Spectral Analysis for a Wave/Plate Transmission System

Abstract

We are concerned with the transmission system of a 1D damped wave equation and a 1D undamped plate equation. Our result reads as follows: the spectrum of the infinitesimal generator of the semigroup associated with the system in question consists merely of an infinite sequence of eigenvalues which are all located in the open left half of the complex plane; the sequence of eigenvalues has the imaginary axis and another vertical line to the left of the imaginary axis as its asymptote lines; the energy of the system under consideration decreases to zero as time goes to infinity.

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1. Introduction

Due to their wide applicability, spectral properties of coupled systems of partial differential equations have been extensively investigated in the literature. Guo and Yung [1] conducted spectral analysis for a 1D thermoelastic system and obtained an energy decay estimate from the spectral property; Lu and Wang [2] studied the spectral property of a wave/Schröodinger transmission system and proved that the associated energy decreases to zero as time approaches infinity, to name just a few of them. In this paper, we are concerned with the spectral property of the transmission system$$\begin{array}{c}\text{(1)}& {\partial }_t^{\mathrm{2}}u-{\partial }_x^{\mathrm{2}}u+a{\partial }_{t}u=\mathrm{0},\hspace{1em}\forall \left(x,t\right)\in \left(\mathrm{0,1}\right)\times \left(\mathrm{0},+\infty \right),{\partial }_t^{\mathrm{2}}w+{\partial }_x^{\mathrm{4}}w=\mathrm{0},\hspace{1em}\forall \left(x,t\right)\in \left(-\mathrm{1,0}\right)\times \left(\mathrm{0},+\infty \right),\\ u\left(\mathrm{1},t\right)={\partial }_x^{\mathrm{2}}w\left(\mathrm{0},t\right)=w\left(-\mathrm{1},t\right)={\partial }_x^{\mathrm{2}}w\left(-\mathrm{1},t\right)=\mathrm{0},\hspace{1em}\forall t\in \left(\mathrm{0},+\infty \right),\\ u\left(\mathrm{0},t\right)=w\left(\mathrm{0},t\right),\\ {\partial }_{x}u\left(\mathrm{0},t\right)=-{\partial }_x^{\mathrm{3}}w\left(\mathrm{0},t\right),\\ \hspace{1em}\forall t\in \left(\mathrm{0},+\infty \right),\\ u\left(·,\mathrm{0}\right)=u^{\mathrm{0}},\\ {\partial }_{t}u\left(·,\mathrm{0}\right)=u^{\mathrm{1}},\\ \hspace{1em}\forall x\in \left(\mathrm{0,1}\right),\\ w\left(·,\mathrm{0}\right)=w^{\mathrm{0}},\\ {\partial }_{t}w\left(·,\mathrm{0}\right)=w^{\mathrm{1}},\\ \hspace{1em}\forall x\in \left(-\mathrm{1,0}\right),\end{array}$$ in which $a>\mathrm{0}$ and the energy in the wave and in the plate can be transferred throughout the interference point $x=\mathrm{0}$ by the relation $u(\mathrm{0},t)=w(\mathrm{0},t)$, ${\partial }_{x}u(\mathrm{0},t)=-{\partial }_x^{\mathrm{3}}w(\mathrm{0},t)$.

One of our motivations to conduct the spectral analysis is to understand better the asymptotic behavior of solutions to (1). Recently, the wave/plate system (or wave/plate transmission system in the case of multidimensions) has been investigated in several references for their stabilization and/or asymptotic property. Ammari and Nicaise [3] proved under a certain geometric condition that the energy of the transmission system of a damped wave equation and a damped plate equation decays exponentially by a multiplier argument. Zhang and Zhang [4] studied the stabilization of transmission coupled wave and Euler-Bernoulli equations on Riemannian manifolds by introducing nonlinear feedback. Hassine [5] proved by the Carleman estimate an energy decay estimate for the transmission system of an undamped Euler-Bernoulli plate and a wave equation with a localized Kelvin-Voigt damping. Gong, Yang and Zhao [6] studied the stabilization of a wave/plate transmission system via a Riemannian geometric approach. Inspired by the afore-mentioned results, and in view of the presence of the damping in the wave equation, we are interested in the large-time behavior of solutions to system (1). And therefore, we are tempted to study the spectral property of the infinitesimal generator of the semigroup associated with system (1). The controllability of wave/plate transmission systems was also investigated in the literature; see [7].

By introducing $v(x,t)=w(-x,t)$ into (1), we are led to the system$$\begin{array}{c}\text{(2)}& {\partial }_t^{\mathrm{2}}u-{\partial }_x^{\mathrm{2}}u+a{\partial }_{t}u=\mathrm{0},\hspace{1em}\forall \left(x,t\right)\in \left(\mathrm{0,1}\right)\times \left(\mathrm{0},+\infty \right),{\partial }_t^{\mathrm{2}}v+{\partial }_x^{\mathrm{4}}v=\mathrm{0},\hspace{1em}\forall \left(x,t\right)\in \left(\mathrm{0,1}\right)\times \left(\mathrm{0},+\infty \right),\\ u\left(\mathrm{1},t\right)={\partial }_x^{\mathrm{2}}v\left(\mathrm{0},t\right)=v\left(\mathrm{1},t\right)={\partial }_x^{\mathrm{2}}v\left(\mathrm{1},t\right)=\mathrm{0},\hspace{1em}\forall \in \left(\mathrm{0},+\infty \right),\\ u\left(\mathrm{0},t\right)=v\left(\mathrm{0},t\right),\\ {\partial }_{x}u\left(\mathrm{0},t\right)={\partial }_x^{\mathrm{3}}v\left(\mathrm{0},t\right),\\ \hspace{1em}\forall t\in \left(\mathrm{0},+\infty \right),\\ u\left(·,\mathrm{0}\right)=u^{\mathrm{0}},\\ {\partial }_{t}u\left(·,\mathrm{0}\right)=u^{\mathrm{1}},\\ \hspace{1em}\forall x\in \left(\mathrm{0,1}\right),\\ v\left(·,\mathrm{0}\right)=v^{\mathrm{0}},\\ {\partial }_{t}v\left(·,\mathrm{0}\right)=v^{\mathrm{1}},\\ \hspace{1em}\forall x\in \left(\mathrm{0,1}\right).\end{array}$$Thus, our main goal boils down to analyzing the spectral property of the infinitesimal generator $A$ of the semigroup associated with system (2). Here $A$ is defined by $$\begin{array}{}\text{(3)}& A\left(\phi ,\tilde{\phi },\psi ,\tilde{\psi }\right)=\left(\tilde{\phi },{\phi }^{\prime \prime }-a\tilde{\phi },\tilde{\psi },-{\psi }^{\left(\mathrm{4}\right)}\right),\hspace{1em}\forall \left(\phi ,\tilde{\phi },\psi ,\tilde{\psi }\right)\in \mathcal{D}\left(A\right),\mathcal{D}\left(A\right)=\left\{\left(\phi ,\tilde{\phi },\psi ,\tilde{\psi }\right)\in \mathcal{H};\tilde{\phi }\left(\mathrm{1}\right)=\tilde{\psi }\left(\mathrm{1}\right)=\tilde{\phi }\left(\mathrm{0}\right)-\tilde{\psi }\left(\mathrm{0}\right)={\psi }^{\prime \prime }\left(\mathrm{0}\right)={\psi }^{\prime \prime }\left(\mathrm{1}\right)={\phi }^{\prime }\left(\mathrm{0}\right)-{\psi }^{\prime \prime \prime }\left(\mathrm{0}\right)=\mathrm{0}\right\},\end{array}$$where $\mathcal{H}$ is the closure of $\mathcal{D}(A)$ in $H^{\mathrm{1}}(\mathrm{0};\mathrm{1})\times L^{\mathrm{2}}(\mathrm{0};\mathrm{1})\times H^{\mathrm{2}}(\mathrm{0};\mathrm{1})\times L^{\mathrm{2}}(\mathrm{0};\mathrm{1})$. Endowed with the sesquilinear form $$\begin{array}{c}\text{(4)}& {\mathcal{H}}^{\mathrm{2}}\ni \left(\left(\phi ,\tilde{\phi },\psi ,\tilde{\psi }\right),\left(f,\tilde{f},g,\tilde{g}\right)\right)\mapsto {{\displaystyle \int }}_{\mathrm{0}}^{\mathrm{1}}\left({\phi }^{\prime }\overline{f^{\prime }}+\tilde{\phi }\overline{\tilde{f}}+{\psi }^{\prime \prime }\overline{g^{\prime \prime }}+\tilde{\psi }\overline{\tilde{g}}\right)dx\in \mathbb{C}\end{array}$$where $\overline{z}$ denotes the complex conjugate of $z\in \mathbb{C}$; $\mathcal{H}$ is a Hilbert space. $\mathcal{H}$ is the natural finite energy space for system (2).

We shall prove that $\mathrm{0}\in \rho (A)$ by using the idea of Green’s functions, thereby proving that $A$ has compact resolvents, shall prove that $\sigma (A)$ is contained in the open left half of the complex plane by proving that $A$ is dissipative in $\mathcal{H}$ and that $\mathrm{i}\mathbb{R}\cap \sigma (A)=\varnothing$;, and shall obtain the spectral asymptotics for $A$.

The rest of the paper is planned as follows. In Section 2, we analyze the spectral property of $A$. In Section 3, we conclude the paper by some remarks.

