundary conditions), and Russell \cite{Ru93}, Liu \& Renardy \cite{LR95}, Liu \& Liu \cite{LL97}, Liu \& Yong \cite{LY98} for proving its analyticity, see also the book of Liu \& Zheng \cite{LZ99} for a survey. In our paper \cite{MR96} we introduced the more general $\alpha$-$\beta$-system (\ref{eq1.1}), (\ref{eq1.2}), in a general Hilbert space $\mathcal{H}$, $S$ self-adjoint, and also proved for $\beta =1/2$ polynomial decay rates of $L^\infty$-norms $\|(- \Delta)^{\eta/2} u (t, \cdot), u_t(t, \cdot), \theta(t, \cdot)\| _{L^\infty (\Omega)}$ of the solutions for $\Omega = \mathbb{R}^n$ or $\Omega$ being an exterior domain, $\mathcal{H} = L^2(\Omega)$ essentially. It was demonstrated that the $\alpha$-$\beta$-system may also describe viscoelastic equations of memory type with even non convolution type kernels for $(\beta = 1/2, \alpha = 0)$, and that it captures features of second-order thermoelasticity for $(\beta = 1/2, \alpha = 1/2)$. In \cite{MR96} the region $D$ of parameters where the system has a smoothing property, \begin{equation} D = \{(\beta, \alpha) | 1 - 2 \beta < \alpha < 2 \beta, \alpha> 2\beta - 1\}; \label{eq1.5} \end{equation} see Figure 1. \noindent \begin{figure}[htb] \centering \setlength{\unitlength}{0.22mm} \begin{picture}(230,270)(50,-5) \put (60,20){\vector(0,1){250}} \put (60,20){\vector(1,0){250}} \put (160,20){\line(-1,2){50}} \put (160,20){\line(1,2){100}} \put (110,120){\line(1,2){50}} \put (110,-5){$\frac{1}{4}$} \put (110,15){\line(0,1){10}} \put (160,-5){$\frac{1}{2}$} \put (160,15){\line(0,1){10}} \put (210,-5){$\frac{3}{4}$} \put (210,15){\line(0,1){10}} \put (260,-5){$1$} \put (300,-5){$\beta$} \put (260,15){\line(0,1){10}} \put (35,70){$\frac{1}{4}$} \put (55,70){\line(1,0){10}} \put (35,120){$\frac{1}{2}$} \put (55,120){\line(1,0){10}} \put (35,170){$\frac{3}{4}$} \put (55,170){\line(1,0){10}} \put (35,220){$1$} \put (35,260){$\alpha$} \put (55,220){\line(1,0){10}} \put (157,26){\line(1,2){97}} \put (154,32){\line(1,2){94}} \put (151,38){\line(1,2){91}} \put (148,44){\line(1,2){88}} \put (145,50){\line(1,2){85}} \put (142,56){\line(1,2){82}} \put (139,62){\line(1,2){79}} \put (136,68){\line(1,2){76}} \put (133,74){\line(1,2){73}} \put (130,80){\line(1,2){70}} \put (127,86){\line(1,2){67}} \put (124,92){\line(1,2){64}} \put (121,98){\line(1,2){61}} \put (118,104){\line(1,2){58}} \put (115,110){\line(1,2){55}} \put (112,116){\line(1,2){52}} \put(60.00,20.00){\dashbox{2.00}(200.00,200.00){}} \put(60.00,20.00){\dashbox{2.00}(100.00,200.00){}} \put(60.00,20.00){\dashbox{2.00}(50.00,100.00){}} \end{picture} %\vspace*{1cm} \caption{Area of smoothing} \end {figure} The $\alpha$-$\beta$-system was independently introduced by Ammar Khodja \& Benabdallah \cite{AB1}. In particular they proved the analyticity of the associated semigroup for $\alpha = 1$ and, if and only if, $3/4 \leq \beta \leq 1$. Also Liu \& Liu \cite{LL97} and Liu \& Yong \cite{LY98} studied general $\alpha$-$\beta$-systems in the Hilbert space case (``bounded domains''), in particular in \cite{LY98} they obtained analyticity in the region \begin{equation} \widetilde {\mathfrak{A}}:= \{(\beta, \alpha) | \alpha > \beta, \alpha \leq 2 \beta - 1/2\}. \label{eq1.6} \end{equation} Shibata \cite{Sh04} obtained the analyticity in $L^p$-spaces, $ 1 < p < \infty$, for the classical thermoelastic plate; i.e., for $(\beta, \alpha) = (1/2, 1/2)$. All but the last one of the above mentioned papers work in Hilbert spaces, none can replace $L^2 (\Omega)$ by $L^p (\Omega)$, $1 < p < \infty$ if $(\beta,\alpha) \neq (1/2, 1/2))$, and none gives (polynomial) decay rates --- if $\beta$ is different from $1/2$. So our goals and new contributions are \begin{itemize} \item To discuss the $\alpha$-$\beta$-system in $L^p (\mathbb{R}^n)$-spaces,$1 < p < \infty$, and to describe the region ${\mathfrak{A}}$ of parameters $(\beta, \alpha)$ of analyticity of the semigroup, and \item To obtain sharp polynomial decay rates for $\| S^{1/2} u(t, \cdot), u_t(t,\cdot),\theta (t, \cdot) \|_{L^q(\Omega)}$ for $2 \leq q \leq \infty$, and $(\beta, \alpha)$ in the analyticity region ${\mathfrak{A}}$, but also for $ 1/4 \leq \beta \leq 3/4$ while $\alpha = 1/2$ (exemplarily). \end{itemize} We shall obtain the following region of analyticity \begin{equation} {\mathfrak{A}} = \{(\beta, \alpha)| \; \alpha \geq \beta, \; \alpha \leq 2 \beta - 1/2\}, \label{eq1.7} \end{equation} see Figure 2 (cp.(\ref{eq1.6})) in proving resolvent estimates in $L^p$-spaces using the theory of parameter-elliptic mixed-order systems by Denk, Mennicken \& Volevich \cite{dmv98}. \noindent \begin{figure}[htb] \centering \vspace*{1cm} \setlength{\unitlength}{0.75\unitlength} \begin{picture}(230,230)(10,20) \put (60,20){\vector(0,1){250}} \put (60,20){\vector(1,0){250}} \put (158,-5){$\frac{1}{2}$} \put (160,15){\line(0,1){10}} \put (208,-5){$\frac{3}{4}$} \put (210,15){\line(0,1){10}} \put (258,-5){$1$} \put (300,-5){$\beta$} \put (260,15){\line(0,1){10}} \put (35,120){$\frac{1}{2}$} \put (55,120){\line(1,0){10}} \put (35,220){$1$} \put (35,260){$\alpha$} \put (55,220){\line(1,0){10}} %\put(250,220){\line(2,-1){10}} %\put(250,220){\line(2,-1){10}} \put(160,120){\line(1,1){100}} \put(160,120){\line(1,2){50}} \put(162.5,125){\line(1,0){2.5}} \put(165,130){\line(1,0){5}} \put(167.5,135){\line(1,0){7.5}} \put(170,140){\line(1,0){10}} \put(172.5,145){\line(1,0){12.5}} \put(175,150){\line(1,0){15}} \put(177.5,155){\line(1,0){17.5}} \put(180,160){\line(1,0){20}} \put(182.5,165){\line(1,0){22.5}} \put(185,170){\line(1,0){5}} \put(207,170){\line(1,0){3}} \put(193,167){$\mathfrak