Remarks on the Green’s Matrix of a General Incompressible Oldroyd-B System

Abstract

In this manuscript, we study a general incompressible Oldroyd-B system and first establish a new interpretation for the Green’s matrix and then establish the pointwise estimates for the Green’s matrix, especially for the high-frequency part, where the dependence on the viscosity constants is carefully analyzed.

1. Introduction

In this manuscript, we consider the general incompressible Oldroyd-B model with damping for viscoelastic flow in ${\mathbb{R}}^3$([1,2,3]):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{𝜕}_t{u}^{ϵ,\mu }+u^{ϵ,\mu }·\nabla u^{ϵ,\mu }+\nabla p^{ϵ,\mu }-ϵ\mathsf{\Delta }{u}^{ϵ,\mu }=\kappa \mathrm{div}{\tau }^{ϵ,\mu },\hfill \\ {𝜕}_t{\tau }^{ϵ,\mu }+u^{ϵ,\mu }·\nabla {\tau }^{ϵ,\mu }-\mu \mathsf{\Delta }{\tau }^{ϵ,\mu }+\beta {\tau }^{ϵ,\mu }=\mathbb{Q}(\nabla u^{ϵ,\mu },{\tau }^{ϵ,\mu })+\alpha \mathbb{D}{u}^{ϵ,\mu },\hfill \\ \mathrm{div}{u}^{ϵ,\mu }=0.\hfill \end{array}\right.\end{array}$$

(1)

Here $u^{ϵ,\mu }:{\mathbb{R}}^3\to {\mathbb{R}}^3$ represents the velocity field of the fluid, $\mathbb{D}{u}^{ϵ,\mu }$ represents the deformation tensor, ${\tau }^{ϵ,\mu }\in {\mathbb{S}}_3$ represents the elastic part of the stress tensor, and $p^{ϵ,\mu }:{\mathbb{R}}^3\to \mathbb{R}$ represents the pressure function. Furthermore,

$$\begin{array}{c}\hfill \mathbb{Q}(\nabla u^{ϵ,\mu },{\tau }^{ϵ,\mu })={\mathsf{\Omega }}^{ϵ,\mu }{\tau }^{ϵ,\mu }-{\tau }^{ϵ,\mu }{\mathsf{\Omega }}^{ϵ,\mu }+b(\mathbb{D}{u}^{ϵ,\mu }{\tau }^{ϵ,\mu }+{\tau }^{ϵ,\mu }\mathbb{D}{u}^{ϵ,\mu })\end{array}$$

(2)

with constant $b\in [-1,1]$. The other physical constants satisfy

$$\begin{array}{c}\hfill ϵ,\mu ,\beta \ge 0,\kappa ,\alpha >0.\end{array}$$

(3)

In particular, $ϵ$ represents the viscosity constant and $\mu$ represents the center-of-mass diffusion coefficient. Here we emphasize and display the dependence on the dissipation constants $ϵ$ and $\mu$, since we focus on the dependence of $ϵ$ and $\mu$ in the Green’s matrix. We summarize previous works on this topic in a table (see Table 1), which are mainly divided into four categories:

Table 1. Well-posedness of incompressible Oldroyd-B model.

$ϵ,\mu >0$	$\beta >0$	[4]: Global strong solutions in ${\mathbb{R}}^2$ in the sense of $u\in L^{\infty }(0,T;H^2)\cap L^2(0,T;H^3)$ and $\sigma \in L^{\infty }(0,T;H^1)\cap L^2(0,T;H^2)$. (The definition of a strong solution may change under differential assumptions on the parameters.)
$ϵ,\mu >0$	$\beta =0$	[5]: Global strong solutions in ${\mathbb{R}}^2$ with a logarithmic dissipation.
$\mu >0$, $ϵ=0$	$\beta >0$	[6]: Global strong solutions with small $\parallel u_0{\parallel }_{H^s}+{\parallel {\tau }_0\parallel }_{H^{s+1}}$ $(s>{\displaystyle \frac{5}{2}})$ in ${\mathbb{R}}^3$. Relevant results: [7,8].
$\mu >0$, $ϵ=0$	$\beta >0$	[1]: Global strong solutions with small $\parallel (u_0,{\tau }_0){\parallel }_{H^1}+{\parallel ({\omega }_0,{\tau }_0)\parallel }_{B_{\infty ,1}^0}$ in ${\mathbb{R}}^2$.
$\mu >0$, $ϵ=0$	$\beta =0$	[9]: Global stability with smallness on $\parallel (u_0,{\tau }_0){\parallel }_{H^s\left({\mathbb{R}}^n\right)}$ with $r>1+{\displaystyle \frac{n}{2}}$ and $n=2,3$ for fractional stress tensor diffusion. Relevant results: [10].
$ϵ>0$, $\mu =0$	$\beta >0$	[11]: Local well-posedness of strong solutions in bounded domain in ${\mathbb{R}}^3$.
$ϵ>0$, $\mu =0$	$\beta >0$	[12]: Global existence of weak solutions with large initial data.
$ϵ>0$, $\mu =0$	$\beta >0$	[13,14]: Global well-posedness with smallness conditions in 3D exterior domains.
$ϵ>0$, $\mu =0$	$\beta >0$	[13,15,16]: Global strong solutions on $L^p$-framework with $\beta >0$.
$ϵ>0$, $\mu =0$	$\beta >0$	[5]: Conditional global regularity with $b=0$ in ${\mathbb{R}}^2$.
$ϵ>0$, $\mu =0$	$\beta =0$	[17]: Global classical solutions with small ${\parallel |\nabla |}^{-1}(u_0,{\tau }_0){\parallel }_{H^3}$ with $\beta =0$ in ${\mathbb{R}}^3$. [18,19]: Improved results.
$ϵ=\mu =0$	$\beta >0$	[20]: Vanishing viscosity limit with small analytic data in ${\mathbb{R}}^3$.

(I) $\mu >0$ and $ϵ>0$ (full dissipation);

(II) $\mu >0$ and $ϵ=0$ (diffusive);

(III) $\mu =0$ and $ϵ>0$ (non-diffusive);

(IV) $\mu =0$ and $ϵ=0$ (fully hyperbolic).

