## Asymptotic Stability Analysis of Solutions to Transport Equations in Structured Bacterial Population Growth

Wu Hongxing ,, Yuan Dengbin, Wang Shenghua

Shangrao Normal University, Jiangxi Shangrao 334001

 基金资助: 国家自然科学基金.  11861053国家自然科学基金.  11461055江西省教育厅科技项目.  GJJ201714上饶市科技项目.  2020L007

 Fund supported: the NSFC.  11861053the NSFC.  11461055the Science and Technology Project of Education Department of Jiangxi Province.  GJJ201714the Science and Technology Project of Shangrao City.  2020L007 Abstract

With the help of linear operator theory, the transport equation with more general boundary condition for the structured equation with bacterial population as background is discussed. By means of resolving operator and comparison operator, it is proved that the corresponding transfer operator spectrum of the transfer equation consists of only a finite number of discrete eigenvalues with finite algebraic multiplicity in band domain $\Gamma_{\alpha, \beta}$. It is proved that the solution of the transfer equation is asymptotically stable when $\psi_0 \in D (A_{H_{\alpha, \beta}})$.

Keywords： Bacterial population ; Transport equation ; Discrete eigenvalues ; Asymptotic stability

Wu Hongxing, Yuan Dengbin, Wang Shenghua. Asymptotic Stability Analysis of Solutions to Transport Equations in Structured Bacterial Population Growth. Acta Mathematica Scientia[J], 2022, 42(3): 807-817 doi:

## 1 相关知识

$\begin{equation} \left\{ \begin{array}{ll} { }\frac{\partial \psi (a, l, t)}{\partial t}=-\frac{\partial \psi(a, l, t)}{\partial a} -\mu(a, l) \psi(a, l, t), \\ \psi(a, l, 0)=\psi_{0}(a, l). \end{array} \right. \end{equation}$

$\begin{equation} \left\{ \begin{array}{ll} { } \frac {\partial\psi}{\partial t}(\mu, v, t)=\frac{\partial\psi}{\partial\mu} (\mu, v, t)-\sigma(\mu, v)\psi(\mu, v, t)+\int_a^b r(\mu, v, v')\psi(\mu, v', t){\rm {d}}v', \\ \psi (\mu, v, 0)=\psi_0(\mu, v). \end{array} \right. \end{equation}$

$\begin{equation} \left\{ \begin{array}{ll} { } \frac {\partial\psi}{\partial t}(\mu, v, t)=-h(v)\frac{\partial\psi}{\partial\mu} (\mu, v, t)-\sigma(\mu, v)\psi(\mu, v, t)\\ { } {\qquad}{\qquad}\; +\int_0^c r(\mu, v, v')\psi(\mu, v', t){\rm {d}}v', \\ \psi (0, v, t)=H\psi(a, v, t), \psi (\mu, v, 0)=\psi_0(\mu, v), \end{array} \right. \end{equation}$

$\begin{equation} H_{\alpha, \beta}\psi(0, v, t)=\alpha \psi(1, v)+\frac{\beta}{h(v)}\int^{c}_{0}\int^{a}_{0}k(\mu', v, v') \psi(\mu', v', t){\rm {d}}\mu'{\rm {d}}v', \end{equation}$

$\begin{equation} X_1=L_1(\Omega), {\quad} \Omega=[0, a]\times[0, c]=I\times J, 0<a<+\infty, 0<c<+\infty. \end{equation}$

$\begin{equation} W=\{ \psi\in X_1 \mid h(v)\frac{\partial\psi}{\partial\mu} \in X_1 \}, {\quad} Y=L_1(J, h(v){\rm {d}}v). \end{equation}$

$\begin{equation} \|\psi\|_{X_1}=\int_0^a\int_0^c|\psi(\mu, v)|{\rm {d}}v {\rm {d}}\mu, \end{equation}$

$\begin{equation} \|\psi\|_{W}=\|\psi\|+ ||h(v)\frac{\partial\psi}{\partial\mu} ||, {\quad} \|\psi\|_{Y}=\int_0^c|\psi(v)| h(v) {\rm {d}}v. \end{equation}$

$\begin{equation} X_1^0=L_1(\Gamma_0;h(v){\rm d}v)=L_1(\{0\}\times\{0, c\};h(v){\rm {d}}v); \end{equation}$

$\begin{equation} X_1^a=L_1(\Gamma_a;h(v){\rm d}v)=L_1(\{a\}\times\{0, c\};h(v){\rm {d}}v). \end{equation}$