A$} \put(187.5,175){\line(1,0){2.5}} \put(207,175){\line(1,0){8}} \put(190,180){\line(1,0){30}} \put(192.5,185){\line(1,0){32.5}} \put(195,190){\line(1,0){35}} \put(197.5,195){\line(1,0){37.5}} \put(200,200){\line(1,0){40}} \put(202.5,205){\line(1,0){42.5}} \put(205,210){\line(1,0){45}} \put(207.5,215){\line(1,0){47.5}} \put(210,220){\line(1,0){50}} \put(164,172.5){\vector(1,-1){15}} \put(87,175){\small $\alpha =2\beta-\frac 12$} \put(232.5,162.5){\vector(-1,1){15}} \put(218,150){\small $\alpha = \beta$} \put(60.00,20.00){\dashbox{2.00}(200.00,200.00){}} \put(60.00,20.00){\dashbox{2.00}(100.00,100.00){}} \put(60.00,20.00){\dashbox{2.00}(150.00,200.00){}} \end{picture} \vspace*{1cm} \caption{Area of analyticity} \end {figure} The polynomial decay estimates will be obtained in applying the Fourier transform and analysing the arising characteristic polynomial \begin{equation} P (\xi, \lambda) := \lambda^3 + \rho^\alpha \lambda^2 + (\rho^{2 \beta} + \rho) \lambda + \rho^{1 + \alpha} \label{eq1.8} \end{equation} carefully, where $\rho := | \xi |^{2\eta}$. In particular we describe the asymptotic expansion as $\lambda \to 0$ (and $\lambda \to \infty$). The results here will also be basic for further investigations of boundary value problems in exterior domains. We remark that the transformation to a first-order system via $(S^{1/2}u,u_t,\theta)$ immediately transfers for the classical case (\ref{eq1.4}), (\ref{eq1.4a}) to the hinged boundary conditions $$ u=\Delta u =\theta =0 \quad \mbox{on the boundary}. $$ For other boundary conditions like the Dirichlet type ones $$ u=\partial_\nu u =\theta =0, $$ where $\partial_\nu$ stands for the normal derivative at the boundary, other transformations like $(u,u_t,\theta)$ will be more appropriate, cp. \cite{Sh04}. We stress that we here obtain information on the Cauchy problem independent of any, and useful for any boundary conditions. Still boundary values problems require a sophisticated analysis as a future task. The paper is organized as follows: In Section 2 we review the relevant parts of the theory of parameter-elliptic mixed-order systems. The application to the $\alpha$-$\beta$-system is given in Section 3. In Section 4 we prove the decay estimates for solutions as time tends to infinity. \section{Remarks on mixed order systems} The theory of mixed order systems usually deals with matrices of partial differential operators. As the generalized thermoelastic plate equation leads to a matrix with pseudo-differential operators with constant symbols, we will formulate the definitions and results for such matrices. It is also possible to consider general pseudo-differential operators (see, for instance, the book of Grubb \cite{grubb} in this context). However, for the present case such general framework is not necessary, and we will deal only with Fourier multipliers. In the following, the letter $\mathcal F$ stands for the Fourier transform in $\mathbb{R}^n$, acting in the Schwartz space of tempered distributions $\mathcal F\colon \mathscr S'(\mathbb{R}^n)\to \mathscr S'(\mathbb{R}^n)$. For a symbol $a(\xi)$ (belonging to some symbol class), the pseudo-differential operator $a(D)$ is defined by $a(D) := \mathcal F^{-1} a(\xi) \mathcal F$. If $a(\xi)$ is homogeneous with respect to $\xi$ of non-negative degree $\mu$, then the pseudo-differential operator $a(D)$ has order $\mu$. In an obvious way, for $r>0$ the order of symbols like $|\xi|^r$ and $1+|\xi|^r$ are equal to $r$. In the following we will consider operator matrices $A(D) = (A_{ij}(D))_{i,j=1,\dots,n}$ where every entry $A_{ij}(D)$ is a Fourier multiplier of the form \[ A_{ij}(D) = |D|^{\alpha_{ij}} = \mathcal F^{-1} |\xi|^{\alpha_{ij}} \mathcal F. \] In this case, $\alpha_{ij} = {\rm ord} A_{ij}(D)$. For a permutation $\pi:\{ 1,\dots,n\}\to\{1,\dots,n\}$ we define \[ R(\pi) := \alpha_{1 \pi(1)} + \dots + \alpha_{n \pi(n)}. \] We set $R := \max_\pi R(\pi)$. Then there exist real numbers $s_1,\dots,s_n$ and $r_1,\dots,r_n$ such that \[ \alpha_{ij} \le s_i + t_j,\quad \sum_{i=1}^n (s_i+t_i) = R. \] For differential operators, this was shown by Volevich in \cite{v63}. The case of non-integer orders follows in exactly the same way. \begin{definition}\label{2.1}\rm The matrix $A(D)$ and the corresponding symbol $A(\xi)$ are called elliptic in the sense of Douglis-Nirenberg (or elliptic mixed order system) if \begin{itemize} \item[(i)] $A(\xi)$ is non-degenerate, i.e. $R = \deg\det A(\xi)$. \item[(ii)] $\det A(\xi)\not=0$ for all $\xi\in \mathbb{R}^n\setminus\{0\}$. \end{itemize} \end{definition} For an elliptic matrix $A(\xi)$ the principal part is defined as \[ A_{ij}^0(\xi) := \begin{cases} A_{ij}(\xi) & \text{ if } \mathop{\rm ord} A_{ij} = s_i+t_j,\\ 0 & \text{otherwise.