Let ${\sigma }^{ϵ,\mu }\doteq \mathbb{P}\mathrm{div}{\tau }^{ϵ,\mu }$, with $\mathbb{P}$ representing the Leray projection operator. Then, the homogeneous linearized system is given as follows.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{𝜕}_t{u}^{ϵ,\mu }-ϵ\mathsf{\Delta }{u}^{ϵ,\mu }-\kappa {\sigma }^{ϵ,\mu }=0,\hfill \\ {𝜕}_t{\sigma }^{ϵ,\mu }-\mu \mathsf{\Delta }{\sigma }^{ϵ,\mu }+\beta {\sigma }^{ϵ,\mu }-{\displaystyle \frac{\alpha }{2}}\mathsf{\Delta }{u}^{ϵ,\mu }=0.\hfill \end{array}\right.\end{array}$$

(4)

We remark here that the above system admits minor differences from that of [21], where $\tilde{\sigma }\doteq {\mathsf{\Lambda }}^{-1}\mathbb{P}\mathrm{div}\tau$. The two mentioned systems admit the same characteristic polynomial and eigenvalues, while there are some slight changes in the Green’s matrix. If $ϵ=0$, then system (4) is reduced to that of [22]. We mainly concentrate on the dependence of the dissipative coefficients $\mu$ and $ϵ$ in the estimates on the Green’s matrix. Without loss of generality, assume that $\mu ,ϵ<1$. Furthermore, one has the Fourier transformation of (4) in the following form:

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\left(\begin{array}{c}{\widehat{u}}^{ϵ,\mu }\\ {\widehat{\sigma }}^{ϵ,\mu }\end{array}\right)=\mathbb{F}\left(\zeta \right)\left(\begin{array}{c}{\widehat{u}}^{ϵ,\mu }\\ {\widehat{\sigma }}^{ϵ,\mu }\end{array}\right),\mathrm{with}\mathbb{F}\left(\zeta \right)=\left(\begin{array}{cc}-{ϵ\left|\zeta \right|}^2{\mathbb{I}}_3& \kappa {\mathbb{I}}_3\\ -{\displaystyle \frac{\alpha }{2}}{\left|\zeta \right|}^2{\mathbb{I}}_3& -\left({\mu \left|\zeta \right|}^2+\beta \right){\mathbb{I}}_3\end{array}\right).\end{array}$$

(5)

Then, combining the arguments in [21,22], the Green’s matrix for system (4) ${\widehat{\mathbb{M}}}_{u,\sigma }(\zeta ,t)=e^{\mathbb{F}\left(\zeta \right)t}$ can be expressed as follows:

$$\begin{array}{c}\hfill {\widehat{\mathbb{M}}}_{u,\sigma }(\zeta ,t)=\left(\begin{array}{cc}\left[{\mathcal{M}}_3(\zeta ,t)-ϵ{\left|\zeta \right|}^2{\mathcal{M}}_1(\zeta ,t)\right]{\mathbb{I}}_3& \kappa {\mathcal{M}}_1(\zeta ,t){\mathbb{I}}_3\\ -{\displaystyle \frac{\alpha }{2}}{\left|\zeta \right|}^2{\mathcal{M}}_1(\zeta ,t){\mathbb{I}}_3& \left[{\mathcal{M}}_2(\zeta ,t)+ϵ{\left|\zeta \right|}^2{\mathcal{M}}_1(\zeta ,t)\right]{\mathbb{I}}_3\end{array}\right),\end{array}$$

(6)

where

$$\begin{array}{c}\hfill {\mathcal{M}}_1(\zeta ,t)={\displaystyle \frac{e^{{\lambda }^{+}t}-e^{{\lambda }^{-}t}}{{\lambda }^{+}-{\lambda }^{-}}},{\mathcal{M}}_2(\zeta ,t)={\displaystyle \frac{{\lambda }^{+}{e}^{{\lambda }^{+}t}-{\lambda }^{-}{e}^{{\lambda }^{-}t}}{{\lambda }^{+}-{\lambda }^{-}}},{\mathcal{M}}_3(\zeta ,t)={\displaystyle \frac{{\lambda }^{+}{e}^{{\lambda }^{-}t}-{\lambda }^{-}{e}^{{\lambda }^{+}t}}{{\lambda }^{+}-{\lambda }^{-}}}.\end{array}$$

(7)

For some related topics on Green’s matrix, the readers are referred to [23,24] and references therein. Now we are in a position to state the motivation of this paper.

• Compared to the methods in [21,22], we intend to give another construction for the Green’s matrix in this manuscript, which is based on careful calculations of the eigenvectors.
• Furthermore, it should be pointed out that Proposition 2.3 in [21] (for Case I: $\mu >0$, $ϵ\ge 0$; for Case II: $\mu >0$, $ϵ\ge 0$) only established the pointwise estimates for the low-frequency part of the Green’s matrix, while, for Proposition 2.1 in [22] and Lemma 3.2 in [16], pointwise estimates for both the low- and the high-frequency parts are established; however, only the subcase $\mu >0,ϵ=0$ or $\mu =0,ϵ>0$ is considered therein. Therefore, motivated by the previous results, this manuscript aims to provide the pointwise estimates for the high-frequency part of the Green’s matrix for general viscosity constants $\mu$ and $ϵ$ in a general form, which can cover the results in [16,21,22], where the dependence on the viscosity constants is carefully analyzed.

The rest of this manuscript is organized as follows. The main results are stated in Section 2 and their proofs are displayed in Section 3.

2. Main Results

We now present the first main result of this paper. Compared to the derivation of the Green’s matrix in [21,22], we tend to give another interpretation based on the calculations of the eigenvector, which can reveal more information about the intrinsic structure of the Green’s matrix.

Theorem 1

(Another interpretation for the Green’s matrix). The Fourier transformation of the solutions to the auxiliary system (4) can be expressed as follows.

$$\begin{array}{c}\hfill \left(\begin{array}{c}{\widehat{u}}^{ϵ,\mu }\\ {\widehat{\sigma }}^{ϵ,\mu }\end{array}\right)\left(t\right)=\left(C_1^{+}{\eta }_1^{+}+C_2^{+}{\eta }_2^{+}+C_3^{+}{\eta }_3^{+}\right)e^{{\lambda }^{+}t}+\left(C_1^{-}{\eta }_1^{-}+C_2^{-}{\eta }_2^{-}+C_3^{-}{\eta }_3^{-}\right)e^{{\lambda }^{-}t},\end{array}$$

(8)

where ${\lambda }_{\pm }$ are the eigenvalues (triple roots) expressed by

$$\begin{array}{c}\hfill {\lambda }_{\pm }={\displaystyle \frac{-\left[\beta +\left(\mu +ϵ\right){\left|\zeta \right|}^2\right]\pm \sqrt{\mathsf{\Delta }}}{2}},\end{array}$$