2. Spectral Property of $A$

In this section, we analyze in detail the spectral property of $A$. We first note that$$\begin{array}{}\text{(5)}& \mathfrak{R}\mathfrak{e}{⟨A\left(\phi ,\tilde{\phi },\psi ,\tilde{\psi }\right),\left(\phi ,\tilde{\phi },\psi ,\tilde{\psi }\right)⟩}_{\mathcal{H}}=-a{‖\tilde{\phi }‖}_{L^{\mathrm{2}}\left(\mathrm{0,1}\right)}^{\mathrm{2}},\hspace{1em}\forall \left(\phi ,\tilde{\phi },\psi ,\tilde{\psi }\right)\in \mathcal{D}\left(A\right),\end{array}$$which implies that $A$ is a dissipative operator in $\mathcal{H}$, and that $\lambda \notin \sigma (A)$ whenever $\mathfrak{R}\mathfrak{e}\lambda >\mathrm{0}$.

Proposition 1.

$\mathrm{0}$ belongs to the resolvent set of $A$, or in symbols, $\mathrm{0}\in \rho (A)$.

Proof.

The statement is equivalent to the well-posedness of the boundary value problem (BVP)$$\begin{array}{c}\text{(6)}& -{\phi }^{\prime \prime }=f,{\psi }^{\left(\mathrm{4}\right)}=g,\\ \hspace{1em}x\in \left(\mathrm{0,1}\right),\\ \phi \left(\mathrm{1}\right)={\psi }^{\prime \prime }\left(\mathrm{0}\right)=\psi \left(\mathrm{1}\right)={\psi }^{\prime \prime }\left(\mathrm{1}\right)=\mathrm{0},\\ \phi \left(\mathrm{0}\right)=\psi \left(\mathrm{0}\right),\\ {\phi }^{\prime }\left(\mathrm{0}\right)={\psi }^{\prime \prime \prime }\left(\mathrm{0}\right).\end{array}$$By utilizing the idea of Green’s functions, we know that $\phi$ and $\psi$ can be written as$$\begin{array}{}\text{(7)}& \phi \left(x\right)=-{{\displaystyle \int }}_{\mathrm{0}}^{\mathrm{1}}\left[c_{\mathrm{11}}\left(x-\xi \right)+c_{\mathrm{12}}+H\left(x-\xi \right)\left({\tilde{c}}_{\mathrm{11}}\left(x-\xi \right)+{\tilde{c}}_{\mathrm{12}}\right)\right]f\left(\xi \right)d\xi +{{\displaystyle \int }}_{\mathrm{0}}^{\mathrm{1}}\left[c_{\mathrm{13}}\left(x-\xi \right)+c_{\mathrm{14}}\right]g\left(\xi \right)d\xi ,\hspace{1em}\forall x\in \left[\mathrm{0,1}\right],\text{(8)}& \psi \left(x\right)=-{{\displaystyle \int }}_{\mathrm{0}}^{\mathrm{1}}\left[c_{\mathrm{21}}{\left(x-\xi \right)}^{\mathrm{3}}+c_{\mathrm{22}}{\left(x-\xi \right)}^{\mathrm{2}}+c_{\mathrm{23}}\left(x-\xi \right)+c_{\mathrm{24}}\right]f\left(\xi \right)d\xi +{{\displaystyle \int }}_{\mathrm{0}}^{\mathrm{1}}\left[c_{\mathrm{25}}{\left(x-\xi \right)}^{\mathrm{3}}+c_{\mathrm{26}}{\left(x-\xi \right)}^{\mathrm{2}}+c_{\mathrm{27}}\left(x-\xi \right)+c_{\mathrm{28}}+H\left(x-\xi \right)\left({\tilde{c}}_{\mathrm{25}}{\left(x-\xi \right)}^{\mathrm{3}}+{\tilde{c}}_{\mathrm{26}}{\left(x-\xi \right)}^{\mathrm{2}}+{\tilde{c}}_{\mathrm{27}}\left(x-\xi \right)+{\tilde{c}}_{\mathrm{28}}\right)\right]g\left(\xi \right)d\xi ,\hspace{1em}\forall x\in \left[\mathrm{0,1}\right],\end{array}$$where the constants $c_{\mathrm{11}}$, $c_{\mathrm{12}}$, ${\tilde{c}}_{\mathrm{11}}$, ${\tilde{c}}_{\mathrm{12}}$, $c_{\mathrm{13}}$, $c_{\mathrm{14}}$, $c_{\mathrm{21}}$, $c_{\mathrm{22}}$, $c_{\mathrm{23}}$, $c_{\mathrm{24}}$, $c_{\mathrm{15}}$, $c_{\mathrm{26}}$, $c_{\mathrm{27}}$, $c_{\mathrm{28}}$, ${\tilde{c}}_{\mathrm{25}}$, ${\tilde{c}}_{\mathrm{26}}$, ${\tilde{c}}_{\mathrm{27}}$, and ${\tilde{c}}_{\mathrm{28}}$, are such that $$\begin{array}{c}\text{(9)}& -{\phi }^{\prime \prime }=f⟹{\tilde{c}}_{\mathrm{12}}=\mathrm{0},\\ \mathrm{2}{\tilde{c}}_{\mathrm{11}}=\mathrm{1},\\ {\psi }^{\left(\mathrm{4}\right)}=g⟹\\ {\tilde{c}}_{\mathrm{28}}=\mathrm{0},\\ \mathrm{4}{\tilde{c}}_{\mathrm{27}}=\mathrm{0},\\ \mathrm{12}{\tilde{c}}_{\mathrm{26}}=\mathrm{0},\\ \mathrm{24}{\tilde{c}}_{\mathrm{25}}=\mathrm{1},\\ \phi \left(\mathrm{1}\right)=\mathrm{0}⟹\\ c_{\mathrm{11}}\left(\mathrm{1}-\xi \right)+c_{\mathrm{12}}+{\tilde{c}}_{\mathrm{11}}\left(\mathrm{1}-\xi \right)+{\tilde{c}}_{\mathrm{12}}=\mathrm{0},\\ c_{\mathrm{13}}\left(\mathrm{1}-\xi \right)+c_{\mathrm{14}}=\mathrm{0},\\ {\psi }^{\prime \prime }\left(\mathrm{0}\right)=\mathrm{0}⟹\\ -\mathrm{6}{c}_{\mathrm{21}}\xi +\mathrm{2}{c}_{\mathrm{22}}=\mathrm{0},\\ -\mathrm{6}{c}_{\mathrm{25}}\xi +\mathrm{2}{c}_{\mathrm{26}}=\mathrm{0},\\ \psi \left(\mathrm{1}\right)=\mathrm{0}⟹\\ c_{\mathrm{21}}{\left(\mathrm{1}-\xi \right)}^{\mathrm{3}}+c_{\mathrm{22}}{\left(\mathrm{1}-\xi \right)}^{\mathrm{2}}+c_{\mathrm{23}}\left(\mathrm{1}-\xi \right)+c_{\mathrm{24}}=\mathrm{0},\\ c_{\mathrm{25}}{\left(\mathrm{1}-\xi \right)}^{\mathrm{3}}+c_{\mathrm{26}}{\left(\mathrm{1}-\xi \right)}^{\mathrm{2}}+c_{\mathrm{27}}\left(\mathrm{1}-\xi \right)+c_{\mathrm{28}}+{\tilde{c}}_{\mathrm{25}}{\left(\mathrm{1}-\xi \right)}^{\mathrm{3}}+{\tilde{c}}_{\mathrm{26}}{\left(\mathrm{1}-\xi \right)}^{\mathrm{2}}+{\tilde{c}}_{\mathrm{27}}\left(\mathrm{1}-\xi \right)+{\tilde{c}}_{\mathrm{28}}=\mathrm{0},\\ {\psi }^{\prime \prime }\left(\mathrm{1}\right)=\mathrm{0}⟹\\ \mathrm{6}{c}_{\mathrm{21}}\left(\mathrm{1}-\xi \right)+\mathrm{2}{c}_{\mathrm{22}}=\mathrm{0},\\ \mathrm{6}{c}_{\mathrm{25}}\left(\mathrm{1}-\xi \right)+\mathrm{2}{c}_{\mathrm{26}}+\mathrm{6}{\tilde{c}}_{\mathrm{25}}\left(\mathrm{1}-\xi \right)+\mathrm{2}{\tilde{c}}_{\mathrm{26}}=\mathrm{0},\\ \phi \left(\mathrm{0}\right)=\psi \left(\mathrm{0}\right)⟹\\ -c_{\mathrm{11}}\xi +c_{\mathrm{12}}-\left(-c_{\mathrm{21}}{\xi }^{\mathrm{3}}+c_{\mathrm{22}}{\xi }^{\mathrm{2}}-c\mathrm{23}\xi +c_{\mathrm{24}}\right)=\mathrm{0},\\ -c_{\mathrm{13}}\xi +c_{\mathrm{14}}-\left(-c_{\mathrm{25}}{\xi }^{\mathrm{3}}+c_{\mathrm{26}}{\xi }^{\mathrm{2}}-c\mathrm{27}\xi +c_{\mathrm{28}}\right)=\mathrm{0},\\ {\phi }^{\prime }\left(\mathrm{0}\right)={\psi }^{\prime \prime \prime }\left(\mathrm{0}\right)⟹\\ c_{\mathrm{11}}-\mathrm{6}{c}_{\mathrm{21}}=\mathrm{0},\\ c_{\mathrm{13}}-\mathrm{6}{c}_{\mathrm{25}}=\mathrm{0},\\ \hspace{1em}\forall \xi \in \left[\mathrm{0,1}\right].\end{array}$$By applying Cramer’s rule, we can deduce$$\begin{array}{c}\text{(10)}& {\tilde{c}}_{\mathrm{12}}=\mathrm{0},{\tilde{c}}_{\mathrm{11}}=\frac{\mathrm{1}}{\mathrm{2}},\\ {\tilde{c}}_{\mathrm{28}}=\mathrm{0},\\ {\tilde{c}}_{\mathrm{27}}=\mathrm{0},\\ {\tilde{c}}_{\mathrm{26}}=\mathrm{0},\\ {\tilde{c}}_{\mathrm{25}}=\frac{\mathrm{1}}{\mathrm{24}},\\ c_{\mathrm{11}}=\mathrm{0},\\ c_{\mathrm{12}}=\mathrm{0},\\ c_{\mathrm{13}}=\frac{\xi -\mathrm{1}}{\mathrm{4}},\\ c_{\mathrm{14}}=\frac{{\left(\xi -\mathrm{1}\right)}^{\mathrm{2}}}{\mathrm{4}},\\ c_{\mathrm{21}}=\mathrm{0},\\ c_{\mathrm{22}}=\mathrm{0},\\ c_{\mathrm{23}}=\frac{\xi -\mathrm{1}}{\mathrm{2}},\\ c_{\mathrm{24}}=-\frac{{\left(\xi -\mathrm{1}\right)}^{\mathrm{2}}}{\mathrm{2}},\\ c_{\mathrm{25}}=\frac{\xi -\mathrm{1}}{\mathrm{24}},\\ c_{\mathrm{26}}=\frac{\xi \left(\xi -\mathrm{1}\right)}{\mathrm{8}},\\ c_{\mathrm{27}}=\frac{\mathrm{1}}{\mathrm{6}}{\xi }^{\mathrm{3}}-\frac{\mathrm{1}}{\mathrm{4}}{\xi }^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}\xi -\frac{\mathrm{1}}{\mathrm{4}},\\ c_{\mathrm{28}}=\frac{\mathrm{1}}{\mathrm{12}}{\xi }^{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{12}}{\xi }^{\mathrm{2}}-\frac{\mathrm{5}}{\mathrm{12}}{\xi }^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}},\\ \hspace{1em}\forall \xi \in \left[\mathrm{0,1}\right].\end{array}$$Plug this into (7) and (8), to yield$$\begin{array}{}\text{(11)}& \phi \left(x\right)=\frac{\mathrm{1}-x}{\mathrm{2}}{{\displaystyle \int }}_{\mathrm{0}}^{x}f\left(\xi \right)d\xi +\frac{\mathrm{1}}{\mathrm{2}}{{\displaystyle \int }}_x^{\mathrm{1}}\left(\mathrm{1}-\xi \right)f\left(\xi \right)d\xi +\frac{\mathrm{1}}{\mathrm{4}}{{\displaystyle \int }}_{\mathrm{0}}^{\mathrm{1}}\left[\left(\xi -\mathrm{1}\right)\left(x-\xi \right)+{\left(\xi -\mathrm{1}\right)}^{\mathrm{2}}\right]g\left(\xi \right)d\xi ,\hspace{1em}x\in \left[\mathrm{0,1}\right],\text{(12)}& \psi \left(x\right)=\frac{\mathrm{1}}{\mathrm{2}}{{\displaystyle \int }}_{\mathrm{0}}^{\mathrm{1}}\left[\left(\xi -\mathrm{1}\right)\left(x-\xi \right)+{\left(\xi -\mathrm{1}\right)}^{\mathrm{2}}\right]f\left(\xi \right)d\xi +\frac{\mathrm{1}}{\mathrm{24}}{{\displaystyle \int }}_{\mathrm{0}}^x\left[\left(\xi -\mathrm{1}\right){\left(x-\xi \right)}^{\mathrm{3}}+\mathrm{3}\xi \left(\xi -\mathrm{1}\right){\left(x-\xi \right)}^{\mathrm{2}}+\left(\mathrm{4}{\xi }^{\mathrm{3}}-\mathrm{6}{\xi }^{\mathrm{2}}+\mathrm{8}\xi -\mathrm{6}\right)\left(x-\xi \right)+\mathrm{2}{\xi }^{\mathrm{3}}+\mathrm{2}{\xi }^{\mathrm{2}}-\mathrm{10}\xi +\mathrm{6}+{\left(x-\xi \right)}^{\mathrm{3}}\right]g\left(\xi \right)d\xi +\frac{\mathrm{1}}{\mathrm{24}}{{\displaystyle \int }}_x^{\mathrm{1}}\left[\left(\xi -\mathrm{1}\right){\left(x-\xi \right)}^{\mathrm{3}}+\mathrm{3}\xi \left(\xi -\mathrm{1}\right){\left(x-\xi \right)}^{\mathrm{2}}+\left(\mathrm{4}{\xi }^{\mathrm{3}}-\mathrm{6}{\xi }^{\mathrm{2}}+\mathrm{8}\xi -\mathrm{6}\right)\left(x-\xi \right)+\mathrm{2}{\xi }^{\mathrm{3}}+\mathrm{2}{\xi }^{\mathrm{2}}-\mathrm{10}\xi +\mathrm{6}\right]g\left(\xi \right)d\xi ,\hspace{1em}x\in \left[\mathrm{0,1}\right],\end{array}$$We deduce from this expression that ${‖\phi ‖}_{H^{\mathrm{2}}(\mathrm{0,1})}^{\mathrm{2}}⩽C[{‖f‖}_{L^{\mathrm{2}}(\mathrm{0,1})}^{\mathrm{2}}+{‖g‖}_{L^{\mathrm{2}}(\mathrm{0,1})}^{\mathrm{2}}]$ and ${‖\psi ‖}_{H^{\mathrm{4}}(\mathrm{0,1})}^{\mathrm{2}}⩽C[{‖f‖}_{L^{\mathrm{2}}(\mathrm{0,1})}^{\mathrm{2}}+{‖g‖}_{L^{\mathrm{2}}(\mathrm{0,1})}^{\mathrm{2}}]$ for some $C>\mathrm{0}$. This means that BVP (6) is well-posed, i.e., $\mathrm{0}\in \rho (A)$.