$\begin{equation} \|\psi\|_{X_1^0}=\int_0^c|\psi(0, v)| h(v) {\rm {d}}v, \; \|\psi\|_{X_1^a}=\int_0^c|\psi(a, v)| h(v) {\rm {d}}v. \end{equation}$

$\begin{equation} \left\{ \begin{array}{ll} { } T_{H_{\alpha, \beta}}\psi (\mu, v) = - h(v)\frac{\partial \psi }{\partial \mu}(\mu, v) - \sigma(\mu, v)\psi (\mu, v), \\ { } D(T_{H_{\alpha, \beta}}) = \left\{\psi \in W \left |\; \psi^0=H_{\alpha, \beta}\psi^a \psi^0=\psi\mid\Gamma_0, \psi^a=\psi\mid\Gamma_a\right.\right\}. \end{array} \right. \end{equation}$

$\begin{equation} (B\psi)(\mu, v)=\int_0^c{r(\mu, v, v')\psi(\mu, v'){\rm {d}}v'}, D(B)=X_1. \end{equation}$

$\begin{equation} A_{H_{\alpha, \beta}}=T_{H_{\alpha, \beta}}+B, {\quad} D(A_{H_{\alpha, \beta}})=D(T_{H_{\alpha, \beta}}). \end{equation}$

$\begin{equation} (\lambda -T_{H_{\alpha, \beta}})\psi=\varphi, \end{equation}$

${\rm Re}\lambda >-\sigma_0$时, 则方程$(1.15)$的解为

$\begin{eqnarray} \psi (\mu, v)=\psi ( 0, v){\rm {e}}^{-\frac{1}{h(v)}\int^\mu_0{(\lambda+\sigma(\mu', v)){\rm {d}}\mu'}} +\frac{1}{h(v)}\int^\mu_0{{\rm {e}}^{-\frac{1}{h(v)}\int^ {\mu}_{\mu'}{(\lambda+\sigma(\xi, v)){\rm {d}}\xi}}\varphi(\mu', v){\rm {d}}\mu'}. \end{eqnarray}$

$\mu =a$, 方程$(1.16)$可化为

$\begin{eqnarray} \psi (a, v) =\psi(0, v){\rm {e}}^{-\frac{1}{h(v)}\int^a_0{(\lambda+\sigma(\mu', v)){\rm {d}}\mu'}} +\frac{1}{h(v)}\int^a_0{{\rm {e}}^{-\frac{1}{h(v)}\int^ {a}_{\mu'}{(\lambda+\sigma(\xi, v)){\rm {d}}\xi}}\varphi(\mu', v){\rm {d}}\mu'}. \end{eqnarray}$

$\begin{equation} \left\|P_\lambda \right\|\leq 1, \left\|Q_\lambda\right\|\leq ({\rm Re}\lambda +{\sigma_0})^{-1}, \left\|D_\lambda\right\|\leq ({\rm Re}\lambda +{\sigma_0})^{-1}, \left\|E_\lambda\right\|\leq ({\rm Re}\lambda +{\sigma_0})^{-1}. \end{equation}$

根据假设, 可构造多项式

$\begin{equation} L(x)=a_1x+a_2x^2+\cdots ++a_nx^n, \end{equation}$

$\begin{equation} || L[B(\lambda -T )^{-1}] || \leq \frac{a_1||B||}{\sigma-\omega (U)}+\frac{a_2||B||^2}{(\sigma-\omega (U))^2} +\cdots +\frac{a_n||B||^n}{(\sigma-\omega (U))^n}. \end{equation}$

$\begin{eqnarray} \mu - L[B(\lambda -T )^{-1}]&=&a_1 (I-[B(\lambda -T )^{-1}])+a_2 (I-[B(\lambda -T )^{-1}]^2){}\\ &&+\cdots +a_n (I-[B(\lambda -T )^{-1}]^n). \end{eqnarray}$

$\begin{eqnarray} [\mu I - L(B(\lambda -T )^{-1})]^{-1}&=&[a_1 (I-[B(\lambda -T )^{-1}]) +a_2 (I-[B(\lambda -T )^{-1}]^2){}\\ &&+\cdots +a_n (I-[B(\lambda -T )^{-1}]^n)]^{-1}{}\\ &=&[ a_1 (I-[B(\lambda -T )^{-1}](a_1 I+a_2 (I+(B(\lambda -T )^{-1})) )+\cdots {}\\ &&+a_n (I+(B(\lambda -T )^{-1})+\cdots +(B(\lambda -T )^{-1})^{n-1}) )]^{-1}. \end{eqnarray}$