}\end{cases} \] Note that the numbers $s_i$ and $t_j$ are defined up to translations of the form \begin{equation} (s_1,\dots,s_n,t_1,\dots,t_n)\mapsto (s_1-\kappa, \dots, s_n-\kappa, t_1+\kappa, \dots, t_n+\kappa). \label{eq2.1} \end{equation} On the diagonal we have the orders \begin{equation} r_i := s_i + t_i \quad (i=1,\dots,n). \label{eq2.2} \end{equation} Now let $A(D)$ be an elliptic mixed order system of the form indicated above. To solve the Cauchy problem $\frac{d}{dt} U = A(D) U$, one can consider the parameter-dependent symbol $\lambda - A(\xi)$ where the complex parameter $\lambda$ belongs to some sector in the complex plane. In the homogeneous case, this is the standard approach to parabolic equations, see, e.g., \cite{AV64}. Here the homogeneity of the determinant $\det(\lambda - A(\xi))$ as a function of $\lambda$ and $\xi$ is essential for resolvent estimates. For mixed order systems of the form $\lambda-A(D)$, however, the definition of parabolicity and parameter-ellipticity is not obvious. In \cite{dmv98} a definition of this notion and several equivalent descriptions can be found. We will recall and slightly generalize some definitions and main results of \cite{dmv98}. We start with the definition of the Newton polygon associated to $A(\xi)$. In the case considered in the present paper, the determinant $P(\xi,\lambda) := \det(\lambda - A(\xi))$ is a polynomial in $\lambda\in\mathbb{C}$ and $|\xi|$ for $\xi\in\mathbb{R}^n$ which can be written in the form \begin{equation} P(\xi,\lambda) = \sum_{\gamma,k} a_{\gamma k} |\xi|^\gamma \lambda^k. \label{eq2.3} \end{equation} Here the exponents $\gamma$ of $|\xi|$ are, in general, non-integer. The Newton polygon $N(P)$ of $P(\xi,\lambda)$ is defined as the convex hull of all points $(\gamma,k)$ for which the coefficient $a_{\gamma k}$ in (\ref{eq2.3}) does not vanish, and the projections of these points onto the coordinate axes. For instance, consider the symbol $\lambda^3 + |\xi|^3 \lambda^2 + |\xi|^{9/2}\lambda$. The associated Newton polygon is the convex hull of the points $(0,3), (3,2), (\frac92,1), (\frac92,0)$ and $(0,0)$. A Newton polygon is called regular if it has no edges parallel to one of the axes but not belonging to this axis. In the following, let $\mathscr L$ be a closed sector in the complex plane with vertex at the origin. The constant $C$ stands for an unspecified constant which may vary from line to line but which is independent of the free variables. The following definition is a slight modification of the definition in \cite{dmv98}. \begin{definition}\label{2.2} \rm a) Let $N(P)$ be the Newton polygon of the symbol \eqref{eq2.3}. Then this symbol is called parameter-elliptic in $\mathscr L$ if there exists a $\lambda_0>0$ such that the inequality \begin{equation} |P(\xi,\lambda) | \ge C W_P(\xi,\lambda) \quad (\lambda\in\mathscr L,\; |\lambda|\ge\lambda_0,\;\xi\in\mathbb{R}^n) \label{eq2.4} \end{equation} holds where $W_P$ denotes the weight function associated to $P$: \begin{equation} W_P(\xi,\lambda) := \sum_{\gamma, k} |\xi|^\gamma |\lambda|^k. \label{eq2.5} \end{equation} The last sum runs over all indices $(\gamma,k)$ which are vertices of the Newton polygon $N(P)$. b) The mixed order system $\lambda-A(D)$ is called parameter-elliptic in $\mathscr L$ if the Newton polygon of $P(\xi,\lambda) := \det(\lambda-A(D))$ is regular and if $P$ is parameter-elliptic in $\mathscr L$. \end{definition} There are several equivalent descriptions of parameter-ellipticity for mixed order systems (see \cite{dmv98}). The Newton polygon approach is a geometric description of the various homogeneities contained in the determinant $P(\xi,\lambda) = \det(\lambda-A(\xi))$. For $r>0$ and a polynomial of the form (\ref{eq2.3}) define the $r$-order of $P$ by \[ d_r(P) := \max\{ \gamma+rk: a_{\gamma k}\not=0\}. \] The $r$-principal part $P_r(\xi,\lambda)$ is given by \[ P_r(\xi,\lambda) := \sum_{\gamma+rk= d_r(P)} a_{\gamma k} |\xi|^\gamma \lambda^k. \] The following result is a straightforward generalization of Theorem 2.2 in \cite{dmv98} where polynomial entries were considered. \begin{theorem}\label{2.3} Let $A(D)$ be a mixed order system and $P(\xi,\lambda) = \det(\lambda-A(\xi))$. Then the following statements are equivalent. \begin{itemize} \item[(a)] The operator matrix $\lambda - A(D)$ is parameter-elliptic in $\mathscr L$. \item[(b)] There exist constants $C>0$, $\lambda_0>0$ such that \begin{equation} \big| P(\xi,\lambda) \big| \ge C \prod_{i=1}^n (|\xi|^{r_i} + |\lambda|) \quad (\lambda\in {\mathscr L}, |\lambda|\ge \lambda_0, \xi\in\mathbb{R}^n). \label{eq2.6} \end{equation} Here the numbers $r_i$ are defined in (\ref{eq2.2}). \item[(c)] For every $r>0$, \begin{equation} P_r(\xi,\lambda)\not=0 \quad (\lambda\in {\mathscr L}\setminus\{0\},\; \xi\in\mathbb{R}^n\setminus\{0\}). \label{eq2.7} \end{equation} \end{itemize} \end{theorem} The condition of parameter-ellipticity is equivalent to a uniform estimate of the entries of the inverse matrix, see \cite{dmv98}, Proposition 3.10. Applying Plancherel's theorem, we immediately obtain $L_2$-estimates for the solution. When we deal with $L_p$-spaces, we want to apply Michlin's theorem. For this we need another estimate which is contained in the following theorem. \begin{theorem}\label{2.4} Let $P(\xi,\lambda)$ be parameter-elliptic in the sector $\mathscr L$ and assume, for simplicity, that $N(P)$ is regular. Let $(\sigma,\kappa)\in\mathbb{R}^2$ be a point belonging to the Newton polygon $N(P)$. Then there exists a $\lambda_0>0$ and for every $\alpha\in\mathbb{N}_0^n$ a constant $C_\alpha= C_\alpha$ such that \begin{equation} \Big|\partial_\xi^\alpha \Big( \frac{|\xi|^\gamma |\lambda|^\kappa}{P(\xi,\lambda)}\Big)\Big| \le C_\alpha|\xi|^{-\alpha}\quad (\xi\in \mathbb{R}^n\setminus\{0\},\; \lambda\in \mathscr L, \; |\lambda|\ge \lambda_0). \label{eq2.8} \end{equation} If $P(\xi,\lambda)\not=0$ for all $\xi\in\mathbb{R}^n$ and $\lambda\in\mathscr L$ with $|\lambda|\ge\epsilon$ for some $\epsilon>0$, then inequality \eqref{eq2.8} holds for all $\xi\in \mathbb{R}^n\setminus\{0\}$ and all $\lambda\in\mathscr L$ with $|\lambda|\ge \epsilon$. \end{theorem} \begin{proof} For a point $(\sigma,\kappa)\in N(P)$ we have, by convexity and Jensen's inequality, \[ |\xi|^\sigma |\lambda|^\kappa \le W_P(\xi,\lambda). \] Let $\alpha=0$. By definition of parameter-ellipticity, there exists a $\lambda_0>0$ such that \[ \big| \;|\xi|^\gamma \lambda^\kappa \big|\le C |P(\xi,\lambda)|\quad ( \lambda\in\mathscr L,\;|\lambda|\ge \lambda_0,\; \xi\in\mathbb{R}^n). \] Thus, the case $\alpha=0$ follows directly from the definition of parameter-ellipticity. Now let $|\alpha|=1$ and assume, without loss of generality, that $\partial_\xi^\alpha = \partial_{\xi_1}$. We have \begin{equation} \big|\partial_{\xi_1}\big( |\xi|^\gamma \lambda^\kappa\big)\big| = \Big|\lambda^\kappa \Big( \sum_{i=1}^n \xi_i^2\Big)^{\frac\gamma 2-1}\cdot\frac\gamma 2\cdot 2\xi_1\Big| = \big|\gamma \cdot \lambda^\kappa \xi_1|\xi|^{\gamma-2}\big| \le C |\lambda|^k |\xi|^{\gamma-1}. \label{eq2.9} \end{equation} In the same way we can estimate \begin{equation} \Big|\partial_{\xi_1} P(\xi,\lambda)\Big|\le W_P(\xi,\lambda) \cdot |\xi|^{-1}. \label{eq2.10} \end{equation} We write \[ \Big|\partial_{\xi_1} \Big( \frac{|\xi|^\gamma \lambda^\kappa}{P(\xi,\lambda)}\Big)\Big| = \Big|\frac{P(\xi,\lambda)\partial_{\xi_1}(|\xi|^\gamma \lambda^\kappa)-|\xi|^\gamma \lambda^\kappa\partial_{\xi_1}P(\xi,\lambda)} {P(\xi,\lambda)^2}\Big|\] and obtain the first statement of the theorem by (\ref{eq2.9}), (\ref{eq2.10}) and the definition of parameter-ellipticity (\ref{eq2.4}). The case of higher derivatives (general $\alpha$) follows by iteration. Now assume that $P(\xi,\lambda)\not=0$ for all $\xi\in\mathbb{R}^n$ and $\lambda\in\mathscr L,\; |\lambda|\ge \epsilon$. By the regularity of $N(P)$, we can write \[ P(\xi,\lambda) = a_{\gamma_0,0} |\xi|^{\gamma_0} + \sum_{\gamma,k,\; k>0} a_{\gamma k} |\xi|^\gamma \lambda^k. \] In the last sum, only exponents of $|\xi|$ appear with $\gamma<\gamma_0$. We obtain \[ \lim_{|\xi|\to\infty} \frac{P(\xi,\lambda)}{|\xi|^{\gamma_0}} = a_{\gamma_0,0}\not=0\quad (\lambda\in \mathscr L, \; \epsilon \le |\lambda|\le \lambda_0).\] In the same way, \[ W_P(\xi,\lambda) = |\xi|^{\gamma_0} + \sum_{\gamma,k,\; k>0} |\xi|^\gamma \lambda^k, \] and \[ \lim_{|\xi|\to\infty} \frac{W_P(\xi,\lambda)}{|\xi|^{\gamma_0}} = 1\quad (\lambda\in \mathscr L, \; \epsilon \le |\lambda|\le \lambda_0). \] Now we use $P(\xi,\lambda)\not=0$ and a compactness argument to see that \begin{align*} |P(\xi,\lambda)| & \ge C |\xi|^{\gamma_0} && (\xi\in\mathbb{R}^n, \lambda\in\mathscr L,\,\epsilon\le |\lambda|\le \lambda_0),\\ |P(\xi,\lambda)| & \ge C W_P(\xi,\lambda) && (\xi\in\mathbb{R}^n, \lambda\in\mathscr L,\,\epsilon\le |\lambda|\le \lambda_0). \end{align*} >From these inequalities we obtain (\ref{eq2.8}) for all $\lambda\in\mathscr L$ with $|\lambda|\ge \epsilon$ in the same way as in the first part of the proof. \end{proof} \begin{remark}\label{2.5} \rm As we can see from the proof of the preceding theorem, we can also estimate \[ \Big|\partial_\xi^\alpha \Big( \frac{\xi^\beta \lambda^\kappa}{P(\xi,\lambda)}\Big)\Big| \le C_\alpha|\xi|^{-\alpha}\quad (\xi\in \mathbb{R}^n\setminus\{0\},\; \lambda\in \mathscr L,\,|\lambda|\ge\lambda_0) \] where now $\beta$ is a multi-index such that $(|\beta|,\kappa)$ belongs to the Newton polygon. \end{remark} \section{Resolvent estimates for the generalized thermoelastic plate equation} To apply the results mentioned above to the generalized linear thermoelastic plate equation (\ref{eq1.1}), (\ref{eq1.2}) we rewrite this equation as a first-order system, setting $U := (S^{1/2}u, \; u_t,\; \theta)^t$. We get \[ U_t = A(D) U := \begin{pmatrix} 0 & S^{1/2} & 0 \\ -S^{1/2} & 0 & S^\beta\\ 0 & -S^\beta & -S^\alpha\end{pmatrix} \; U. \] The symbol of this system is given by \[ A(\xi) = \begin{pmatrix} 0 & \rho^{1/2} & 0\\ -\rho^{1/2} & 0 & \rho^{\beta} \\ 0 & -\rho^{\beta} & - \rho^{\alpha} \end{pmatrix} \] with $\rho := |\xi|^{2\eta}$. Thus we have ${\rm ord} A_{ij}(\xi) \le s_i+t_j$ with \begin{equation} s := 2\eta\cdot \begin{pmatrix} \frac12\\ 2\beta-\alpha \\ \beta\end{pmatrix},\quad t := 2\eta\cdot\begin{pmatrix} \frac12+\alpha-2\beta\\ 0\\ \alpha-\beta\end{pmatrix}. \label{eq3.1} \end{equation} Consequently, the weight vector is given by \begin{equation} \begin{pmatrix} r_1\\ r_2\\ r_3\end{pmatrix} = 2\eta\begin{pmatrix} 1+\alpha-2\beta\\ 2\beta-\alpha\\ \alpha\end{pmatrix}. \label{eq3.2} \end{equation} With the order vectors $s$ and $t$ defined as above, the matrix $A(\xi)$ coincides with its principal part. The determinant of this system equals \begin{equation} P(\xi,\lambda) := \det(\lambda-A(\xi)) = \lambda^3 + \lambda^2 \rho^\alpha + \lambda(\rho^{2\beta} + \rho) + \rho^{1+\alpha}. \label{eq3.3} \end{equation} >From (\ref{eq3.3}) we can see that the Newton polygon $N(P)$ is the convex hull of the points \[ (0,3),\; (2\eta\alpha,2),\; (4\eta\beta,1),\; (2\eta+2\eta\alpha,0),\; (0,0) \] (see Figure 