Remark 2.

Since the embedding $\mathcal{D}(A)\subset \mathcal{H}$ is compact, $A$ has compact resolvents. And therefore, $\sigma (A)$ consists merely of a sequence of eigenvalues.

Next, we shall prove that $\sigma (A)$ is contained in the open left half complex plane.

Proposition 3.

$\sigma (A)$ is contained in the open left half of complex plane, or in symbols, $\sigma (A)\subset {\mathbb{C}}^{-}$, where ${\mathbb{C}}^{-}=\left\{\lambda \in \mathbb{C};\mathfrak{R}\mathfrak{e}\hspace{0.17em}\lambda <\mathrm{0}\right\}$.

Proof.

By using the observation (5), it suffices to show that $\mathrm{i}\mathbb{R}\cap \sigma (A)=\varnothing$. Indeed, $\mathrm{0}\notin \sigma (A)$. Otherwise, there exists, by Remark 2, a nonzero quadruple $(\phi ,\tilde{\phi },\psi ,\tilde{\psi })$ such that $A(\phi ,\tilde{\phi },\psi ,\tilde{\psi })=(\tilde{\phi },{\phi }^{\prime \prime }-a\tilde{\phi },\tilde{\psi },-{\psi }^{(\mathrm{4})})=\mathrm{0}(\phi ,\tilde{\phi },\psi ,\tilde{\psi })$. We have necessarily $\tilde{\phi }=\mathrm{0}$, $\tilde{\psi }=\mathrm{0}$, ${\phi }^{\prime \prime }=\mathrm{0}$ and ${\psi }^{(\mathrm{4})}=\mathrm{0}$. And therefore, $\phi$ and $\psi$ assume the forms $\phi (x)=c_{\mathrm{11}}x+c_{\mathrm{12}}$ and $\psi (x)=c_{\mathrm{21}}{x}^{\mathrm{3}}+c_{\mathrm{22}}{x}^{\mathrm{2}}+c_{\mathrm{23}}x+c_{\mathrm{24}}$, respectively. Recalling the boundary conditions to which $\phi$ and $\psi$ are subject, we have further $$\begin{array}{c}\text{(13)}& \phi \left(\mathrm{1}\right)=\mathrm{0}⟹c_{\mathrm{11}}+c_{\mathrm{12}}=\mathrm{0},\text{,}\\ \psi \left(\mathrm{1}\right)=\mathrm{0}⟹\\ c_{\mathrm{21}}+c_{\mathrm{22}}+c_{\mathrm{23}}+c_{\mathrm{24}}=\mathrm{0},\\ \phi \left(\mathrm{0}\right)-\psi \left(\mathrm{0}\right)=\mathrm{0}⟹\\ c_{\mathrm{12}}-c_{\mathrm{24}}=\mathrm{0},\\ {\psi }^{\prime \prime }\left(\mathrm{0}\right)=\mathrm{0}⟹\\ \mathrm{2}{c}_{\mathrm{22}}=\mathrm{0},\\ {\psi }^{\prime \prime }\left(\mathrm{1}\right)=\mathrm{0}⟹\\ \mathrm{6}{c}_{\mathrm{21}}+\mathrm{2}{c}_{\mathrm{22}}=\mathrm{0},\\ {\phi }^{\prime }\left(\mathrm{0}\right)-{\psi }^{\prime \prime \prime }\left(\mathrm{0}\right)=\mathrm{0}⟹\\ c_{\mathrm{11}}-\mathrm{6}{c}_{\mathrm{21}}=\mathrm{0}.\end{array}$$It is ready to check that$$\begin{array}{}\text{(14)}& \mathrm{d}\mathrm{e}\mathrm{t}\left(\begin{array}{cccccc}\mathrm{1}& \mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}& \mathrm{1}& \mathrm{1}& \mathrm{1}\\ \mathrm{0}& \mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}& -\mathrm{1}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{2}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{6}& \mathrm{2}& \mathrm{0}& \mathrm{0}\\ \mathrm{1}& \mathrm{0}& -\mathrm{6}& \mathrm{0}& \mathrm{0}& \mathrm{0}\end{array}\right)=-\mathrm{12}\ne \mathrm{0}.\end{array}$$Then $c_{\mathrm{11}}=c_{\mathrm{12}}=c_{\mathrm{21}}=c_{\mathrm{22}}=c_{\mathrm{23}}=c_{\mathrm{24}}=\mathrm{0}$, that is, $\phi =\mathrm{0}$ and $\psi =\mathrm{0}$. This, together with the observation that $\tilde{\phi }=\mathrm{0}$ and $\tilde{\psi }=\mathrm{0}$, contradicts the fact that the quadruple $(\phi ,\tilde{\phi },\psi ,\tilde{\psi })$ is nonzero. That is, $\mathrm{0}\in \rho (A)$.