$\begin{eqnarray} & &[(a_1 I+a_2 (I+(B(\lambda -T )^{-1}) )+\cdots +a_n (I+(B(\lambda -T )^{-1})\\ & &+\cdots +(B(\lambda -T )^{-1})^{n-1}))]^{-1}\cdot[ \mu I - L(B(\lambda -T )^{-1})]^{-1}\\ & =&(I-B(\lambda -T ))^{-1}. \end{eqnarray}$

$\begin{equation} \Delta_\lambda= (\lambda -T )^{-1}[I-B(\lambda -T )^{-1}]^{-1}, A_\lambda=\lambda I-A=\lambda I-T-B. \end{equation}$

$\begin{equation} A_\lambda \Delta_\lambda=(\lambda -A)(\lambda -T-B)^{-1}=I. \end{equation}$

$\begin{equation} [B(\lambda_k -T)^{-1}]\varphi=\varphi. \end{equation}$

$\begin{equation} (\mu -L[B(\lambda_k -T)^{-1}])\varphi=0, \end{equation}$

$\begin{equation} (L[B(\lambda_k-T)^{-1}\mu^{-1}])\varphi=\varphi, \end{equation}$

$\Gamma_0 =\{\lambda\in {\bf C}, {\rm Re}\lambda > \lambda_0 \}$.

$\begin{eqnarray} (D_\lambda B\varphi)(v)=\int^{c}_{0}\int^{a}_{0}\frac{\eta(\mu)\alpha(v)}{h(v)}{\rm {e}}^{-\frac{1}{h(v)}\int^{a}_{\mu}{(\lambda+\sigma(\xi, v)){\rm {d}}\xi}} \beta(v')\varphi(\mu, v){\rm {d}}\mu {\rm {d}}v' =L_\lambda R , \end{eqnarray}$

$\begin{equation} \int_E|(L_\lambda \varphi)(v)|v {\rm {d}}v\leq || \eta ||^1_\infty ||\varphi||\int_E|\alpha(v)|v{\rm {d}}v, \end{equation}$

根据定理$2.1$可得算子$(\lambda -T_{H_{\alpha, \beta}})^{-1}B $$L_1 空间上是弱紧的, 故可得到 [(\lambda -T_{H_{\alpha, \beta}})^{-1}B]^2$$ L_1$空间上是弱紧的.因此, 考虑复多项式$L(x)=ax^2 (x\neq 0)$, 则算子$(\lambda -T_{H_{\alpha, \beta}})^{-1}L[(\lambda -T_{H_{\alpha, \beta}})^{-1}] $$\Gamma_0 是紧的, 于是又由定理2.1可得 \Gamma_{\alpha, \beta} 由至多可数个具有限代数重数的离散本征值所组成.不妨设 \begin{equation} \sigma (A_{H_{\alpha, \beta}}) \cap \Gamma_0=\{\lambda_1, \lambda_2, \cdots , \lambda_n\}. \end{equation} \delta_1=\sup \{{\rm Re}\lambda, \lambda\in \sigma(A_{H_{\alpha, \beta}}), {\rm Re}\lambda < \lambda_0+ \varepsilon \}, \delta_2=\min \{{\rm Re}\lambda_j, 1\leq j \leq n\} , 则存在 \delta (\delta_1<\delta<\delta_2) , 当 \psi_0 \in D(A_{H_{\alpha, \beta}}) , 可得方程(2.18)–(2.19)成立. 定理2.4 若扰动算子B非负正则的, 当 { }\lim\limits_{{\rm Re}\lambda\rightarrow \lambda_0} r_\sigma(P_\lambda H_{0, \beta})>1 时, 则迁移算子 A_{H_{\alpha, \beta}} 的谱 \sigma(A_{H_{\alpha, \beta}}) 在带域 \Gamma_{\alpha, \beta} 仅由有限个具有限代数重数的离散本征值组成. 根据方程(1.23)得到 \begin{equation} \psi^a = P_\lambda {H_{\alpha, 0}} \psi^a +P_\lambda {H_{0, \beta}} \psi^a + D_\lambda\varphi, \end{equation} 因为当 \lambda > \lambda_0$$ I-P_\lambda {H_{0, \beta}}$有界可逆算子, 于是方程(2.21)可化为

$\begin{equation} \psi^a = F_\lambda \psi^a +L_\lambda\varphi, \end{equation}$

$\begin{equation} ||(F_\lambda)^3 || \leq ||(I-P_{\lambda_1} {H_{0, \beta}})^{-1} || ||P_{\lambda_1} {H_{0, \beta}}(F_\lambda)^2||. \end{equation}$