(9)

with $\mathsf{\Delta }={\left[\left(\mu +ϵ\right){\left|\zeta \right|}^2+\beta \right]}^2-4{\left|\zeta \right|}^2\left[ϵ\left({\mu \left|\zeta \right|}^2+\beta \right)+{\displaystyle \frac{\alpha \kappa }{2}}\right]$, and the following linear independent corresponding eigenvectors:

$$\begin{array}{ccc}\hfill & & {\eta }_1^{+}=\left(\begin{array}{c}{\displaystyle \frac{\kappa }{{\lambda }^{+}+ϵ{\left|\zeta \right|}^2}}{e}_1\\ e_1\end{array}\right),{\eta }_2^{+}=\left(\begin{array}{c}{\displaystyle \frac{\kappa }{{\lambda }^{+}+ϵ{\left|\zeta \right|}^2}}{e}_2\\ e_2\end{array}\right),{\eta }_3^{+}=\left(\begin{array}{c}{\displaystyle \frac{\kappa }{{\lambda }^{+}+ϵ{\left|\zeta \right|}^2}}{e}_3\\ e_3\end{array}\right),\mathrm{for}{\lambda }^{+},\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill & & {\eta }_1^{-}=\left(\begin{array}{c}{\displaystyle \frac{\kappa }{{\lambda }^{-}+ϵ{\left|\zeta \right|}^2}}{e}_1\\ e_1\end{array}\right),{\eta }_2^{-}=\left(\begin{array}{c}{\displaystyle \frac{\kappa }{{\lambda }^{-}+ϵ{\left|\zeta \right|}^2}}{e}_2\\ e_2\end{array}\right),{\eta }_3^{-}=\left(\begin{array}{c}{\displaystyle \frac{\kappa }{{\lambda }^{-}+ϵ{\left|\zeta \right|}^2}}{e}_3\\ e_3\end{array}\right),\mathrm{for}{\lambda }^{-},\hfill \end{array}$$

(11)

where $e_1={(1,0,0)}^T$, $e_2={(0,1,0)}^T$, and $e_3={(0,0,1)}^T$, and the constants $C_{\pm }^i$, $i=1,2,3$ are determined by the following linear equations involving the initial data:

$$\begin{array}{ccc}\hfill & & C_i^{+}={\displaystyle \frac{({\lambda }^{+}+{ϵ\left|\zeta \right|}^2\left)\right({\lambda }^{-}+{ϵ\left|\zeta \right|}^2)}{\kappa ({\lambda }^{-}-{\lambda }^{+})}}{\widehat{u}}_0^i-{\displaystyle \frac{{\lambda }^{+}+ϵ{\left|\zeta \right|}^2}{{\lambda }^{-}-{\lambda }^{+}}}{\widehat{\sigma }}_0^i,\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill & & C_i^{-}={\widehat{\sigma }}_0^i-C_i^{+}=-{\displaystyle \frac{({\lambda }^{+}+{ϵ\left|\zeta \right|}^2\left)\right({\lambda }^{-}+{ϵ\left|\zeta \right|}^2)}{\kappa ({\lambda }^{-}-{\lambda }^{+})}}{\widehat{u}}_0^i+{\displaystyle \frac{{\lambda }^{-}+ϵ{\left|\zeta \right|}^2}{{\lambda }^{-}-{\lambda }^{+}}}{\widehat{\sigma }}_0^i,\hfill \end{array}$$

(13)

for $i=1,2,3$.

Remark 1.

It should be pointed out that if we substitute (10)–(13) into (8), then one can obtain the Green’s matrix (6) by direct calculations.

Now we state the second result of this paper.

Theorem 2

(Pointwise estimates for the high-frequency part). There exists a positive constant $R=R(\alpha ,\kappa ,\beta ,|\mu -ϵ\left|\right)$ such that, for any $R\le \left|\zeta \right|$ and multi-index γ with $\left|\gamma \right|=k$, the following statements hold.

(a) For the case $\mu >ϵ$, it holds that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |{𝜕}_{\zeta }^{\gamma }{\mathcal{M}}_1(\zeta ,t)|& \le {C\left|\zeta \right|}^{-(k+2)}\left(e^{-{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t-{\displaystyle \frac{1}{2}}ϵ{\left|\zeta \right|}^{2}t}+e^{-\beta t-{\displaystyle \frac{1}{2}}\mu {\left|\zeta \right|}^{2}t}\right),\hfill \\ \hfill |{𝜕}_{\zeta }^{\gamma }{\mathcal{M}}_2(\zeta ,t)|& \le C\left({ϵ\left|\zeta \right|}^{-k}+{\left|\zeta \right|}^{-k-2}\right)e^{-{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t-{\displaystyle \frac{1}{2}}ϵ{\left|\zeta \right|}^{2}t}\hfill \\ & +C\left({\mu \left|\zeta \right|}^{-k}+{\left|\zeta \right|}^{-k-2}\right)e^{-\beta t-{\displaystyle \frac{1}{2}}\mu {\left|\zeta \right|}^{2}t},\hfill \\ \hfill |{𝜕}_{\zeta }^{\gamma }{\mathcal{M}}_3(\zeta ,t)|& \le {C\left|\zeta \right|}^{-k}\left(e^{-{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t-{\displaystyle \frac{1}{2}}ϵ{\left|\zeta \right|}^{2}t}+e^{-\beta t-{\displaystyle \frac{1}{2}}\mu {\left|\zeta \right|}^{2}t}\right),\hfill \end{array}\end{array}$$

(14)

for some generic positive constants C depending on k, α, κ, β, and $\mu -ϵ$. Here ${𝜕}_{\zeta }^{\gamma }={𝜕}_{{\zeta }_1}^{{\gamma }_1}{𝜕}_{{\zeta }_2}^{{\gamma }_2}{𝜕}_{{\zeta }_3}^{{\gamma }_3}$, with $\left|\gamma \right|={\gamma }_1+{\gamma }_2+{\gamma }_3=k$.