It remains to show that $\mathrm{i}\mu \notin \sigma (A)$ for every $\mu \in \mathbb{R}\setminus \{\mathrm{0}\}$. Let $(\phi ,\tilde{\phi },\psi ,\tilde{\psi })\in \mathcal{D}(A)$ be such that $A(\phi ,\tilde{\phi },\psi ,\tilde{\psi })=\mathrm{i}\mu (\phi ,\tilde{\phi },\psi ,\tilde{\psi })$ for some $\mu \in \mathbb{R}\setminus \{\mathrm{0}\}$. We have $⟨A(\phi ,\tilde{\phi },\psi ,\tilde{\psi }),(\phi ,\tilde{\phi },\psi ,\tilde{\psi }){⟩}_{\mathcal{H}}=\mathrm{i}\mu {‖(\phi ,\tilde{\phi },\psi ,\tilde{\psi })‖}_{\mathcal{H}}^{\mathrm{2}}$ which implies $\mathfrak{R}\mathfrak{e}⟨A(\phi ,\tilde{\phi },\psi ,\tilde{\psi }),(\phi ,\tilde{\phi },\psi ,\tilde{\psi }){⟩}_{\mathcal{H}}=\mathrm{0}$. This, together with (5), implies $\tilde{\phi }(x)=\mathrm{0}$, $x\in [\mathrm{0,1}]$. Thus, the equation $A(\phi ,\tilde{\phi },\psi ,\tilde{\psi })=\mathrm{i}\mu (\phi ,\tilde{\phi },\psi ,\tilde{\psi })$ is reduced to the boundary value problem$$\begin{array}{c}\text{(15)}& {\psi }^{\left(\mathrm{4}\right)}={\mu }^{\mathrm{2}}\psi \hspace{1em}\mathrm{i}\mathrm{n}\left(\mathrm{0,1}\right),\psi \left(\mathrm{0}\right)={\psi }^{\prime \prime }\left(\mathrm{0}\right)={\psi }^{\prime \prime }\left(\mathrm{1}\right)={\psi }^{\prime \prime \prime }\left(\mathrm{0}\right)=\mathrm{0}.\end{array}$$The solution to BVP (15) can be written as $$\begin{array}{}\text{(16)}& \psi \left(x\right)={\widehat{c}}_{\mathrm{1}}\cosh \left(x\sqrt{\left|\mu \right|}\right)+{\widehat{c}}_{\mathrm{2}}\sinh \left(x\sqrt{\left|\mu \right|}\right)+{\widehat{c}}_{\mathrm{3}}\cos \left(x\sqrt{\left|\mu \right|}\right)+{\widehat{c}}_{\mathrm{4}}\sin \left(x\sqrt{\left|\mu \right|}\right)\end{array}$$where ${\widehat{c}}_{\mathrm{1}},{\widehat{c}}_{\mathrm{2}},{\widehat{c}}_{\mathrm{3}}$, and ${\widehat{c}}_{\mathrm{4}}$ are chosen so that $$\begin{array}{c}\text{(17)}& {\widehat{c}}_{\mathrm{1}}+{\widehat{c}}_{\mathrm{3}}=\mathrm{0},{\widehat{c}}_{\mathrm{1}}-{\widehat{c}}_{\mathrm{3}}=\mathrm{0},\\ {\widehat{c}}_{\mathrm{1}}\cosh \left(\sqrt{\left|\mu \right|}\right)+{\widehat{c}}_{\mathrm{2}}\sinh \left(\sqrt{\left|\mu \right|}\right)-{\widehat{c}}_{\mathrm{3}}\cos \left(\sqrt{\left|\mu \right|}\right)-{\widehat{c}}_{\mathrm{4}}\sin \left(\sqrt{\left|\mu \right|}\right)=\mathrm{0},\\ {\widehat{c}}_{\mathrm{2}}-{\widehat{c}}_{\mathrm{4}}=\mathrm{0}.\end{array}$$We deduce from the first two equations that ${\widehat{c}}_{\mathrm{1}}={\widehat{c}}_{\mathrm{3}}=\mathrm{0}$. Substitute this into the third equation, use the fact that $\sinh (\sqrt{\left|\mu \right|})>\sqrt{\left|\mu \right|}>\sin (\sqrt{\left|\mu \right|})$ for $\mu \in \mathbb{R}\setminus \{\mathrm{0}\}$, and conduct some routine calculations, to obtain ${\widehat{c}}_{\mathrm{2}}={\widehat{c}}_{\mathrm{4}}=\mathrm{0}$. Therefore, $\psi (x)=\mathrm{0}$, $x\in [\mathrm{0,1}]$. To sum, $(\phi ,\tilde{\phi },\psi ,\tilde{\psi })=\mathrm{0}$. This means that $\mathrm{i}\mu \notin \sigma (A)$ for every $\mu \in \mathbb{R}\setminus \{\mathrm{0}\}$. The proof is complete.

Proposition 4.

Let $\lambda \in \mathbb{R}\setminus \{\mathrm{0}\}$. Then $\lambda \in \sigma (A)$ if and only if $\lambda$ satisfies the equation $\Sigma (\lambda )=\mathrm{0}$, where$$\begin{array}{}\text{(18)}& \Sigma \left(\lambda \right)=\lambda \sqrt{\mathrm{i}\lambda }\sinh \left(\sqrt{\lambda \left(\lambda +a\right)}\right)\left[\cosh \left(\sqrt{\mathrm{i}\lambda }\right)\sinh \left(\mathrm{i}\sqrt{\mathrm{i}\lambda }\right)-\mathrm{i}\cosh \left(\mathrm{i}\sqrt{\mathrm{i}\lambda }\right)\sinh \left(\sqrt{\mathrm{i}\lambda }\right)\right]+\mathrm{2}\mathrm{i}\sqrt{\lambda \left(\lambda +a\right)}\cosh \left(\sqrt{\lambda \left(\lambda +a\right)}\right)\sinh \left(\mathrm{i}\sqrt{\mathrm{i}\lambda }\right)\sinh \left(\sqrt{\mathrm{i}\lambda }\right).\end{array}$$

Proof.