$\begin{equation} \lim\limits_{\lambda\rightarrow \infty}\sigma_{(F_\lambda)} =0. \end{equation}$

$\begin{equation} \lambda\in\sigma_p(A_{H_{\alpha, \beta}})\neq \emptyset \Leftrightarrow 1\in \sigma(F_\lambda ), \end{equation}$

$\begin{equation} \lambda\in\sigma_p(A_{H_{\alpha, \beta}})\neq \emptyset \Leftrightarrow \lim\limits_{\lambda\rightarrow \lambda_0}r_{\sigma}(F_\lambda) >1. \end{equation}$

$\begin{equation} F_\lambda \geq P_\lambda H_{0, \beta}, {\quad} \lim\limits_{\lambda\rightarrow \lambda_0}r_{\sigma}(F_\lambda)\geq \lim\limits_{\lambda\rightarrow \lambda_0}r_{\sigma}(P_\lambda H_{0, \beta}) \end{equation}$

$\begin{equation} ||V_{H_{\alpha, \beta}}(t)(I-P)||_X \leq M {\rm {e}}^{(\varepsilon+{\rm Re}\lambda_{n+1})t} , {\quad} t>0, \end{equation}$

对任意的$\varepsilon> 0$, 令$\Gamma_{n, \varepsilon}=\{ \lambda\in {\bf C }: {\rm Re}\lambda > \beta_{n, \varepsilon} -\frac{\varepsilon}{2} \}, \beta_{n, \varepsilon}= {\rm Re}\lambda_{n+1 }+\varepsilon$, 定义函数

$\begin{equation} \Xi(\lambda)=(\lambda -T_{H_{\alpha, \beta}})^{-1}[B(\lambda -T_{H_{\alpha, \beta}})^{-1} ]^m B(\lambda -A_{H_{\alpha, \beta}})^{-1}(I-P)\psi. \end{equation}$

$\omega>\lambda_0$, 则由定理2.2知存在$C(\omega)$, 使得$|{\rm Im}\lambda| || (\lambda -T_{H_{\alpha, \beta}})^{-1}B ||$在区间$\{\lambda\in {\bf C }, {\rm Re}\lambda \geq \omega , |{\rm Im}\lambda|\geq C(\omega) \}$上是有界的.又因为$|| (\lambda -T_{H_{\alpha, \beta}})^{-1}||$在区间$\{\lambda\in {\bf C }, {\rm Re}\lambda \geq \omega , |{\rm Im}\lambda|\geq C(\omega) \}$一致有界.于是得到

$\begin{eqnarray} &&|{\rm Im}\lambda| || (\lambda -T_{H_{\alpha, \beta}})^{-1} B(\lambda -T_{H_{\alpha, \beta}})^{-1}B (\lambda -A_{H_{\alpha, \beta}})^{-1} || \\ &\leq&|{\rm Im}\lambda| ||(\lambda -T_{H_{\alpha, \beta}})^{-1}B||^2 ||(\lambda -A_{H_{\alpha, \beta}})^{-1} ||. \end{eqnarray}$

$\begin{equation} ||\Xi(\lambda)||\leq \frac{\eta}{|Im \lambda |^{r_0}} \end{equation}$

$\Gamma_{n, \varepsilon}$一致成立.由于$\Xi(\lambda)$$\Gamma_{n, \varepsilon}$上是解析的, 故对任意$t\geq 0$, 可得

$\begin{equation} \alpha(t)=\frac{1}{2\pi {\rm i}}\int_{\tau -{\rm i}\infty}^{\tau +{\rm i}\infty} {\rm {e}}^{\lambda t} \Xi(\lambda){\rm {d}}\lambda. \end{equation}$

$\begin{equation} \int_{0}^{\infty} {\rm {e}}^{-\lambda t}\alpha(t){\rm {d}}t= \Xi(\lambda), \end{equation}$

$\begin{equation} W(t)(I-P)\psi=V_{H_{\alpha, \beta}}(t)(I-P)\psi-\sum\limits_{k\geq 0}^{m}U_k(t)(I-P)\psi. \end{equation}$

$\begin{equation} \int_0^\infty {\rm {e}}^{-\lambda t}U_k(t)\psi {\rm {d}}t= (\lambda -T_{H_{\alpha, \beta}})^{-1}[B(\lambda -T_{H_{\alpha, \beta}})^{-1} ]^k \psi, \end{equation}$