(b) For the case $ϵ>\mu$, it holds that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |{𝜕}_{\zeta }^{\gamma }{\mathcal{M}}_1(\zeta ,t)|& \le {C\left|\zeta \right|}^{-(k+2)}\left(e^{-{\displaystyle \frac{1}{2}}ϵ{\left|\zeta \right|}^{2}t}+e^{-{\displaystyle \frac{\alpha \kappa }{4(ϵ-\mu )}}t-\beta t-{\displaystyle \frac{1}{2}}\mu {\left|\zeta \right|}^{2}t}\right),\hfill \\ \hfill |{𝜕}_{\zeta }^{\gamma }{\mathcal{M}}_2(\zeta ,t)|& \le C\left({\mu \left|\zeta \right|}^{-k}+{\left|\zeta \right|}^{-k-2}\right)e^{-\left({\displaystyle \frac{\alpha \kappa }{4(ϵ-\mu )}}+\beta \right)t-{\displaystyle \frac{1}{2}}\mu {\left|\zeta \right|}^{2}t}\hfill \\ & +C\left({ϵ\left|\zeta \right|}^{-k}+{\left|\zeta \right|}^{-k-2}\right)e^{-{\displaystyle \frac{1}{2}}ϵ{\left|\zeta \right|}^{2}t},\hfill \\ \hfill |{𝜕}_{\zeta }^{\gamma }{\mathcal{M}}_3(\zeta ,t)|& \le {C\left|\zeta \right|}^{-k}\left(e^{-{\displaystyle \frac{1}{2}}ϵ{\left|\zeta \right|}^{2}t}+e^{-{\displaystyle \frac{\alpha \kappa }{4(ϵ-\mu )}}t-\beta t-{\displaystyle \frac{1}{2}}\mu {\left|\zeta \right|}^{2}t}\right),\hfill \end{array}\end{array}$$

(15)

for some generic positive constants C depending on k, α, κ, β, and $ϵ-\mu$.

(c) For the case $ϵ=\mu$, the roots ${\lambda }_{\pm }\left(r\right)$ of $P(·,r)$ satisfy

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathbf{Re}\left({\lambda }_{\pm }\left(r\right)\right)=-\left(\mu r^2+{\displaystyle \frac{\beta }{2}}\right),\end{array}\end{array}$$

(16)

which implies that the components ${\mathcal{M}}_i(\xi ,t)$$(i=1,2,3)$ enjoy similar properties to the heat kernel in high-frequency part.

3. Proofs of Main Theorems

Firstly, we show the proof of Theorem 1 The main tools are matrix theory and the theory of linear ordinary differential equations.

Proof. 

The proof of (9) is very close to the calculations in [21,22] and thus we omit it here. Furthermore, we derive (10) and (11) directly from the linear equations $({\lambda }_{\pm }{\mathbb{I}}_6-\mathbb{A})x=0$ and the fact induced by elementary row operations that

$$\begin{array}{c}\hfill {\lambda }_{\pm }{\mathbb{I}}_6-\mathbb{A}\sim \left(\begin{array}{cc}\left({\lambda }_{\pm }+ϵ{\left|\zeta \right|}^2\right){\mathbb{I}}_3& -\kappa {\mathbb{I}}_3\\ {\displaystyle \frac{\alpha }{2}}{\left|\zeta \right|}^2{\mathbb{I}}_3& \left[{\lambda }_{\pm }+{\left(\mu \right|\zeta |}^2+\beta )\right]{\mathbb{I}}_3\end{array}\right)\sim \left(\begin{array}{cc}\left({\lambda }_{\pm }+ϵ{\left|\zeta \right|}^2\right){\mathbb{I}}_3& -\kappa {\mathbb{I}}_3\\ 0& 0\end{array}\right),\end{array}$$

where we have used the following fact, ${\lambda }_{\pm }+{\left(\mu \right|\zeta |}^2+\beta )+{\displaystyle \frac{\frac{\alpha \kappa }{2}{\left|\zeta \right|}^2}{{\lambda }_{\pm }+ϵ{\left|\zeta \right|}^2}}=0$, by applying Vieta’s theorem and

$$\begin{array}{c}\hfill {\lambda }^{+}+{\lambda }^{-}=-{(\mu +ϵ)\left|\zeta \right|}^2-\beta ,{\lambda }^{+}{\lambda }^{-}={\left|\zeta \right|}^2\left[ϵ\left({\mu \left|\zeta \right|}^2+\beta \right)+{\displaystyle \frac{\alpha \kappa }{2}}\right].\end{array}$$

(17)

Following [25], we obtain expression (8) by directly applying the theory of linear ordinary differential equations. Finally, we take $t=0$ in (8) and get

$$\begin{array}{c}\hfill {\widehat{u}}_0={\displaystyle \frac{\kappa }{{\lambda }^{+}+ϵ{\left|\zeta \right|}^2}}\left(\begin{array}{c}{C}_1^{+}\\ C_2^{+}\\ C_3^{+}\end{array}\right)+{\displaystyle \frac{\kappa }{{\lambda }^{-}+ϵ{\left|\zeta \right|}^2}}\left(\begin{array}{c}{C}_1^{-}\\ C_2^{-}\\ C_3^{-}\end{array}\right),{\widehat{\sigma }}_0=\left(\begin{array}{c}{C}_1^{+}+C_1^{-}\\ C_2^{+}+C_1^{-}\\ C_3^{+}+C_3^{-}\end{array}\right),\end{array}$$

(18)

which yields (12)–(13). Therefore, we finish the proof of Theorem 1. □

Secondly, we show the proof of Theorem 2 The main tools are complex analysis and series theory.

Proof. 

First of all, (9) yields

$$\begin{array}{c}\hfill {\lambda }^{+}={\displaystyle \frac{-{2\left|\zeta \right|}^2\left[ϵ\left({\mu \left|\zeta \right|}^2+\beta \right)+\frac{\alpha \kappa }{2}\right]}{\left[{(\mu +ϵ)\left|\zeta \right|}^2+\beta \right]+\sqrt{\mathsf{\Delta }}}},\end{array}$$

(19)

which implies that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{{\lambda }^{+}}{{\left|\zeta \right|}^2}}=& {\displaystyle \frac{-2\left[ϵ\left(\mu +\frac{\beta }{{\left|\zeta \right|}^2}\right)+\frac{\alpha \kappa }{2}\frac{1}{{\left|\zeta \right|}^2}\right]}{(\mu +ϵ)+\frac{\beta }{{\left|\zeta \right|}^2}+\sqrt{{\left[(\mu +ϵ)+\frac{\beta }{{\left|\zeta \right|}^2}\right]}^2-4\left[ϵ\left(\mu +\frac{\beta }{{\left|\zeta \right|}^2}\right)+\frac{\alpha \kappa }{2}\frac{1}{{\left|\zeta \right|}^2}\right]}}}\hfill \\ \hfill \to & {\displaystyle \frac{-2ϵ\mu }{(\mu +\epsilon )+|\mu -ϵ|}}=\left\{\begin{array}{c}-ϵ,\mathrm{if}ϵ<\mu ,\hfill \\ -\mu ,\mathrm{if}ϵ\ge \mu ,\hfill \end{array}\right.\hfill \end{array}\end{array}$$