We should note that $\lambda \in \sigma (A)$ if and only if the following boundary value problem admits a nonzero solution: $$\begin{array}{c}\text{(19)}& {\lambda }^{\mathrm{2}}\phi -{\phi }^{\prime \prime }+a\lambda \phi =\mathrm{0}\hspace{1em}\mathrm{i}\mathrm{n}\left(\mathrm{0,1}\right),{\lambda }^{\mathrm{2}}\psi +{\psi }^{\left(\mathrm{4}\right)}=\mathrm{0}\hspace{1em}\mathrm{i}\mathrm{n}\left(\mathrm{0,1}\right),\\ \phi \left(\mathrm{1}\right)={\psi }^{\prime \prime }\left(\mathrm{0}\right)=\psi \left(\mathrm{1}\right)={\psi }^{\prime \prime }\left(\mathrm{1}\right)=\mathrm{0},\\ \phi \left(\mathrm{0}\right)=\psi \left(\mathrm{0}\right),\\ {\phi }^{\prime }\left(\mathrm{0}\right)={\psi }^{\prime \prime \prime }\left(\mathrm{0}\right).\end{array}$$By the classical theory of ODEs, $(\phi ,\psi )$ can be written as $$\begin{array}{c}\text{(20)}& \phi \left(x\right)=c_{\mathrm{1}}\cosh \left(\left(\mathrm{1}-x\right)\sqrt{\lambda \left(\lambda +a\right)}\right)+c_{\mathrm{2}}\sinh \left(\left(\mathrm{1}-x\right)\sqrt{\lambda \left(\lambda +a\right)}\right),\psi \left(x\right)=c_{\mathrm{3}}\cosh \left(\left(\mathrm{1}-x\right)\sqrt{\mathrm{i}\lambda }\right)+c_{\mathrm{4}}\sinh \left(\left(\mathrm{1}-x\right)\sqrt{\mathrm{i}\lambda }\right)+c_{\mathrm{5}}\cosh \left(\left(\mathrm{1}-x\right)\mathrm{i}\sqrt{\mathrm{i}\lambda }\right)+c_{\mathrm{6}}\sinh \left(\left(\mathrm{1}-x\right)\mathrm{i}\sqrt{\mathrm{i}\lambda }\right).\end{array}$$The coefficients $c_{\mathrm{1}},c_{\mathrm{2}},c_{\mathrm{3}},c_{\mathrm{4}},c_{\mathrm{5}},c_{\mathrm{6}}$ are determined by the boundary conditions on $\phi$ and $\psi$. More precisely, we have$$\begin{array}{c}\text{(21)}& \phi \left(\mathrm{1}\right)=\mathrm{0}⟹c_{\mathrm{1}}=\mathrm{0},\\ \psi \left(\mathrm{1}\right)=\mathrm{0}⟹\\ c_{\mathrm{3}}+c_{\mathrm{5}}=\mathrm{0},\\ {\psi }^{\prime \prime }\left(\mathrm{1}\right)=\mathrm{0}⟹\\ c_{\mathrm{3}}\mathrm{i}\lambda -c_{\mathrm{5}}\mathrm{i}\lambda =\mathrm{0},\\ {\psi }^{\prime \prime }\left(\mathrm{0}\right)=\mathrm{0}⟹\\ c_{\mathrm{4}}\mathrm{i}\lambda \sinh \left(\sqrt{\mathrm{i}\lambda }\right)-c_{\mathrm{6}}\mathrm{i}\lambda \sinh \left(\mathrm{i}\sqrt{\mathrm{i}\lambda }\right)=\mathrm{0},\\ \phi \left(\mathrm{0}\right)-\psi \left(\mathrm{0}\right)=\mathrm{0}⟹\\ c_{\mathrm{2}}\sinh \left(\sqrt{\lambda \left(\lambda +a\right)}\right)-c_{\mathrm{4}}\sinh \left(\sqrt{\mathrm{i}\lambda }\right)-c_{\mathrm{6}}\sinh \left(\mathrm{i}\sqrt{\mathrm{i}\lambda }\right)=\mathrm{0},\\ {\phi }^{\prime }\left(\mathrm{0}\right)-{\psi }^{\prime \prime \prime }\left(\mathrm{0}\right)=\mathrm{0}⟹\\ -c_{\mathrm{2}}\sqrt{\lambda \left(\lambda +a\right)}\cosh \left(\sqrt{\lambda \left(\lambda +a\right)}\right)+c_{\mathrm{4}}\mathrm{i}\lambda \sqrt{\mathrm{i}\lambda }\cosh \left(\sqrt{\mathrm{i}\lambda }\right)+c_{\mathrm{6}}\lambda \sqrt{\mathrm{i}\lambda }\cosh \left(\mathrm{i}\sqrt{\mathrm{i}\lambda }\right)=\mathrm{0}.\end{array}$$By the related classical theory from Linear Algebra, (21) has a nontrivial solution if and only if $$\begin{array}{}\text{(22)}& \det \left(\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 1& 0\\ 0& 0& \mathrm{i}\lambda & 0& -\mathrm{i}\lambda & 0\\ 0& 0& 0& \mathrm{i}\lambda \sinh \left(\sqrt{\mathrm{i}\lambda }\right)& 0& -\mathrm{i}\lambda \sinh \left(\mathrm{i}\sqrt{\mathrm{i}\lambda }\right)\\ 0& \sinh \left(\sqrt{\lambda \left(\lambda +a\right)}\right)& 0& -\sinh \left(\sqrt{\mathrm{i}\lambda }\right)& 0& -\sinh \left(\mathrm{i}\sqrt{\mathrm{i}\lambda }\right)\\ 0& -\sqrt{\lambda \left(\lambda +a\right)}\cosh \left(\sqrt{\lambda \left(\lambda +a\right)}\right)& 0& \mathrm{i}\lambda \sqrt{\mathrm{i}\lambda }\cosh \left(\sqrt{\mathrm{i}\lambda }\right)& 0& \lambda \sqrt{\mathrm{i}\lambda }\cosh \left(\mathrm{i}\sqrt{\mathrm{i}\lambda }\right)\end{array}\right)=\mathrm{2}\mathrm{i}{\lambda }^{\mathrm{2}}\Sigma \left(\lambda \right)=\mathrm{0}.\end{array}$$By recalling that $\lambda \ne \mathrm{0}$, we infer that the proof is complete.

Theorem 5.

$\sigma (A)$ is contained in the open left half complex plane ${\mathbb{C}}^{-}$ and consists merely of eigenvalues $\{{\lambda }_{\mathrm{1}k},{\lambda }_{\mathrm{2}\mathcal{l}}\}$ of $A$. And moreover, the eigenvalues are distributed as follows: $$\begin{array}{c}\text{(23)}& {\lambda }_{\mathrm{1}k}=\frac{-a\pm \mathrm{i}\sqrt{\mathrm{4}{k}^{\mathrm{2}}{\pi }^{\mathrm{2}}-a^{\mathrm{2}}}}{\mathrm{2}}+\mathcal{O}\left(k^{-\mathrm{1}/\mathrm{2}}\right)\hspace{1em}\mathrm{a}\mathrm{s}k\to +\infty \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}k\in \mathbb{N},{\lambda }_{\mathrm{2}\mathcal{l}}=\pm {\pi }^{\mathrm{2}}{\left(\mathcal{l}+\frac{\mathrm{1}}{\mathrm{4}}\right)}^{\mathrm{2}}\mathrm{i}+\mathcal{O}\left({\left|\mathcal{l}\right|}^{-\mathrm{1}/\mathrm{2}}\right)\hspace{1em}\mathrm{a}\mathrm{s}\left|\mathcal{l}\right|\to +\infty \text{\hspace{0.17em}\hspace{0.17em}with\hspace{0.17em}\hspace{0.17em}}\mathcal{l}\in \mathbb{Z}.\end{array}$$

Proof.

Recall that $\sigma (A)\subset {\mathbb{C}}^{-}$. We shall first consider the situation $\lambda \in {\mathbb{D}}_{\mathrm{1}}\cup {\mathbb{D}}_{\mathrm{2}}$ where ${\mathbb{D}}_{\mathrm{1}}=\{\lambda \in {\mathbb{C}}^{-};\mathrm{a}\mathrm{r}\mathrm{g}\hspace{0.17em}\lambda \in [\mathrm{3}\pi /\mathrm{4},\pi ]\}$ and ${\mathbb{D}}_{\mathrm{2}}=\{\lambda \in {\mathbb{C}}^{-};\mathrm{a}\mathrm{r}\mathrm{g}\hspace{0.17em}\lambda \in [\pi /\mathrm{2},\mathrm{3}\pi /\mathrm{4}]\}$. For the sake of simplification, we would use $\lambda =\left|\lambda \right|e^{\mathrm{i}\theta }$ for complex numbers. We next study the eigenvalues of $A$ in ${\mathbb{D}}_{\mathrm{1}}\cup {\mathbb{D}}_{\mathrm{2}}$.

Case  1 ($\lambda \in {\mathbb{D}}_{\mathrm{1}}$). By applying Taylor’s expansion, we have$$\begin{array}{}\text{(24)}& e^{\sqrt{\lambda \left(\lambda +a\right)}}=e^{\sqrt{\lambda \left(\lambda +a\right)}}=e^{-\lambda \sqrt{\mathrm{1}+a/\lambda }}=e^{-\lambda \left[\mathrm{1}+a/\mathrm{2}\lambda +\mathcal{O}\left({\left|\lambda \right|}^{-\mathrm{2}}\right)\right]}=e^{-\lambda -a/\mathrm{2}+\mathcal{O}\left({\left|\lambda \right|}^{-\mathrm{1}}\right)}\end{array}$$for $\lambda$ with sufficiently large modules. Besides, we have$$\begin{array}{}\text{(25)}& e^{\sqrt{\mathrm{i}\lambda }}=\exp \left(\sqrt{\left|\lambda \right|}{e}^{\mathrm{i}\left(\theta /\mathrm{2}-\mathrm{3}\pi /\mathrm{4}\right)}\right)=\exp \left(\sqrt{\left|\lambda \right|}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\theta }{\mathrm{2}}-\frac{\mathrm{3}\pi }{\mathrm{4}}\right)\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{i}\sqrt{\left|\lambda \right|}\sin \left(\frac{\theta }{\mathrm{2}}-\frac{\mathrm{3}\pi }{\mathrm{4}}\right)\right),\text{(26)}& e^{\mathrm{i}\sqrt{\mathrm{i}\lambda }}=\exp \left(\sqrt{\left|\lambda \right|}\cos \left(\frac{\theta }{\mathrm{2}}-\frac{\pi }{\mathrm{4}}\right)\right)\exp \left(\mathrm{i}\sqrt{\left|\lambda \right|}\sin \left(\frac{\theta }{\mathrm{2}}-\frac{\pi }{\mathrm{4}}\right)\right).\end{array}$$Therefore,$$\begin{array}{c}\text{(27)}& \underset{\left|\lambda \right|\to +\infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\left|e^{\sqrt{\lambda \left(\lambda +a\right)}}\right|=+\infty ,\underset{\left|\lambda \right|\to +\infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\left|e^{\sqrt{\mathrm{i}\lambda }}\right|=+\infty ,\\ \underset{\left|\lambda \right|\to +\infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\left|e^{\mathrm{i}\sqrt{\mathrm{i}\lambda }}\right|=+\infty .\end{array}$$By analyzing the definition (18) of $\Sigma (\lambda )$, we have $$\begin{array}{}\text{(28)}& \underset{\lambda \in {\mathbb{D}}_{\mathrm{1}},\left|\lambda \right|\to +\infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\Sigma \left(\lambda \right)}{\lambda \sqrt{\mathrm{i}\lambda }{e}^{\sqrt{\lambda \left(\lambda +a\right)}+\sqrt{\mathrm{i}\lambda }+\mathrm{i}\sqrt{\mathrm{i}\lambda }}}=\frac{\mathrm{1}-\mathrm{i}}{\mathrm{8}}\ne \mathrm{0}.\end{array}$$This, together with (27), implies that $\lambda \in {\mathbb{D}}_{\mathrm{1}}$ cannot be a solution to the equation $\Sigma (\lambda )=\mathrm{0}$, or equivalently, $\lambda$ cannot be an eigenvalue of $A$, whenever its module $\left|\lambda \right|$ is sufficiently large.