3). \begin{figure}[htb] \begin{center} \setlength{\unitlength}{0.7mm} \begin{picture}(120,90)(-5,-2) \put(0,0){\vector(1,0){115}} \put(0,0){\vector(0,1){70}} \put(-5,19){1} \put(-5,39){2} \put(-5,59){3} \put(-5,-1){0} \put(-1,-7){0} \put(-2,20){\line(1,0){4}} \put(-2,40){\line(1,0){4}} \put(-2,60){\line(1,0){4}} \put(-0.8,59.2){{$\scriptstyle \bullet$}} \put(36.7,39.2){{$\scriptstyle \bullet$}} \put(-0.8,-0.8){{$\scriptstyle \bullet$}} \put(69.2,19.2){{$\scriptstyle \bullet$}} \put(86.7,-0.8){{$\scriptstyle \bullet$}} \put(70,-2){\line(0,1){4}} \put(37.5,-2){\line(0,1){4}} \put(50,-2){\line(0,1){4}} \put(87.5,-2){\line(0,1){4}} \put(36,-7){$2\eta\alpha$} \put(68,-7){$4\eta\beta$} \put(49,-7){$2\eta$} \put(82,-7){$2\eta+2\eta\alpha$} \put(0,60){\rotatebox{-28}{\line(1,0){43}}} \put(37.5,40){\rotatebox{-31.5}{\line(1,0){38}}} \put(70,20){\rotatebox{-49}{\line(1,0){27}}} \put(-23,65){{\small power of $\lambda$}} \put(110,-7){{\small power of $\xi$}} \put(25,20){$N(P)$} \end{picture} \end{center} \caption{The Newton polygon of the mixed order system $\lambda-A(\xi)$} \end{figure} \begin{lemma}\label{3.1} Assume that $(\beta,\alpha)\in\mathfrak A$, i.e. that \begin{equation} \alpha\ge\beta\quad\text{and}\quad 2\beta-\alpha\ge\frac12. \label{eq3.4} \end{equation} Then the matrix $\lambda-A(D)$ is parameter-elliptic in $\mathbb{C}_+ := \{\lambda \in \mathbb{C}: \mathop{\rm Re}\lambda \ge 0\}$. \end{lemma} \begin{proof} We will check the conditions of Theorem \ref{2.3} (c). Let us first assume $\alpha>\beta$ and $2\beta-\alpha>\frac12$. Then we have $r_1 2\eta\alpha,\\ P_r(\xi,\lambda) & = \lambda^3+\lambda^2\rho^\alpha, && r=2\eta\alpha,\\ P_r(\xi,\lambda) & = \lambda^2\rho^\alpha, && 4\eta\beta-2\eta\alpha < r < 2\eta\alpha,\\ P_r(\xi,\lambda) & = \lambda^2\rho^\alpha + \lambda\rho^{2\beta}, && r= 4\eta\beta - 2\eta\alpha,\\ P_r(\xi,\lambda) & = \lambda\rho^{2\beta},&& 2\eta+2\eta\alpha-4\eta\beta < r < 4\eta\beta-2\eta\alpha,\\ P_r(\xi,\lambda) & = \lambda \rho^{2\beta}+\rho^{1+\alpha},&& r= 2\eta+2\eta\alpha-4\eta\beta,\\ P_r(\xi,\lambda) & =\rho^{1+\alpha},&& 0\frac12$ and $\alpha=\beta$ we have $r_10$ such that for all $\lambda \in \mathbb{C}_+$, $|\lambda|\ge\lambda_0$, the equation $(\lambda-A(D))U = F$ has a unique solution $U = (v, w,\theta)^t\in Y$ for every $F = (f,g,h)^t\in X$. Moreover, the estimate \[ |\lambda|\cdot \| U \|_X + \|U\|_{D(A)}\le C \| F \|_X \] holds for all $\lambda\in\mathbb{C}_+$ with $|\lambda|\ge\lambda_0$. \end{theorem} \begin{proof} We already know from Lemma \ref{3.1} that $\lambda - A(D)$ is parameter-elliptic in $\Sigma_\epsilon$. In particular, for $\xi\in\mathbb{R}^n$ and large $\lambda\in\Sigma_\epsilon$, the determinant $P(\xi,\lambda) = \det(\lambda-A(\xi))$ does not vanish. \textbf{(i)} First we show that there exists a $\lambda_0>0$ and for every multi-index $\gamma\in\mathbb{N}_0^n$ a constant $C_\gamma$ such that \begin{equation} \Big|\partial_\xi^\gamma \big[\lambda M_X(\xi) (\lambda-A(\xi))^{-1} \cdot M_X(\xi)^{-1} \big] \Big|\le C_\gamma |\xi|^{-|\gamma|} \quad (\xi\in\mathbb{R}^n\setminus\{0\}, \lambda\in\Sigma_\epsilon,\,|\lambda|\ge\lambda_0). \label{3.8} \end{equation} Here the diagonal matrix $M_X(\xi)$ is defined as \begin{equation} M_X(\xi):= \begin{pmatrix} (1+\rho)^{\beta-\frac12} & 0 & 0\\ 0 & (1+\rho)^{\alpha-\beta} & 0 \\ 0 & 0 & 1\end{pmatrix} \label{3.9} \end{equation} where we again have set $\rho=|\xi|^{2\eta}$. To prove the inequality above, we compute the inverse matrix explicitly. We have \[ (\lambda-A(\xi))^{-1} = \frac1{P(\xi,\lambda)} \begin{pmatrix} \lambda(\lambda+\rho^\alpha)+\rho^{2\beta} & \rho^{\frac12}(\lambda+\rho^{\alpha}) & \rho^{\beta+\frac12}\\[2pt] -\rho^{\frac12}(\lambda+\rho^\alpha) & \lambda(\lambda+\rho^\alpha) & \lambda\rho^\beta\\[2pt] \rho^{\beta+\frac12} & -\lambda\rho^\beta & \lambda^2+\rho\end{pmatrix}. \] Therefore, the the matrix $\lambda M_X(\xi) (\lambda-A(\xi))^{-1} \cdot M_X(\xi)^{-1}$ is given by $P(\xi,\lambda)^{-1}$ times \begin{equation} \begin{pmatrix} \lambda^2(\lambda+\rho^\alpha)+\lambda\rho^{2\beta} & \lambda\rho^{\frac12}(1+\rho)^{2\beta-\alpha-\frac12} (\lambda+\rho^{\alpha}) & \lambda\rho^{\beta+\frac12}(1+\rho)^{\beta-\frac12}\\[2pt] -\lambda\rho^{\frac12}(1+\rho)^{\frac12+\alpha-2\beta}(\lambda+\rho^\alpha) & \lambda^2(\lambda+\rho^\alpha) & \lambda^2\rho^\beta(1+\rho)^{\alpha-\beta}\\[2pt] \lambda\rho^{\beta+\frac12}(1+\rho)^{-\beta+\frac12} & -\lambda^2\rho^\beta(1+\rho)^{\beta-\alpha} & \lambda(\lambda^2+\rho)\end{pmatrix}. \label{eq3.10} \end{equation} We want to apply Theorem \ref{2.4} to every component of this matrix. As an example, let us consider the left lower corner \begin{equation} \lambda \rho^{\beta+\frac12} (1+\rho)^{-\beta+\frac12}. \label{eq3.11} \end{equation} It is easily seen that $(\frac\rho{1+\rho})^\beta$ is a Fourier multiplier in the sense that for every multi-index $\gamma\in\mathbb{N}_0^n$ \[ \Big| \partial_\xi^\gamma \Big( \frac\rho{1+\rho}\Big)^\beta\Big| \le C_\gamma |\xi|^{-|\gamma|} \quad (\xi\in\mathbb{R}^n\setminus\{0\}). \] Therefore, it suffices to consider $\lambda \rho^{\frac12}(1+\rho)^{\frac12}$ instead of \ref{eq3.11}. In the same way, $\frac{1+\rho^{1/2}}{(1+\rho)^{1/2}}$ is a Fourier multiplier, so we have to estimate $\lambda\rho^{\frac12} + \lambda \rho$. These two terms correspond to the pairs $(2,1)$ and $(4,1)$ of exponents. Both points belong to the Newton polygon $N(P)$, and by Theorem \ref{2.4} the desired result follows. The same proof works for all components of the matrix (\ref{eq3.10}). More precisely, we get