(20)

as $\left|\zeta \right|\to \infty$. Therefore, one has

$$\begin{array}{c}\hfill \begin{array}{cc}& {\lambda }^{+}+ϵ{\left|\zeta \right|}^2\hfill \\ \hfill =& {\displaystyle \frac{-\left[\left(\mu -ϵ\right){\left|\xi \right|}^2+\beta \right]+\sqrt{{\left[\left(\mu +ϵ\right){\left|\xi \right|}^2+\beta \right]}^2-4{\left|\xi \right|}^2\left[ϵ\left({\mu \left|\xi \right|}^2+\beta \right)+\frac{\alpha \kappa }{2}\right]}}{2}}\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}·{\displaystyle \frac{-{\left[\left(\mu -ϵ\right){\left|\xi \right|}^2+\beta \right]}^2+{\left[\left(\mu +ϵ\right){\left|\xi \right|}^2+\beta \right]}^2-4{\left|\xi \right|}^2\left[ϵ\left({\mu \left|\xi \right|}^2+\beta \right)+\frac{\alpha \kappa }{2}\right]}{\left[\left(\mu -ϵ\right){\left|\xi \right|}^2+\beta \right]+\sqrt{{\left[\left(\mu +ϵ\right){\left|\xi \right|}^2+\beta \right]}^2-4{\left|\xi \right|}^2\left[ϵ\left({\mu \left|\xi \right|}^2+\beta \right)+\frac{\alpha \kappa }{2}\right]}}}\hfill \\ \hfill =& -{\displaystyle \frac{\alpha \kappa }{(\mu -ϵ)+\frac{\beta }{{\left|\zeta \right|}^2}+\sqrt{{\left[(\mu +ϵ)+\frac{\beta }{{\left|\zeta \right|}^2}\right]}^2-4\left[ϵ\left(\mu +\frac{\beta }{{\left|\zeta \right|}^2}\right)+\frac{\alpha \kappa }{{2\left|\zeta \right|}^2}\right]}}}\hfill \\ \hfill \to & -{\displaystyle \frac{\alpha \kappa }{(\mu -ϵ)+|\mu -ϵ|}}=-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}},\mathrm{as}\left|\zeta \right|\to \infty ,\mathrm{if}ϵ<\mu .\hfill \end{array}\end{array}$$

(21)

and similarly,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\lambda }^{+}+\mu {\left|\zeta \right|}^2\to -{\displaystyle \frac{\alpha \kappa }{2(ϵ-\mu )}}-\beta ,\mathrm{if}ϵ>\mu .\end{array}\end{array}$$

(22)

If $\mu =ϵ$, then we find

$$\begin{array}{c}\hfill \mathsf{\Delta }\left(\zeta \right)={\left[{(\mu +ϵ)\left|\zeta \right|}^2+\beta \right]}^2-{4\left|\zeta \right|}^2\left[ϵ\left({\mu \left|\zeta \right|}^2+\beta \right)+{\displaystyle \frac{\alpha \kappa }{2}}\right]={\beta }^2-2\alpha \kappa {\left|\zeta \right|}^2<0,\end{array}$$

(23)

as $\left|\zeta \right|\to \infty$. Then one can easily get $\mathbf{Re}\left({\lambda }_{\pm }\right)=-\left({\mu \left|\zeta \right|}^2+{\displaystyle \frac{\beta }{2}}\right)$.

Thus, we can easily prove the following lemma, which describes the characteristics of the eigenvalues.

Lemma 1. 

(1) In the case $ϵ<\mu$, the roots ${\lambda }_{\pm }\left(r\right)$ of $P(·,r)$ satisfy, as r tends to infinity,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\lambda }^{+}\left(r\right)=-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-ϵr^2+O\left(r^{-2}\right),{\lambda }^{-}\left(r\right)={\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-\beta -\mu r^2+O\left(r^{-2}\right).\end{array}\end{array}$$

(24)

(2) In the case $ϵ>\mu$, the roots ${\lambda }_{\pm }\left(r\right)$ of $P(·,r)$ satisfy, as r tends to infinity,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\lambda }^{+}\left(r\right)=-{\displaystyle \frac{\alpha \kappa }{2(ϵ-\mu )}}-\beta -\mu r^2+O\left(r^{-2}\right),{\lambda }^{-}\left(r\right)={\displaystyle \frac{\alpha \kappa }{2(ϵ-\mu )}}-ϵr^2+O\left(r^{-2}\right).\end{array}\end{array}$$

(25)

(3) In the case $ϵ=\mu$, the roots ${\lambda }_{\pm }\left(r\right)$ of $P(·,r)$ satisfy

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathbf{Re}\left({\lambda }_{\pm }\left(r\right)\right)=-\left(\mu r^2+{\displaystyle \frac{\beta }{2}}\right),\end{array}\end{array}$$

(26)

Now we continue the proof of Theorem 2. For simplicity, we only discuss the first case $ϵ<\mu$.

Define contours ${\mathsf{\Gamma }}_i$, $i=1,2,3$, to be circles of radius

$$R={\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}<{\displaystyle \frac{1}{2}}\left(-{\displaystyle \frac{\alpha \kappa }{\mu -ϵ}}+\beta +(\mu -ϵ)r^2\right),$$

with large $r^2>{\displaystyle \frac{\frac{3\alpha \kappa }{2(\mu -ϵ)}-\beta }{\mu -ϵ}}={\displaystyle \frac{3\alpha \kappa -2\beta (\mu -ϵ)}{2{(\mu -ϵ)}^2}}$ in the complex plane, centered at $-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-ϵr^2$, ${\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-\beta -\mu r^2$, and 0, respectively.

Step One: Estimates on ${\mathcal{G}}_1(t,r)$. Denote by P the characteristic polynomial of the linear system (5):

$$\begin{array}{c}\hfill P(z,r)=z^2+\left[\left(\mu +ϵ\right)r^2+\beta \right]z+r^2\left[ϵ\left(\mu r^2+\beta \right)+{\displaystyle \frac{\alpha \kappa }{2}}\right].\end{array}$$

(27)

By applying Cauchy’s integral formula, one has

$$\begin{array}{c}\hfill {\mathcal{M}}_1(r,t)={\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1\cup {\Gamma }_2}{\displaystyle \frac{e^{tz}}{(z-{\lambda }^{+})(z-{\lambda }^{-})}}dz={\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1\cup {\Gamma }_2}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz.\end{array}$$

(28)

Applying a change of variable $w=z+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}+ϵr^2$ yields