Case  2 ($\lambda \in {\mathbb{D}}_{\mathrm{2}}$). We deduce from (24), (25), and (26) that $$\begin{array}{}\text{(29)}& \left|e^{-\sqrt{\lambda \left(\lambda +a\right)}}\right|=\left|e^{\lambda +a/\mathrm{2}+\mathcal{O}\left({\left|\lambda \right|}^{-\mathrm{1}}\right)}\right|⩽C_{\mathrm{1}},\end{array}$$for some $C_{\mathrm{1}}>\mathrm{0}$, $$\begin{array}{c}\text{(30)}& \left|e^{-\sqrt{\mathrm{i}\lambda }}\right|=\left|\exp \left(-\sqrt{\left|\lambda \right|}\cos \left(\frac{\theta }{\mathrm{2}}-\frac{\mathrm{3}\pi }{\mathrm{4}}\right)\right)\right|⩽\mathrm{1},\left|e^{-\mathrm{i}\sqrt{\mathrm{i}\lambda }}\right|=\exp \left(-\sqrt{\left|\lambda \right|}\cos \left(\frac{\theta }{\mathrm{2}}-\frac{\pi }{\mathrm{4}}\right)\right)⩽e^{-\sqrt{\mathrm{2}\left|\lambda \right|}/\mathrm{2}}.\end{array}$$Divide the both hand sides of the equation $\Sigma (\lambda )=\mathrm{0}$ by $\lambda \sqrt{\mathrm{i}\lambda }{e}^{\sqrt{\lambda (\lambda +a)}+\sqrt{\mathrm{i}\lambda }+\mathrm{i}\sqrt{\mathrm{i}\lambda }}$, and conduct some routine calculations, to obtain $$\begin{array}{}\text{(31)}& \left(\mathrm{1}-e^{-\mathrm{2}\sqrt{\lambda \left(\lambda +a\right)}}\right)\left(\mathrm{i}-e^{-\mathrm{2}\sqrt{\mathrm{i}\lambda }}\right)=\mathcal{O}\left({\left|\lambda \right|}^{-\mathrm{1}/\mathrm{2}}\right)\end{array}$$as $\left|\lambda \right|\to +\infty$ with $\lambda \in {\mathbb{D}}_{\mathrm{2}}$. Note that the equation $\left(\mathrm{1}-e^{-\mathrm{2}\sqrt{\lambda (\lambda +a)}}\right)\left(\mathrm{i}-e^{-\mathrm{2}\sqrt{\mathrm{i}\lambda }}\right)=\mathrm{0}$ with the restriction $\lambda \in {\mathbb{D}}_{\mathrm{2}}$ has the solutions ${\widehat{\lambda }}_{\mathrm{1}k}=(-a+\mathrm{i}\sqrt{\mathrm{4}{k}^{\mathrm{2}}{\pi }^{\mathrm{2}}-a^{\mathrm{2}}})/\mathrm{2}$ with $k\in \mathbb{N}$, and ${\widehat{\lambda }}_{\mathrm{2}\mathcal{l}}={\pi }^{\mathrm{2}}{\left(\mathcal{l}+\mathrm{1}/\mathrm{4}\right)}^{\mathrm{2}}\mathrm{i}$ with $\mathcal{l}\in \mathbb{Z}$. By applying Rouché’s theorem (see [1]) and Proposition 4, we obtain the spectral asymptotics for $A$: ${\lambda }_{\mathrm{1}k}=(-a+\mathrm{i}\sqrt{\mathrm{4}{k}^{\mathrm{2}}{\pi }^{\mathrm{2}}-a^{\mathrm{2}}})/\mathrm{2}+\mathcal{O}(k^{-\mathrm{1}/\mathrm{2}})$ as $k\to \infty$ with $k\in \mathbb{N}$, and ${\lambda }_{\mathrm{2}\mathcal{l}}={\pi }^{\mathrm{2}}{\left(\mathcal{l}+\mathrm{1}/\mathrm{4}\right)}^{\mathrm{2}}\mathrm{i}+\mathcal{O}(|\mathcal{l}{|}^{-\mathrm{1}/\mathrm{2}})$ as $\mathcal{l}\to \infty$ with $\mathcal{l}\in \mathbb{Z}$.

Lastly, noting that $\lambda \in \sigma (A)$ iff $\overline{\lambda }\in \sigma (A)$, we know that there are eigenvalues of $A$ which are distributed as follows: ${\lambda }_{\mathrm{1}k}=(-a-\mathrm{i}\sqrt{\mathrm{4}{k}^{\mathrm{2}}{\pi }^{\mathrm{2}}-a^{\mathrm{2}}})/\mathrm{2}+\mathcal{O}(k^{-\mathrm{1}/\mathrm{2}})$ as $k\to \infty$ with $k\in \mathbb{N}$, and ${\lambda }_{\mathrm{2}\mathcal{l}}=-{\pi }^{\mathrm{2}}{\left(\mathcal{l}+\mathrm{1}/\mathrm{4}\right)}^{\mathrm{2}}\mathrm{i}+\mathcal{O}(|\mathcal{l}{|}^{-\mathrm{1}/\mathrm{2}})$ as $\mathcal{l}\to \infty$ with $\mathcal{l}\in \mathbb{Z}$. This, together with the afore-conducted analysis, implies that the proof is complete.

Theorem 6.

The natural energy functional associated with system (1)$$\begin{array}{}\text{(32)}& E\left(t\right)=\frac{\mathrm{1}}{\mathrm{2}}{{\displaystyle \int }}_{\mathrm{0}}^{\mathrm{1}}\left({\left|{\partial }_{t}u\left(x,t\right)\right|}^{\mathrm{2}}+{\left|{\partial }_{x}u\left(x,t\right)\right|}^{\mathrm{2}}\right)dx+\frac{\mathrm{1}}{\mathrm{2}}{{\displaystyle \int }}_{-\mathrm{1}}^{\mathrm{0}}\left({\left|{\partial }_{t}w\left(x,t\right)\right|}^{\mathrm{2}}+{\left|{\partial }_x^{\mathrm{2}}w\left(x,t\right)\right|}^{\mathrm{2}}\right)dx\end{array}$$decreases to $\mathrm{0}$ as $t\to +\infty$, $(u,v)$ is a solution to system (1).

Proof.

Let us define the natural energy functional associated with$$\begin{array}{}\text{(33)}& \tilde{E}\left(t\right)=\frac{\mathrm{1}}{\mathrm{2}}{{\displaystyle \int }}_{\mathrm{0}}^{\mathrm{1}}\left({\left|{\partial }_{t}u\left(x,t\right)\right|}^{\mathrm{2}}+{\left|{\partial }_{x}u\left(x,t\right)\right|}^{\mathrm{2}}\right)dx+\frac{\mathrm{1}}{\mathrm{2}}{{\displaystyle \int }}_{\mathrm{0}}^{\mathrm{1}}\left({\left|{\partial }_{t}v\left(x,t\right)\right|}^{\mathrm{2}}+{\left|{\partial }_x^{\mathrm{2}}v\left(x,t\right)\right|}^{\mathrm{2}}\right)dx,\hspace{1em}t\in \left[\mathrm{0},+\infty \right).\end{array}$$Recalling the relation $v(x,t)=w(-x,t)$, and by applying integration by change-of-variable, we have $E(t)=\tilde{E}(t)$ for all $t\in [\mathrm{0},+\infty )$. But we have also $$\begin{array}{}\text{(34)}& \left(u\left(·,t\right),{\partial }_{t}u\left(·,t\right),v\left(·,t\right),{\partial }_{t}v\left(·,t\right)\right)=e^{tA}\left(u^{\mathrm{0}},u^{\mathrm{1}},v^{\mathrm{0}},v^{\mathrm{1}}\right),\hspace{1em}\forall t\in \left[\mathrm{0},+\infty \right).\end{array}$$By, we have $$\begin{array}{}\text{(35)}& \underset{t\to +\infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{‖e^{tA}\left(u^{\mathrm{0}},u^{\mathrm{1}},v^{\mathrm{0}},v^{\mathrm{1}}\right)‖}_{\mathcal{H}}^{\mathrm{2}}=\mathrm{0}.\end{array}$$And therefore,$$\begin{array}{}\text{(36)}& \underset{t\to +\infty }{\mathrm{l}\mathrm{i}\mathrm{m}}E\left(t\right)=\underset{t\to +\infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\tilde{E}\left(t\right)=\frac{\mathrm{1}}{\mathrm{2}}\underset{t\to +\infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{‖\left(u\left(·,t\right),{\partial }_{t}u\left(·,t\right),v\left(·,t\right),{\partial }_{t}v\left(·,t\right)\right)‖}_{\mathcal{H}}^{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}}\underset{t\to +\infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{‖e^{tA}\left(u^{\mathrm{0}},u^{\mathrm{1}},v^{\mathrm{0}},v^{\mathrm{1}}\right)‖}_{\mathcal{H}}^{\mathrm{2}}=\mathrm{0}.\end{array}$$The proof is complete.

3. Conclusion

We investigated in this paper the transmission system (1) of a 1D damped wave equation and a 1D undamped plate equation for its spectral property. We proved that the spectrum of the associated infinitesimal generator consists merely with eigenvalues which lie to the left of the imaginary axis and have two distinct vertical lines as its asymptote lines in which the one on the right hand side is the imaginary axis. We obtained also a byproduct: the energy $E(t)$ of system (1) decreases to zero as time approaches infinity. The first natural question reads: Can one obtain the optimal decay rate for the energy functional $E(t)$ defined by (32)? We would try to answer this question in our future work.