the following exponents (in the order of appearance in the matrix (\ref{eq3.10})): \[ \begin{aligned} &(0,3), (2\eta\alpha,2), (4\eta\beta,1); && (4\eta\beta-2\eta\alpha,2), (4\eta\beta,1); && (4\eta\beta, 1);\\ &(2\eta+2\eta\alpha-4\eta\beta, 2), (2\eta+4\eta\alpha-4\eta\beta,1); && (0,3), (2\eta\alpha,2); && (2\eta\alpha,2);\\ &(2\eta,1); && (4\eta\beta-2\eta\alpha,2); && (0,3), (2\eta,1). \end{aligned} \] Due to our conditions on the parameters $\alpha$ and $\beta$, all points appearing in this list are in the interior or on the boundary of the Newton polygon $N(P)$. In Figure 4 the points are marked with ``$\scriptstyle\blacksquare$". \begin{figure}[ht] \setlength{\unitlength}{0.8mm} \begin{center} \begin{picture}(140,85)(-14,-5) \put(0,0){\vector(1,0){115}} \put(0,0){\vector(0,1){70}} \put(-5,19){1} \put(-5,39){2} \put(-5,59){3} \put(-5,-1){0} \put(-1,-7){0} \put(-2,20){\line(1,0){4}} \put(-2,40){\line(1,0){4}} \put(-2,60){\line(1,0){4}} \put(-1,59){{$\scriptstyle\blacksquare$}} \put(39,39){{$\scriptstyle\blacksquare$}} % (4 alpha,2) \put(35,45){$(2\eta\alpha,2)$} \put(69,19){{$\scriptstyle\blacksquare$}} % (8 beta,1) \put(65,25){$(4\eta\beta,1)$} \put(29,39){{$\scriptstyle\blacksquare$}}% (8 beta- 4 alpha,2) \put(20,25){$(4\eta\beta-2\eta\alpha,2)$}\put(30,28){\vector(0,1){10}} \put(59,19){{$\scriptstyle\blacksquare$}}% (4+8 alpha-8beta,1) \put(40,7){$(2\eta+4\eta\alpha-4\eta\beta,1)$}\put(60,12){\vector(0,1){6}} \put(19,39){{$\scriptstyle\blacksquare$}}% (4+4alpha-8beta,2) \put(10,57){$(2\eta+2\eta\alpha-4\eta\beta,2)$}\put(20,55){\vector(0,-1){13}} \put(-0.8,-0.8){{$\scriptstyle \bullet$}} \put(70,-2){\line(0,1){4}} \put(40,-2){\line(0,1){4}}\put(50,-2){\line(0,1){4}} \put(87.5,-2){\line(0,1){4}} \put(37,-7){$2\eta\alpha$} \put(68,-7){$4\eta\beta$} \put(49,-7){4} \put(82,-7){$2\eta+2\eta\alpha$} \put(0,60){\rotatebox{-26}{\line(1,0){45}}} \put(40,40){\rotatebox{-33}{\line(1,0){37}}} \put(70,20){\rotatebox{-49}{\line(1,0){27}}} \put(-20,65){{\small power of $\lambda$}}\put(110,-7){{\small power of $\xi$}} \end{picture} \end{center} \caption{Exponents appearing in the matrix \eqref{eq3.10}.} \end{figure} \textbf{(ii)} Now we apply Michlin's theorem which tells us that for $U := (\lambda- A(D))^{-1} F$ (being defined for $\lambda\in \mathbb{C}_+$ with $|\lambda|\ge \lambda_0$) the inequality \[ |\lambda| \; \|M_X(D) U \|_{L_p} \le \| M_X(D) F \|_{L_p}\] holds. Due to the definition of the space $X$, this can be reformulated as \[ |\lambda|\cdot \|U\|_X\le C\|F\|_X.\] \textbf{(iii)} Now we show that for any $F\in X$ the formula \[ U := (\lambda-A(\xi))^{-1} F\] defines a solution in the space $Y$. This can be shown in exactly the same way as the resolvent estimate but now we have to apply Michlin's theorem to the coefficients of the matrix \[ M_Y(\xi) (\lambda-A(\xi))^{-1} M_X(\xi)^{-1}. \] Here we set \[ M_Y(\xi) := \begin{pmatrix} (1+\rho)^{\frac12+\alpha-\beta} & 0 & 0\\ 0 & (1+\rho)^\beta & 0 \\ 0 & 0 & (1+\rho)^\alpha \end{pmatrix}. \] The matrix $M_Y(\xi) (\lambda-A(\xi))^{-1} M_X(\xi)^{-1}$ is given by $P(\xi,\lambda)^{-1}$ times \[ \begin{pmatrix} (1+\rho)^{1+\alpha-2\beta}[\lambda(\lambda+\rho^\alpha)+\rho^{2\beta}] & \rho^{\frac12}(1+\rho)^{\frac12} (\lambda+\rho^{\alpha}) & \rho^{\beta+\frac12}(1+\rho)^{\frac12 +\alpha-\beta}\\[2pt] -\rho^{\frac12}(1+\rho)^{\frac12}(\lambda+\rho^\alpha) & \lambda(1+\rho)^{2\beta-\alpha}(\lambda+\rho^\alpha) & \lambda\rho^{\beta}(1+\rho)^\beta\\[2pt] \rho^{\beta+\frac12}(1+\rho)^{\alpha-\beta+\frac12} & -\lambda\rho^{\beta}(1+\rho)^\beta & (1+\rho)^\alpha(\lambda^2+\rho) \end{pmatrix}. \] As before, we can see that all exponents appearing in this matrix belong to $N(P)$, and applying Theorem \ref{2.2} and Michlin's theorem, we obtain the desired inequality \[ \| U \|_{Y} \le C \|F\|_X. \] \end{proof} As a consequence we obtain the following result. \begin{theorem} The semigroup associated to $A$ is analytic in the region $\mathfrak{A}$. \end{theorem} We note that the choice of the spaces $X$ and $Y$ above is in some sense unique. More precisely, consider the spaces \[ X = W_p^{2\eta t_1}(\mathbb{R}^n) \times W_p^{2\eta t_2}(\mathbb{R}^n) \times W_p^{2\eta t_3}(\mathbb{R}^n) \] and \[ Y = W_p^{2\eta s_1}(\mathbb{R}^n) \times W_p^{2\eta s_2}(\mathbb{R}^n) \times W_p^{2\eta s_3}(\mathbb{R}^n) \] with $t_i, s_i\in\mathbb{R}$. We are looking for indices $s_i, t_i$ such that the following conditions are satisfied: \begin{itemize} \item[(i)] $A(D)\colon X\to Y$ is well defined and continuous. \item[(ii)] $\lambda (\lambda-A(D))^{-1}\colon X\to X$ is continuous for $\lambda\in\mathbb{C}_+$, $|\lambda|\ge\lambda_0$ with norm bounded by a constant independent of $\lambda$. \item[(iii)] $(\lambda-A(D))^{-1}\colon X\to Y$ is continuous for $\lambda\in\mathbb{C}_+$, $|\lambda|\ge\lambda_0$ with norm bounded by a constant independent of $\lambda$. \end{itemize} By Michlin's theorem, this implies that the corresponding symbols are bounded for $|\xi|\to\infty$. Setting \[ M_X(\xi) := \begin{pmatrix} (1+\rho)^{t_1} & 0 & 0 \\ 0 & (1+\rho)^{t-2} & 0 \\ 0 & 0 & (1+\rho)^{t_3}\end{pmatrix} \] and \[ M_Y(\xi) := \begin{pmatrix} (1+\rho)^{t_1} & 0 & 0 \\ 0 & (1+\rho)^{t-2} & 0 \\ 0 & 0 & (1+\rho)^{t_3}\end{pmatrix},\] we see that that (i)-(iii) implies that the following matrices are bounded for $|\xi|\to\infty$ and $\lambda\in\Sigma_\epsilon$, $|\lambda|\ge \lambda_0$: \begin{align*} N_1(\xi) & := M_X(\xi) A(\xi) M_Y(\xi)^{-1},\\ N_2(\xi) & := \lambda M_X(\xi)(\lambda-A(\xi))^{-1} M_X(\xi),\\ N_3(\xi) & := M_Y(\xi) (\lambda- A(\xi))^{-1} M_X(\xi). \end{align*} The boundedness of $N_1$ for large $|\xi|$ implies that every exponent of $\xi$ appearing in this matrix is less or equal to 0. For instance, in the first row and second column of $N_1(\xi)$ we have the coefficient $(1+\rho)^{t_1}\rho^{1/2}(1+\rho)^{-s_2}$. The boundedness implies \[ t_1-s_2+\frac12\le 0. \] In the same way we get an inequality for every non-zero coefficient of $N_1$. Both $N_2$ and $N_3$ have the form $\frac1{P(\xi,\lambda)} \widetilde N_i(\xi,\lambda)$ ($i=2,3$) where the coefficients of the matrix $\widetilde N_i$ are sums of terms of the form $\rho^\sigma\lambda^k$. From the properties of the Newton polygon, it can easily be seen that an expression of the form \[ \frac{\rho^\sigma\lambda^k}{P(\xi,\lambda)}\] can only be bounded for $|\xi|\to\infty$ if $(2\eta\sigma,k)$ belongs to the Newton polygon. For instance, in the first row and second column of $\widetilde N_2$ we have the exponents $(4t_1-4t_2+2,2)$ and $(4t_1-4t_2+4\alpha+2,1)$. These points belong to $N(P)$ if \[ t_1-t_2+\frac12\le\alpha\quad\text{ and }\quad t_1-t_2+\alpha+\frac12\le 2\beta. \] Altogether we obtain a set of inequalities for $s_i$ and $t_i$, $i=1,2,3$. Note that for a solution $s_i, t_i$ also $s_i+\tau, t_i+\tau$ with arbitrary $\tau\in\mathbb{R}$ is a solution. If we assume $t_3=0$, a simple but lengthy calculation shows that \begin{align*} s_1 &= 2+4\alpha-4\beta,& s_2 &= 4\beta, & s_3 & = 4\alpha,\\ t_1 &= 4\beta -2, & t_2 & = 4\alpha-4\beta,& t_3 & = 0 \end{align*} is the only solution of all inequalities. \begin{remark} \rm For simplicity of presentation, we did not consider lower-order perturbations of the operator $S$ in \eqref{eq1.1}, \eqref{eq1.2}. The results of Theorems 3.4 and 3.5, however, are stable under such perturbations. This can be seen by standard methods of elliptic theory based on Sobolev space inequalities, cp., e.g., \cite{ADF}, Section~2. We also want to point out that the notion of parameter-ellipticity is based on the $r$-principal parts of $\det\big(\lambda -A(\xi)\big)$, see Theorem 2.3. These $r$-principal parts are unchanged if we add lower-order perturbations. Thus, parameter-ellipticity is stable under lower-order perturbations, too. \end{remark} \section{Decay rates} Denoting by $$ \big( v (t, \xi) := (\mathcal{F}u (t, \cdot)) (\xi), \psi (t, \xi) := (\mathcal{F} \theta (t, \cdot)) (\xi)\big) $$ the Fourier transform of the solution $(u, \theta)$ to the $\alpha$-$\beta$-system (\ref{eq1.1}), (\ref{eq1.2}) with initial conditions (\ref{eq1.3}), it is easy to see that both $v$ and $\psi$ satisfy the following third-order equation (without loss of generality, $ a = b = d = g = 1$ again) \begin{equation} w_{t t t} + \rho^\alpha w_{tt} + (\rho^{2 \beta} + \rho) w_{t} + \rho^{1 + \alpha} w = 0,\label{eq4.1} \end{equation} where $\rho := | \xi |^{2 \eta}$, and with initial conditions \begin{equation} w (0, \cdot) = w_{0} := \mathcal{F} u_{0},\quad w_{t} (0, \cdot) = \mathcal{F} u_{1}, \quad w_{tt} (0, \cdot) = \mathcal{F} u_{tt} (0, \cdot) \label{eq4.2} \end{equation} (cp. \cite{MR96}). Then $w$ is given by \begin{equation} w (t, \xi) = \sum_{j=1}^3 b_{j}(\rho) e^{\lambda j (\rho) t}, \label{eq4.3} \end{equation} where $\lambda j (\rho), j = 1, 2, 3,$ are the roots of the characteristic equation \begin{equation} P (\lambda, \rho) = \lambda^3 + \rho^\alpha \lambda^2 + (\rho^{2 \beta} + \rho) \lambda + \rho^{1 + \alpha} = 0 \label{eq4.4} \end{equation} and $$b_{j} (\rho) := \sum_{k = 0}^2 b_{j}^k (\rho) w_{k} (\rho)$$ with \[ b_{j}^0 := \frac{\prod_{l \neq j} \lambda_{e}}{\prod_{l \neq j} (\lambda_{j} - \lambda_{e})},\quad b_{j}^1 := \frac{\sum_{l \neq j} \lambda_{e}}{\prod_{l \neq j} (\lambda_{j} - \lambda_{e})}, \quad b_{j}^2 := \frac{1}{\prod_{l \neq j} (\lambda_{j} - \lambda_{e})}. \] The study of the asymptotic behavior of $\lambda_{j}(\rho)$ when $\rho \to\infty$ gives the information on the smoothing effect of the solutions, that is: if for any $j = 1, 2, 3$ the real part of $\lambda_{j}$ tends to infinity as $\rho \to \infty$, then the smoothing property holds; if there exists a subscript $j$ for which the real part does not tend to infinity, then the solution cannot be more regular than the initial data, cp. \cite{MR96}. On the other hand, the behavior of the real part of $\lambda_{j}$ as $\rho \to 0$ gives the information on the asymptotic behavior of the solution as time goes to infinity, such as uniform polynomial decay rates. In the following theorem the equalities are to be read up to terms of lower/higher order in $\rho$ as $\rho \to \infty/0$. \begin{theorem}\quad\\ (1) As $\rho\to\infty$, in the interior of $\mathfrak{A}$ we have: \begin{align*} &\mbox{For } \alpha=1:\quad \lambda_1 = k_1\rho^\alpha,\quad \lambda_{2,3} = k_{2a,2b} \rho^{2\beta-\alpha}\\ &\mbox{For } \alpha=\beta> \frac{1}{2}: \quad \lambda_1 = k_3\rho^{1-\alpha},\quad \lambda_{2,3} = k_4 \rho^{\alpha} \pm {\rm i} k_5\rho^\alpha\\ &\mbox{For } \alpha=\beta= \frac{1}{2}: \quad \lambda_1 = k_6\rho^{1/2},\quad \lambda_{2,3} = k_7 \rho^{1/2} \pm {\rm i} k_8\rho^{1/2}\\ &\mbox{For } \frac{1}{2} < \alpha=2\beta - \frac{1}{2}: \quad \lambda_1 = k_9\rho^{\alpha},\quad \lambda_{2,3} = k_{10} \rho^{1/2} \pm {\rm