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\hfill \\ & ={\displaystyle \frac{e^{-\left(\frac{\alpha \kappa }{2(\mu -ϵ)}+ϵr^2\right)t}}{2\pi i}}{\int }_{{\Gamma }_3}{\displaystyle \frac{e^{tw}}{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}\right)\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}+\beta \right)+(\mu -ϵ)w{r}^2}}dw\hfill \\ & ={\displaystyle \frac{e^{-\left(\frac{\alpha \kappa }{2(\mu -ϵ)}+ϵr^2\right)t}}{2\pi i}}{\int }_{{\Gamma }_3}{\displaystyle \frac{e^{tw}}{(\mu -ϵ)w{r}^2}}·{\displaystyle \frac{1}{1+\frac{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}\right)\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}+\beta \right)}{(\mu -ϵ)w{r}^2}}}dw\hfill \\ & ={\displaystyle \frac{e^{-\left(\frac{\alpha \kappa }{2(\mu -ϵ)}+ϵr^2\right)t}}{2\pi i}}\sum _{j=0}^{\infty }{(-1)}^j{r}^{-2(j+1)}{\int }_{{\Gamma }_3}{e}^{tw}{\displaystyle \frac{{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}\right)}^j{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}+\beta \right)}^j}{{\left[(\mu -ϵ)w\right]}^{j+1}}}dw,\hfill \end{array}\end{array}$$

(29)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill P\left(z\right)=& {\left(w-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-ϵr^2\right)}^2+\left[\left(\mu +ϵ\right)r^2+\beta \right]\left(w-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-ϵr^2\right)\hfill \\ \hfill & +r^2\left[ϵ\left(\mu r^2+\beta \right)+{\displaystyle \frac{\alpha \kappa }{2}}\right]\hfill \\ \hfill =& \left(w-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-ϵr^2\right)\left(w-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}+\mu r^2+\beta \right)+r^2\left[ϵ\left(\mu r^2+\beta \right)+{\displaystyle \frac{\alpha \kappa }{2}}\right]\hfill \\ \hfill =& \left(w-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}\right)\left(w-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}+\beta \right)+\mu r^2\left(w-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}\right)\hfill \\ & -ϵr^2\left(w-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}+\mu r^2+\beta \right)+r^2\left[ϵ\left(\mu r^2+\beta \right)+{\displaystyle \frac{\alpha \kappa }{2}}\right]\hfill \\ \hfill =& \left(w-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}\right)\left(w-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}+\beta \right)+(\mu -ϵ)w{r}^2.\hfill \end{array}\end{array}$$

(30)

Let $c_j\left(t\right)={\int }_{{\Gamma }_3}{e}^{tw}{\displaystyle \frac{{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}\right)}^j{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}+\beta \right)}^j}{{\left[(\mu -ϵ)w\right]}^{j+1}}}dw$ and observe that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |c_j\left(t\right)|& \le {\displaystyle \frac{2\pi \alpha \kappa }{4(\mu -ϵ)}}{e}^{{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t}{\displaystyle \frac{{\left(\frac{3}{4}\frac{\alpha \kappa }{\mu -ϵ}\right)}^j{\left(\frac{3}{4}\frac{\alpha \kappa }{\mu -ϵ}+\beta \right)}^j}{{(\mu -ϵ)}^{j+1}{\left(\frac{\alpha \kappa }{4(\mu -ϵ)}\right)}^{j+1}}}\hfill \\ & ={\displaystyle \frac{2\pi }{\mu -ϵ}}{e}^{{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t}{\left({\displaystyle \frac{3}{\mu -ϵ}}\right)}^j{\left({\displaystyle \frac{3}{4}}{\displaystyle \frac{\alpha \kappa }{\mu -ϵ}}+\beta \right)}^j.\hfill \end{array}\end{array}$$

(31)

Then, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^k{\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\hfill \\ \hfill =& {\displaystyle \frac{e^{-\frac{\alpha \kappa }{2(\mu -ϵ)}t}}{2\pi i}}\sum _{j=0}^{\infty }\sum _{l=0}^k\left(\begin{array}{c}k\\ l\end{array}\right)\left[{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^l{e}^{-ϵr^{2}t}\right]{(-1)}^j{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^{k-l}\left[r^{-2(j+1)}\right]c_j\left(t\right)\hfill \\ \hfill =& {\displaystyle \frac{e^{-\frac{\alpha \kappa }{2(\mu -ϵ)}t}}{2\pi i}}\sum _{j=0}^{\infty }\sum _{l=0}^k\left(\begin{array}{c}k\\ l\end{array}\right)\left[{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^l{e}^{-ϵr^{2}t}\right]{(-1)}^{j+k-l}{\displaystyle \frac{(2j+1+k-l)!}{(2j+1)!}}{r}^{-2(j+1)-(k-l)}{c}_j\left(t\right).\hfill \end{array}\end{array}$$

(32)

With the help of (31) and the following estimate,

$$\begin{array}{c}\hfill \left|{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^l{e}^{-ϵr^{2}t}\right|\le C\left(l\right)r^{-l}{e}^{-{\displaystyle \frac{1}{2}}ϵr^{2}t},\mathrm{if}ϵ>0,\end{array}$$

(33)

we have, for $r^2\ge max\left\{{\displaystyle \frac{3\alpha \kappa -2\beta (\mu -ϵ)}{2{(\mu -ϵ)}^2}}+1,{\displaystyle \frac{6(k+1)}{\mu -ϵ}}\left({\displaystyle \frac{3}{4}}{\displaystyle \frac{\alpha \kappa }{\mu -ϵ}}+\beta \right)\right\}$,

$$\begin{array}{c}\hfill \begin{array}{cc}& \left|{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^k{\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\right|\hfill \\ \hfill \le & C\left(k\right)r^{-k-2}{e}^{-{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t-{\displaystyle \frac{1}{2}}ϵr^{2}t}\sum _{j=0}^{\infty }\sum _{l=0}^k\left(\begin{array}{c}k\\ l\end{array}\right){\displaystyle \frac{(2j+1+k-l)!}{(2j+1)!}}{\left[{\displaystyle \frac{3}{\mu -ϵ}}\left({\displaystyle \frac{3}{4}}{\displaystyle \frac{\alpha \kappa }{\mu -ϵ}}+\beta \right)r^{-2}\right]}^j\hfill \\ \hfill \le & C\left(k\right)r^{-k-2}{e}^{-{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t-{\displaystyle \frac{1}{2}}ϵr^{2}t}.\hfill \end{array}\end{array}$$

(34)

In the case $ϵ=0$, we directly obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^k{\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\hfill \\ \hfill =& {\displaystyle \frac{e^{-\frac{\alpha \kappa }{2(\mu -ϵ)}t}}{2\pi i}}\sum _{j=0}^{\infty }{(-1)}^j{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^k\left[r^{-2(j+1)}\right]c_j\left(t\right)\hfill \\ \hfill =& {\displaystyle \frac{e^{-\frac{\alpha \kappa }{2(\mu -ϵ)}t}}{2\pi i}}\sum _{j=0}^{\infty }{(-1)}^{j+k}{\displaystyle \frac{(2j+1+k)!}{(2j+1)!}}{r}^{-2(j+1)-k}{c}_j\left(t\right).\hfill \end{array}\end{array}$$