Another interesting question is: Can the method be carried out for wave/plate equations in higher dimensional domains to obtain the same results as in this paper? The answer is: It seems to be possible to use the idea of this paper to conduct spectral analysis for systems posed in certain special higher dimensional domains, but would need much more complicated calculations to obtain similar results as in 1D cases. We explain here this claim in certain detail. Let us consider a very special 2D transmission coupled system of wave and plate equations, that is,$$\begin{array}{c}\text{(37)}& {\partial }_t^{\mathrm{2}}u-\Delta u+a{\partial }_{t}u=\mathrm{0},\hspace{1em}\forall \left(x,y,t\right)\in {\Omega }^{+}\times \left(\mathrm{0},+\infty \right),{\partial }_t^{\mathrm{2}}w+{\Delta }^{\mathrm{2}}w=\mathrm{0},\hspace{1em}\forall \left(x,y,t\right)\in {\Omega }^{-}\times \left(\mathrm{0},+\infty \right),\\ u\left(\mathrm{1},y,t\right)={\partial }_x^{\mathrm{2}}w\left(\mathrm{0},y,t\right)=w\left(-\mathrm{1},y,t\right)={\partial }_x^{\mathrm{2}}w\left(-\mathrm{1},y,t\right)=\mathrm{0},\hspace{1em}\forall \left(y,t\right)\in {\left(\mathrm{0,1}\right)}_y\times \left(\mathrm{0},+\infty \right),\\ u\left(\mathrm{0},y,t\right)=w\left(\mathrm{0},y,t\right),\\ {\partial }_{x}u\left(\mathrm{0},y,t\right)=-{\partial }_x^{\mathrm{3}}w\left(\mathrm{0},y,t\right),\\ \hspace{1em}\forall \left(y,t\right)\in {\left(\mathrm{0,1}\right)}_y\times \left(\mathrm{0},+\infty \right),\\ u\left(x,\mathrm{0},t\right)=u\left(x,\mathrm{1},t\right)=\mathrm{0},\hspace{1em}\forall \left(x,t\right)\in {\left(\mathrm{0,1}\right)}_x\times \left(\mathrm{0},+\infty \right),\\ w\left(x,\mathrm{0},t\right)={\partial }_y^{\mathrm{2}}w\left(x,\mathrm{0},t\right)=w\left(x,\mathrm{1},t\right)={\partial }_y^{\mathrm{2}}w\left(x,\mathrm{1},t\right)=\mathrm{0},\hspace{1em}\forall \left(x,t\right)\in {\left(-\mathrm{1,0}\right)}_x\times \left(\mathrm{0},+\infty \right),\\ u\left(·,\mathrm{0}\right)=u^{\mathrm{0}},\\ {\partial }_{t}u\left(·,\mathrm{0}\right)=u^{\mathrm{1}},\\ \hspace{1em}\forall \left(x,y\right)\in {\Omega }^{+},\\ w\left(·,\mathrm{0}\right)=w^{\mathrm{0}},\\ {\partial }_{t}w\left(·,\mathrm{0}\right)=w^{\mathrm{1}},\\ \hspace{1em}\forall \left(x,y\right)\in {\Omega }^{-},\end{array}$$where ${\Omega }^{-}=[-\mathrm{1,0}{]}_x\times [\mathrm{0,1}{]}_y$, ${\Omega }^{+}=[\mathrm{0,1}{]}_x\times [\mathrm{0,1}{]}_y$, $\Delta ={\partial }_x^{\mathrm{2}}+{\partial }_y^{\mathrm{2}}$, ${\Delta }^{\mathrm{2}}={\partial }_x^{\mathrm{4}}+{\partial }_y^{\mathrm{4}}$, $\Omega ={\Omega }^{+}\cup {\Omega }^{-}$, $u(x,y,t)$ and $w(x,y,t)$. By introducing $v(x,y,t)=w(-x,y,t)$, we obtain the auxiliary system$$\begin{array}{c}\text{(38)}& {\partial }_t^{\mathrm{2}}u-\Delta u+a{\partial }_{t}u=\mathrm{0},\hspace{1em}\forall \left(x,y,t\right)\in {\Omega }^{+}\times \left(\mathrm{0},+\infty \right),{\partial }_t^{\mathrm{2}}v+{\Delta }^{\mathrm{2}}v=\mathrm{0},\hspace{1em}\forall \left(x,y,t\right)\in {\Omega }^{+}\times \left(\mathrm{0},+\infty \right),\\ u\left(\mathrm{1},y,t\right)={\partial }_x^{\mathrm{2}}v\left(\mathrm{0},y,t\right)=v\left(\mathrm{1},y,t\right)={\partial }_x^{\mathrm{2}}v\left(\mathrm{1},y,t\right)=\mathrm{0},\hspace{1em}\forall \left(y,t\right)\in {\left(\mathrm{0,1}\right)}_y\times \left(\mathrm{0},+\infty \right),\\ u\left(\mathrm{0},y,t\right)=v\left(\mathrm{0},y,t\right),\\ {\partial }_{x}u\left(\mathrm{0},y,t\right)={\partial }_x^{\mathrm{3}}v\left(\mathrm{0},y,t\right),\\ \hspace{1em}\forall \left(y,t\right)\in {\left(\mathrm{0,1}\right)}_y\times \left(\mathrm{0},+\infty \right),\\ u\left(x,\mathrm{0},t\right)=u\left(x,\mathrm{1},t\right)=\mathrm{0},\hspace{1em}\forall \left(x,t\right)\in {\left(\mathrm{0,1}\right)}_x\times \left(\mathrm{0},+\infty \right),\\ v\left(x,\mathrm{0},t\right)={\partial }_y^{\mathrm{2}}v\left(x,\mathrm{0},t\right)=v\left(x,\mathrm{1},t\right)={\partial }_y^{\mathrm{2}}v\left(x,\mathrm{1},t\right)=\mathrm{0},\hspace{1em}\forall \left(x,t\right)\in {\left(\mathrm{0,1}\right)}_x\times \left(\mathrm{0},+\infty \right),\\ u\left(·,\mathrm{0}\right)=u^{\mathrm{0}},\\ {\partial }_{t}u\left(·,\mathrm{0}\right)=u^{\mathrm{1}},\\ \hspace{1em}\forall \left(x,y\right)\in {\Omega }^{+},\\ v\left(·,\mathrm{0}\right)=v^{\mathrm{0}},\\ {\partial }_{t}v\left(·,\mathrm{0}\right)=v^{\mathrm{1}},\\ \hspace{1em}\forall \left(x,y\right)\in {\Omega }^{+}.\end{array}$$The spectral study of (37) is equivalent to that of (38). And thus we are led to the characteristic problem $$\begin{array}{c}\text{(39)}& {\lambda }^{\mathrm{2}}\phi -\Delta \phi +a\lambda \phi =\mathrm{0},\hspace{1em}\forall \left(x,y\right)\in {\Omega }^{+},{\lambda }^{\mathrm{2}}\psi +{\Delta }^{\mathrm{2}}\psi =\mathrm{0},\hspace{1em}\forall \left(x,y\right)\in {\Omega }^{+},\\ \phi \left(\mathrm{1},y\right)={\partial }_x^{\mathrm{2}}\psi \left(\mathrm{0},y\right)=\psi \left(\mathrm{1},y\right)={\partial }_x^{\mathrm{2}}\psi \left(\mathrm{1},y\right)=\mathrm{0},\hspace{1em}\forall y\in {\left(\mathrm{0,1}\right)}_y,\\ \phi \left(\mathrm{0},y\right)=\psi \left(\mathrm{0},y\right),\\ {\partial }_x\phi \left(\mathrm{0},y\right)={\partial }_x^{\mathrm{3}}\psi \left(\mathrm{0},y\right),\\ \hspace{1em}\forall y\in {\left(\mathrm{0,1}\right)}_y,\\ \phi \left(x,\mathrm{0}\right)=\phi \left(x,\mathrm{1}\right)=\mathrm{0},\hspace{1em}\forall x\in {\left(\mathrm{0,1}\right)}_x,\\ \psi \left(x,\mathrm{0}\right)={\partial }_y^{\mathrm{2}}\psi \left(x,\mathrm{0}\right)=\psi \left(x,\mathrm{1}\right)={\partial }_y^{\mathrm{2}}\psi \left(x,\mathrm{1}\right)=\mathrm{0},\hspace{1em}\forall x\in {\left(\mathrm{0,1}\right)}_x.\end{array}$$