i} k_{11}\rho^{1/2} \end{align*} The constant coefficients $k_{m} > 0, m = 1, \dots, 11,$ can be given explicitly, they are not the same in all of $\mathfrak{A}$, but can jump coming from the interior to the boundary; the same holds for the positive constants $r_{m}$ below. \\[2pt] (2) As $\rho\to 0$, In the interior of $\mathfrak{A}$, we have: \begin{align*} &\mbox{For } \alpha=1:\quad \lambda_1 = r_1 \rho^\alpha,\quad \lambda_{2,3} = r_{2} \rho^{\alpha + 2\beta -1} \pm {\rm i} r_{3}\lambda^{1/2}\\ &\mbox{For } \alpha=\beta> \frac{1}{2}: \quad \lambda_1 = r_4\rho^{\alpha},\quad \lambda_{2,3} = r_5 \rho^{3\alpha-1} \pm {\rm i} r_6\rho^{1/2}\\ &\mbox{For } \alpha=\beta= \frac{1}{2}: \quad \lambda_1 = r_7\rho^{\alpha},\quad \mathop{\rm Re}\lambda_{2,3} = r_8\rho^{1/2}\\ &\mbox{For } \frac{1}{2} < \alpha=2\beta - \frac{1}{2}: \quad \lambda_1 = r_9 \rho^{\alpha},\quad \lambda_{2,3} = r_{10} \rho^{2\alpha - 1/2} \pm {\rm i} r_{11}\rho^{1/2} \end{align*} (3) As $\rho \to 0$, for $\beta = 1/2,\;0 \leq \alpha \leq 1$, we have $$\mathop{\rm Re}\lambda_{j} = \begin{cases} c_{j}\rho^\alpha , & \frac 1 2 \leq \alpha \leq 1\\ c_{j}\rho^{1 - \alpha} , & 0 \leq \alpha \leq 1/2 \end{cases} $$ where $c_{j}$ denotes (different) positive constants.\\[2pt] (4) For $\alpha = 1/2, \; \frac 1 4 \leq \beta \leq 1/2$, we have $\mathop{\rm Re}\lambda_{j} = c_{j}\rho^{1/2}$\\[2pt] (5) For $\alpha = 1/2, \; \frac 1 2 \leq \beta \leq 3/4$, we have $\mathop{\rm Re}\lambda_{j} = c_{j}\rho^{2 \beta - 1/2}$ \end{theorem} \begin{proof} The roots $\lambda_{j}$ of the cubic polynomial $P (\lambda, \rho)$ in (\ref{eq4.4}) are given by \begin{gather} \lambda_{1} = \frac 1 3 (\rho^\alpha + B - D^{1/3}), \label{eq4.5} \\ \lambda_{2, 3} = \frac 1 3 \rho^\alpha + \frac 1 6 (D^{1/3} - B) \mp \frac{\sqrt{3}}{6}(D^{1/3} + B), \label{eq4.6} \end{gather} where \begin{equation} B := \frac{3_{\rho} - \rho^{2 \alpha} + 3\rho^{2 \beta}}{D^{1/3}} \label{eq4.7} \end{equation} and \begin{equation} \begin{aligned} D &:= \frac 1 2 \Big(- 2 \rho^{3 \alpha} - 18 \rho^{1 + \alpha} + 9 \rho^{\alpha + 2 \beta} \\ &\quad + \sqrt{4(3 \rho - \rho^{2 \alpha} + 3 \rho^{2 \beta})^3 + (- 2 \rho^{3 \alpha} - 18 \rho^{1 + \alpha} + 9 \rho^{\alpha + 2 \beta})^2} \Big) \end{aligned} \label{eq4.8} \end{equation} By a straightforward but lengthy analysis of the terms in (\ref{eq4.4})--(\ref{eq4.7}) as $\rho \to \infty$ and $\rho \to 0$, respectively, studying the different cases (1), (2), (4), (5), we arrive at the expansion claimed in Theorem 4.1. Claim (3) was already given in \cite{MR96}. Combining the representation of the solution with the asymptotic properties given in Theorem 4.1 for $\rho \to 0$, we can conclude \end{proof} \begin{theorem} For the solution $(\psi, \theta)$ to the $\alpha$-$\beta$-system (\ref{eq1.1}), (\ref{eq1.2}), (\ref{eq1.3}) we have for $2 \leq q \leq \infty, \frac 1 q + \frac{1}{q'} = 1,$ and $t > 0$: $$ \| ((- \Delta)^\eta u (t, \cdot), u_t(t,\cdot),\theta(t, \cdot)) \| _{L^ q(\mathbb{R}^n)} \leq c_{q, n}t^{- \frac{n}{2 \eta d}(1- \frac 2 q)} \| ( (- \Delta)^\eta u_{0},\;u_1, \theta_{0}) \| _{L^{q \prime}(\mathbb{R}^n)} $$ where $c_{q, n}$ is a positive constant at most depending on $q$ and the space dimension $n$, and where $d$ is given as follows:\\ (1) For $(\beta, \alpha) \in \mathfrak{A}:\quad d =\alpha$.\\ (2) For $\beta = \frac 1 2, 0 \leq \alpha \leq 1 :\quad d = \left\{ \begin{array}{cc} \alpha, & \frac1 2 \leq \alpha \leq 1\\ 1 - \alpha, & 0 \leq \alpha \leq 1/2 \end{array}\right.$. \\[2pt] (3) For $\alpha = \frac1 2, \frac 1 4 \leq \beta \leq 1/2 :\quad d = \frac 1 2$.\\ (4) For $\alpha = 1/2, \frac 1 2 \leq \beta \leq \frac 3 4 :\quad d = 2 \beta - \frac 1 2$. \end{theorem} \begin{proof} Claim (2) was already proved in \cite{MR96}. For the remaining cases the $L^\infty$-decay $(q = \infty)$ follows in a standard way from the asymptotic expansions given in Theorem 4.1, see e.g. \cite{MR95} or \cite{Ra92}. The $L^2$-``decay'' $(q = 2 = q\prime)$ is given by the dissipation of the system, so the claims follow by interpolation.\\ \end{proof} As already remarked in \cite{MR96} we again see the very special r\^ole of $(\beta, \alpha) = (\frac 1 2, \frac 1 2)$, i.e. of the classical thermoelastic plate system. It is also interesting to notice the (non-) dependence of the decay rate on the parameter $\beta$. We studied case (3) and (4), respectively case (4) and (5) in Theorem 4.1, just exemplarily. Further cases for $(\beta, \alpha)$ can be treated similarly.\\ The analysis of the roots as given in (\ref{eq4.4}), (\ref{eq4.5}) gives three real roots $\lambda_{1}, \lambda_{2}, \lambda_{3}$ for $(\beta, \alpha) \in \mathfrak{A}$, fitting to the analyticity there. On the other hand, another asymptotic analysis for $\alpha > \beta \geq \frac 1 2, \; \alpha > 2 \beta - \frac1 2$, i.e. for $(\beta, \alpha)$ {\it outside} the region $\mathfrak{A}$, yields \begin{equation} \Big| \frac{{\rm Im} \lambda_{2/3}}{\mathop{\rm Re}\lambda_{2/3}} \Big| = \rho^{\alpha - 2 \beta + 1/2} + \mbox{l.o.t.}\; \to \infty \quad {\mbox as }\; \rho \to \infty \label{eq4.9} \end{equation} excluding analyticity because of a non-sectorial operator appearing. $\mathfrak{A}$ is expected to be {\it the} analyticity region. Finally we remark that the next step is to consider domains $\Omega \subsetneqq \mathbb{R}^n$ with boundaries. 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