(35)

This, combined with (31), implies

$$\begin{array}{c}\hfill \begin{array}{cc}& \left|{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^k{\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\right|\hfill \\ \hfill \le & C\left(k\right)e^{-{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t}\sum _{j=0}^{\infty }{\displaystyle \frac{(2j+1+k)!}{(2j+1)!}}{r}^{-2-k}{\left[{\displaystyle \frac{3}{\mu -ϵ}}\left({\displaystyle \frac{3}{4}}{\displaystyle \frac{\alpha \kappa }{\mu -ϵ}}+\beta \right)r^{-2}\right]}^j\hfill \\ \hfill \le & C\left(k\right)r^{-k-2}{e}^{-{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t},\hfill \end{array}\end{array}$$

(36)

for $r^2\ge max\left\{{\displaystyle \frac{3\alpha \kappa -2\beta (\mu -ϵ)}{2{(\mu -ϵ)}^2}}+1,{\displaystyle \frac{6(k+1)}{\mu -ϵ}}\left({\displaystyle \frac{3}{4}}{\displaystyle \frac{\alpha \kappa }{\mu -ϵ}}+\beta \right)\right\}$. This is consistent with (35) in the case $ϵ=0$.

Furthermore, applying again a change of variable $v=z-{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}+\beta +\mu r^2$ yields

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_2}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\hfill \\ \hfill =& {\displaystyle \frac{e^{-\left(-\frac{\alpha \kappa }{2(\mu -ϵ)}+\beta +\mu r^2\right)t}}{2\pi i}}{\int }_{{\Gamma }_3}{\displaystyle \frac{e^{tv}}{\left(v+\frac{\alpha \kappa }{2(\mu -ϵ)}-\beta \right)\left(v+\frac{\alpha \kappa }{2(\mu -ϵ)}\right)-(\mu -ϵ)r^{2}v}}dv\hfill \\ \hfill =& -{\displaystyle \frac{e^{\left(\frac{\alpha \kappa }{2(\mu -ϵ)}-\beta -\mu r^2\right)t}}{2\pi i}}{\int }_{{\Gamma }_3}{\displaystyle \frac{e^{tv}}{(\mu -ϵ)r^{2}v}}{\displaystyle \frac{1}{1-\frac{\left(v+\frac{\alpha \kappa }{2(\mu -ϵ)}-\beta \right)\left(v+\frac{\alpha \kappa }{2(\mu -ϵ)}\right)}{(\mu -ϵ)r^{2}v}}}dv\hfill \\ \hfill =& -{\displaystyle \frac{e^{\left(\frac{\alpha \kappa }{2(\mu -ϵ)}-\beta -\mu r^2\right)t}}{2\pi i}}\sum _{j=0}^{\infty }{r}^{-2(j+1)}{\int }_{{\Gamma }_3}{e}^{tv}{\displaystyle \frac{{\left(v+\frac{\alpha \kappa }{2(\mu -ϵ)}-\beta \right)}^j{\left(v+\frac{\alpha \kappa }{2(\mu -ϵ)}\right)}^j}{{\left[(\mu -ϵ)v\right]}^{j+1}}}dv.\hfill \end{array}\end{array}$$

(37)

Here

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill P\left(v\right)=& {\left(v+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-\beta -\mu r^2\right)}^2+\left[\left(\mu +ϵ\right)r^2+\beta \right]\left(v+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-\beta -\mu r^2\right)\hfill \\ \hfill & +r^2\left[ϵ\left(\mu r^2+\beta \right)+{\displaystyle \frac{\alpha \kappa }{2}}\right]\hfill \\ \hfill =& \left(v+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-\beta -\mu r^2\right)\left(v+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}+ϵr^2\right)+r^2\left[ϵ\left(\mu r^2+\beta \right)+{\displaystyle \frac{\alpha \kappa }{2}}\right]\hfill \\ \hfill =& \left(v+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-\beta \right)\left(v+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}\right)-\mu r^2\left(v+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}\right)\hfill \\ & +ϵr^2\left(v+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-\beta -\mu r^2\right)+r^2\left[ϵ\left(\mu r^2+\beta \right)+{\displaystyle \frac{\alpha \kappa }{2}}\right]\hfill \\ \hfill =& \left(v+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}-\beta \right)\left(v+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}\right)-(\mu -ϵ)r^{2}v.\hfill \end{array}\end{array}$$

(38)

Note that ${\tilde{c}}_j\left(t\right)={\int }_{{\Gamma }_3}{e}^{tv}{\displaystyle \frac{{\left(v+\frac{\alpha \kappa }{2(\mu -ϵ)}-\beta \right)}^j{\left(v+\frac{\alpha \kappa }{2(\mu -ϵ)}\right)}^j}{{\left[(\mu -ϵ)v\right]}^{j+1}}}$ induces the same estimates as $c_j\left(t\right)$ in (31). Then, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^k{\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_2}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\hfill \\ \hfill =& -{\displaystyle \frac{e^{\left(\frac{\alpha \kappa }{2(\mu -ϵ)}-\beta \right)t}}{2\pi i}}\sum _{j=0}^{\infty }\sum _{l=0}^k\left(\begin{array}{c}k\\ l\end{array}\right)\left[{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^l{e}^{-\mu r^{2}t}\right]{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^{k-l}\left[r^{-2(j+1)}\right]{\tilde{c}}_j\left(t\right)\hfill \\ \hfill =& -{\displaystyle \frac{e^{\left(\frac{\alpha \kappa }{2(\mu -ϵ)}-\beta \right)t}}{2\pi i}}\sum _{j=0}^{\infty }\sum _{l=0}^k\left(\begin{array}{c}k\\ l\end{array}\right)\left[{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^l{e}^{-\mu r^{2}t}\right]{(-1)}^{k-l}{\displaystyle \frac{(2j+1+k-l)!}{(2j+1)!}}{r}^{-2(j+1)-(k-l)}{\tilde{c}}_j\left(t\right).\hfill \end{array}\end{array}$$

(39)

This, combined with (37) and the estimate $\left|{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^l{e}^{-\mu r^{2}t}\right|\le C\left(l\right)r^{-l}{e}^{-{\displaystyle \frac{2}{3}}\mu r^{2}t}$, directly implies that

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left|{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^k{\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_2}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\right|\le C\left(k\right)r^{-k-2}{e}^{\left({\displaystyle \frac{3\alpha \kappa }{4(\mu -ϵ)}}-\beta -{\displaystyle \frac{2}{3}}\mu r^2\right)t}\le C\left(k\right)r^{-k-2}{e}^{-\beta t-{\displaystyle \frac{1}{2}}\mu r^{2}t},\end{array}\end{array}$$

(40)

for r satisfies

$$\begin{array}{c}\hfill r^2\ge max\left\{{\displaystyle \frac{3\alpha \kappa -2\beta (\mu -ϵ)}{2{(\mu -ϵ)}^2}}+1,{\displaystyle \frac{6(k+1)}{\mu -ϵ}}\left({\displaystyle \frac{3}{4}}{\displaystyle \frac{\alpha \kappa }{\mu -ϵ}}+\beta \right),{\displaystyle \frac{9\alpha \kappa }{2\mu (\mu -ϵ)}}\right\}.\end{array}$$

(41)

Finally, combining (35) and (40) together, (14)₁ follows.