Consider the auxiliary characteristic problems $$\begin{array}{c}\text{(40)}& -\frac{d^{\mathrm{2}}}{d{y}^{\mathrm{2}}}\phi +\mu \phi =\mathrm{0},\hspace{1em}\forall y\in {\left(\mathrm{0,1}\right)}_y,\phi \left(\mathrm{0}\right)=\phi \left(\mathrm{1}\right)=\mathrm{0},\\ \frac{d^{\mathrm{4}}}{d{y}^{\mathrm{4}}}\psi +\zeta \psi =\mathrm{0},\hspace{1em}\forall y\in {\left(\mathrm{0,1}\right)}_y,\\ v\left(\mathrm{0}\right)=\frac{d^{\mathrm{2}}}{d{y}^{\mathrm{2}}}v\left(\mathrm{0}\right)=v\left(\mathrm{1}\right)=\frac{d^{\mathrm{2}}}{d{y}^{\mathrm{2}}}v\left(\mathrm{1}\right)=\mathrm{0}.\end{array}$$ By recalling the classical results concerning PDE theory, we know that the former problem has the following pairs of nontrivial “normalized” solutions$$\begin{array}{c}\text{(41)}& {\mu }_k=k^{\mathrm{2}}{\pi }^{\mathrm{2}},{\phi }_{\mathrm{1}k}\left(y\right)=\sqrt{\mathrm{2}}\sin \left(k\pi y\right),\end{array}$$and that the latter problem has the following pairs of nontrivial “normalized” solutions$$\begin{array}{c}\text{(42)}& {\zeta }_k=k^{\mathrm{4}}{\pi }^{\mathrm{4}},{\psi }_{\mathrm{1}k}\left(y\right)=\sqrt{\mathrm{2}}\sin \left(k\pi y\right).\end{array}$$Let us recall that $\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}(\{{\phi }_{\mathrm{1}k}(y)\})=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}(\{{\psi }_{\mathrm{1}k}(y)\})$ is dense in $L^{\mathrm{2}}((\mathrm{0,1}{)}_y)$, and that $L^{\mathrm{2}}({\Omega }^{+})=L^{\mathrm{2}}((\mathrm{0,1}{)}_x)\overline{\otimes }{L}^{\mathrm{2}}((\mathrm{0,1}{)}_y)$, here and hereafter, we denote $\overline{\otimes }$ by the closure of the algebraic tensor product two Hilbert spaces. Let $(\phi ,\psi ,\lambda )$ be a solution triple to the problem (39). By using the results recalled just now, we know that $$\begin{array}{c}\text{(43)}& \phi \left(x,y\right)={\displaystyle \sum }_{k=\mathrm{1}}^{\infty }{\phi }_{\mathrm{2}k}\left(x\right){\phi }_{\mathrm{1}k}\left(y\right),\psi \left(x,y\right)={\displaystyle \sum }_{k=\mathrm{1}}^{\infty }{\psi }_{\mathrm{2}k}\left(x\right){\psi }_{\mathrm{1}k}\left(y\right).\end{array}$$Plug this into (39), and proceed by some routine calculations, to obtain $$\begin{array}{c}\text{(44)}& {\displaystyle \sum }_{k=\mathrm{1}}^{\infty }\left({\lambda }^{\mathrm{2}}{\phi }_{\mathrm{2}k}\left(x\right)-{\phi }_{\mathrm{2}k}^{\prime \prime }\left(x\right)+{\mu }_k{\phi }_{\mathrm{2}k}\left(x\right)+a\lambda {\phi }_{\mathrm{2}k}\left(x\right)\right){\phi }_{\mathrm{1}k}\left(y\right)=\mathrm{0},\hspace{1em}\forall \left(x,y\right)\in {\Omega }^{+},{\displaystyle \sum }_{k=\mathrm{1}}^{\infty }\left({\lambda }^{\mathrm{2}}{\psi }_{\mathrm{2}k}\left(x\right)+{\phi }_{\mathrm{2}k}^{\left(\mathrm{4}\right)}\left(x\right)+{\zeta }_k{\psi }_{\mathrm{2}k}\left(x\right)\right){\psi }_{\mathrm{1}k}\left(y\right)=\mathrm{0},\hspace{1em}\forall \left(x,y\right)\in {\Omega }^{+},\\ {\displaystyle \sum }_{k=\mathrm{1}}^{\infty }{\phi }_{\mathrm{2}k}\left(\mathrm{1}\right){\phi }_{\mathrm{1}k}\left(y\right)={\displaystyle \sum }_{k=\mathrm{1}}^{\infty }{\psi }_{\mathrm{2}k}^{\prime \prime }\left(\mathrm{0}\right){\psi }_{\mathrm{1}k}\left(y\right)={\displaystyle \sum }_{k=\mathrm{1}}^{\infty }{\psi }_{\mathrm{2}k}\left(\mathrm{1}\right){\psi }_{\mathrm{1}k}\left(y\right)={\displaystyle \sum }_{k=\mathrm{1}}^{\infty }{\psi }_{\mathrm{2}k}^{\prime \prime }\left(\mathrm{1}\right){\psi }_{\mathrm{1}k}\left(y\right)=\mathrm{0},\hspace{1em}\forall y\in {\left(\mathrm{0,1}\right)}_y,\\ {\displaystyle \sum }_{k=\mathrm{1}}^{\infty }{\phi }_{\mathrm{2}k}\left(\mathrm{0}\right){\phi }_{\mathrm{1}k}\left(y\right)={\displaystyle \sum }_{k=\mathrm{1}}^{\infty }{\psi }_{\mathrm{2}k}\left(\mathrm{0}\right){\psi }_{\mathrm{1}k}\left(y\right),\\ {\displaystyle \sum }_{k=\mathrm{1}}^{\infty }{\phi }_{\mathrm{2}k}^{\prime }\left(\mathrm{0}\right){\phi }_{\mathrm{1}k}\left(y\right)={\displaystyle \sum }_{k=\mathrm{1}}^{\infty }{\psi }_{\mathrm{2}k}^{\prime \prime \prime }\left(\mathrm{0}\right){\psi }_{\mathrm{1}k}\left(y\right),\hspace{1em}\forall y\in {\left(\mathrm{0,1}\right)}_y.\end{array}$$By utilizing the relation ${\phi }_{\mathrm{1}k}={\psi }_{\mathrm{1}k}$, and the fact that $\{{\phi }_{\mathrm{1}k}(y)\}=\{{\psi }_{\mathrm{1}k}(y)\}$ is an orthonormal basis for $L^{\mathrm{2}}((\mathrm{0,1}{)}_y)$, we have$$\begin{array}{c}\text{(45)}& {\lambda }^{\mathrm{2}}{\phi }_{\mathrm{2}k}\left(x\right)-{\phi }_{\mathrm{2}k}^{\prime \prime }\left(x\right)+{\mu }_k{\phi }_{\mathrm{2}k}\left(x\right)+a\lambda {\phi }_{\mathrm{2}k}\left(x\right)=\mathrm{0},\hspace{1em}\forall x\in {\left(\mathrm{0,1}\right)}_x,{\lambda }^{\mathrm{2}}{\psi }_{\mathrm{2}k}\left(x\right)+{\phi }_{\mathrm{2}k}^{\left(\mathrm{4}\right)}\left(x\right)+{\zeta }_k{\psi }_{\mathrm{2}k}\left(x\right)=\mathrm{0},\hspace{1em}\forall x\in {\left(\mathrm{0,1}\right)}_x,\\ {\phi }_{\mathrm{2}k}\left(\mathrm{1}\right)={\psi }_{\mathrm{2}k}^{\prime \prime }\left(\mathrm{0}\right)={\psi }_{\mathrm{2}k}\left(\mathrm{1}\right)={\psi }_{\mathrm{2}k}^{\prime \prime }\left(\mathrm{1}\right)=\mathrm{0},\\ {\phi }_{\mathrm{2}k}\left(\mathrm{0}\right)={\psi }_{\mathrm{2}k}\left(\mathrm{0}\right),\\ {\phi }_{\mathrm{2}k}^{\prime }\left(\mathrm{0}\right)={\psi }_{\mathrm{2}k}^{\prime \prime \prime }\left(\mathrm{0}\right).\end{array}$$By an argument similar to the one used in this paper, we can show that for each $k=\mathrm{1,2},\dots$, the eigenproblem (45) has a countably infinite many solutions, and eigenvalues are located in the open left half of the complex plane and can be divided into two subgroups; one subgroup has the imaginary axis as its asymptote line, while the other has a line parallel to the imaginary axis as its asymptote line.

The above example shows that our method used in this paper can be carried out in certain special higher order systems. But it should be stressed again that the calculations could be much more complicated.

Disclosure

This piece of work is credited to Chengqiang Wang.

Conflicts of Interest

There are no conflicts of interest in this paper.

Acknowledgments

The author is supported by NSFC (no. 11701050 and no. 11571244), by SCJYT Program (no. 18ZB0098) of Sichuan Province, China, and by XJPY Program (no. CS18ZD07) and XJJG Program (# 2017JG13) of Chengdu Normal University.

References

[1] B. Z. Guo, S. P. Yung, "Asymptotic behavior of the eigenfrequency of a one-dimensional linear thermoelastic system," Journal of Mathematical Analysis and Applications, vol. 213 no. 2, pp. 406-421, DOI: 10.1006/jmaa.1997.5544, 1997.

[2] L. Lu, J.-M. Wang, "Transmission problem of Schröodinger and wave equation with viscous damping," Applied Mathematics Letters, vol. 54,DOI: 10.1016/j.aml.2015.11.002, 2016.

[3] K. Ammari, S. Nicaise, "Stabilization of a transmission wave/plate equation," Journal of Differential Equations, vol. 249 no. 3, pp. 707-727, DOI: 10.1016/j.jde.2010.03.007, 2010.

[4] W. Zhang, Z. Zhang, "Stabilization of transmission coupled wave and euler-bernoulli equations on riemannian manifolds by nonlinear feedbacks," Journal of Mathematical Analysis and Applications, vol. 422 no. 2, pp. 1504-1526, DOI: 10.1016/j.jmaa.2014.09.044, 2015.

[5] F. Hassine, "Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping," Discrete and Continuous Dynamical Systems - Series B, vol. 21 no. 6, pp. 1757-1774, DOI: 10.3934/dcdsb.2016021, 2016.

[6] B. Gong, F. Yang, X. Zhao, "Stabilization of the transmission wave/plate equation with variable coefficients," Journal of Mathematical Analysis and Applications, vol. 455 no. 2, pp. 947-962, DOI: 10.1016/j.jmaa.2017.06.014, 2017.

[7] L. Deng, Z. Zhang, "Controllability for transmission wave/plate equations on Riemannian manifolds," Systems & Control Letters, vol. 91, pp. 48-54, DOI: 10.1016/j.sysconle.2016.02.016, 2016.