Step Two: Estimates on ${\mathcal{M}}_2(t,r)$. By applying Cauchy’s integral formula, we have

$$\begin{array}{c}\hfill {\mathcal{M}}_2(r,t)={\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1\cup {\Gamma }_2}{\displaystyle \frac{z{e}^{tz}}{P(z,r)}}dz.\end{array}$$

(42)

Applying a change of variable $w=z+{\displaystyle \frac{\alpha \kappa }{2(\mu -ϵ)}}+ϵr^2$ implies

$$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1}{\displaystyle \frac{z{e}^{tz}}{P(z,r)}}dz\hfill \\ \hfill =& {\displaystyle \frac{e^{-\left(\frac{\alpha \kappa }{2(\mu -ϵ)}+ϵr^2\right)t}}{2\pi i}}{\int }_{{\Gamma }_3}{\displaystyle \frac{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}-ϵr^2\right)e^{tw}}{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}\right)\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}+\beta \right)+(\mu -ϵ)w{r}^2}}dw\hfill \\ \hfill =& I_1+I_2,\hfill \end{array}\end{array}$$

(43)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I_1=& {\displaystyle \frac{e^{-\left(\frac{\alpha \kappa }{2(\mu -ϵ)}+ϵr^2\right)t}}{2\pi i}}{\int }_{{\Gamma }_3}{\displaystyle \frac{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}\right)e^{tw}}{(\mu -ϵ)w{r}^2}}·{\displaystyle \frac{1}{1+\frac{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}\right)\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}+\beta \right)}{(\mu -ϵ)w{r}^2}}}dw\hfill \\ \hfill =& {\displaystyle \frac{e^{-\left(\frac{\alpha \kappa }{2(\mu -ϵ)}+ϵr^2\right)t}}{2\pi i}}\sum _{j=0}^{\infty }{(-1)}^j{r}^{-2(j+1)}{\int }_{{\Gamma }_3}{e}^{tw}{\displaystyle \frac{{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}\right)}^{j+1}{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}+\beta \right)}^j}{{\left[(\mu -ϵ)w\right]}^{j+1}}}dw.\hfill \end{array}\end{array}$$

(44)

Note that $d_j\left(t\right)={\int }_{{\Gamma }_3}{e}^{tw}{\displaystyle \frac{{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}\right)}^{j+1}{\left(w-\frac{\alpha \kappa }{2(\mu -ϵ)}+\beta \right)}^j}{{\left[(\mu -ϵ)w\right]}^{j+1}}}dw$ induces similar estimates to $c_j\left(t\right)$, which, as discussions before in (31), implies

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left|{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^k{I}_1\right|\le C\left(k\right)r^{-k-2}{e}^{-{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t-{\displaystyle \frac{1}{2}}ϵr^{2}t},\end{array}\end{array}$$

(45)

for suitably large r. Furthermore, in the case $ϵ>0$, the definition

$$\begin{array}{c}\hfill \begin{array}{c}\hfill I_2=-ϵr^2{\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\end{array}\end{array}$$

(46)

and (35) directly imply that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \left|{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^k{I}_2\right|=& \left|\sum _{l=0}^k\left(\begin{array}{c}k\\ l\end{array}\right){\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^{k-l}\left(ϵr^2\right){\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^l\left({\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\right)\right|\hfill \\ \hfill \le & C\left(k\right)ϵr^{-k}{e}^{-{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t-{\displaystyle \frac{1}{2}}ϵr^{2}t}.\hfill \end{array}\end{array}$$

(47)

In conclusion, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left|{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^k{\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\right|\le C\left(k\right)\left(ϵr^{-k}+r^{-k-2}\right)e^{-{\displaystyle \frac{\alpha \kappa }{4(\mu -ϵ)}}t-{\displaystyle \frac{1}{2}}ϵr^{2}t},\end{array}\end{array}$$

(48)

On the other hand, we can perform similar calculations to prove

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left|{\left({\displaystyle \frac{𝜕}{𝜕r}}\right)}^k{\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_2}{\displaystyle \frac{e^{tz}}{P(z,r)}}dz\right|\le C\left(k\right)\left(\mu r^{-k}+r^{-k-2}\right)e^{-\beta t-{\displaystyle \frac{1}{2}}\mu r^{2}t},\end{array}\end{array}$$

(49)

for suitably large r. Then the estimate on ${\mathcal{G}}_2(t,\zeta )$ is completed.

Step Three: Estimate on ${\mathcal{G}}_3(t,r)$. Combining (9) and Cauchy’s integral formula, one has

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{M}}_3(r,t)=& {\displaystyle \frac{{\lambda }^{+}{e}^{{\lambda }^{-}t}-{\lambda }^{-}{e}^{{\lambda }^{+}t}}{{\lambda }^{+}-{\lambda }^{-}}}\hfill \\ \hfill =& {\displaystyle \frac{\left(\left[\left(\mu +ϵ\right)r^2+\beta \right]+{\lambda }_{+}\right)e^{{\lambda }_{+}t}-\left(\left[\left(\mu +ϵ\right)r^2+\beta \right]+{\lambda }_{-}\right)e^{{\lambda }_{-}t}}{{\lambda }_{+}-{\lambda }_{-}}}\hfill \\ \hfill =& {\displaystyle \frac{1}{2\pi i}}{\int }_{{\Gamma }_1\cup {\Gamma }_2}{\displaystyle \frac{\left(\left[\left(\mu +ϵ\right)r^2+\beta \right]+z\right)e^{tz}}{P(z,r)}}dz.\hfill \end{array}\end{array}$$

(50)

One can obtain the estimate for ${\mathcal{M}}_3(\zeta ,t)$ by applying a procedure similar to the one used in Steps One and Two. Then we complete the proof of Theorem 2. □