$\begin{equation} || U_k(t)|| \leq {\rm {e}}^{(\omega(U)+\varepsilon)t} {M_1}^{k+1}\frac{|| B ||^k t^k}{k!} , \end{equation}$

$\begin{equation} || U_k(t)|| \leq {M_1}{\rm {e}}^{(\omega(U)+\varepsilon)t}. \end{equation}$

$\begin{eqnarray} &&\int_0^\infty {\rm {e}}^{-\lambda t}W(t)(I-P)\psi {\rm {d}}t{}\\ &=& (\lambda -A_{H_{\alpha, \beta}})^{-1}(I-P)\psi-\sum _{k=0}^m (\lambda -T_{H_{\alpha, \beta}})^{-1}[B(\lambda -T_{H_{\alpha, \beta}})^{-1} ]^k(I-P)\psi. \end{eqnarray}$

$\begin{equation} (\lambda -A_{H_{\alpha, \beta}})^{-1}=\sum\limits_{k=0}^\infty (\lambda -T_{H_{\alpha, \beta}})^{-1}[B(\lambda -T_{H_{\alpha, \beta}})^{-1} ]^k, \end{equation}$

$\begin{eqnarray} \int_0^\infty {\rm {e}}^{-\lambda t}W(t)(I-P)\psi {\rm {d}}t &=&\sum _{k=m+1}^\infty (\lambda -T_{H_{\alpha, \beta}})^{-1}[B(\lambda -T_{H_{\alpha, \beta}})^{-1} ]^k(I-P)\psi \\ &=&(\lambda -T_{H_{\alpha, \beta}})^{-1}[B(\lambda -T_{H_{\alpha, \beta}})^{-1} ]^m B(\lambda -A_{H_{\alpha, \beta}})^{-1}(I-P)\psi. {\qquad} \end{eqnarray}$

$\begin{equation} \Xi(\lambda)=\int_0^\infty {\rm {e}}^{-\lambda t}W(t)(I-P)\psi {\rm {d}}t. \end{equation}$

$\begin{equation} W(t)(I-P)\psi=\alpha(t). \end{equation}$

$\begin{eqnarray} \alpha(t)&=& \frac{1}{2\pi {\rm i}} \lim\limits_{y\rightarrow +\infty} \Big( \int_{\beta_{n, \varepsilon} -{\rm i}y}^{\beta_{n, \varepsilon} +{\rm i}y}{\rm {e}}^{-\lambda t} \Xi(\lambda){\rm {d}}\lambda \\ && + \int_{\beta_{n, \varepsilon}}^{\tau}{\rm {e}}^{(x+{\rm i}y)t} \Xi(x+{\rm i}y){\rm {d}}x - \int_{\beta_{n, \varepsilon}}^{\tau}{\rm {e}}^{(x-{\rm i}y)t} \Xi(x-{\rm i}y){\rm {d}}x \Big). \end{eqnarray}$

$\begin{equation} \lim\limits_{y\rightarrow +\infty}\int_{\beta_{n, \varepsilon}}^{\tau}{\rm {e}}^{(x+{\rm i}y)t} \Xi(x+{\rm i}y){\rm {d}}x =\lim\limits_{y\rightarrow +\infty}\int_{\beta_{n, \varepsilon}}^{\tau}{\rm {e}}^{(x-{\rm i}y)t} \Xi(x-{\rm i}y){\rm {d}}x. \end{equation}$

$\begin{equation} \alpha(t)=\frac{1}{2\pi i}\int_{\beta_{n, \varepsilon} -{\rm i}y}^{\beta_{n, \varepsilon} +{\rm i}y} {\rm {e}}^{\lambda t}\Xi(\lambda){\rm {d}}\lambda, \end{equation}$

$\begin{eqnarray} ||\alpha(t)||\leq \frac{1}{2\pi } {\rm {e}}^{\beta_{n, \varepsilon} t}\int_{-\infty}^{+\infty}|| \Xi(\beta_{n, \varepsilon} +{\rm i}y) ||{\rm {d}}y. \end{eqnarray}$

$\begin{eqnarray} ||\alpha(t)||\leq \frac{\eta}{2\pi |{\rm Im}\lambda_{n+1}|^{r_0}} {\rm {e}}^{\beta_{n, \varepsilon}t}. \end{eqnarray}$

$\begin{eqnarray} || V_{\alpha, \beta}(t)(I-P) ||&\leq & || W(t)(I-P) || + \sum\limits_{k\geq 0}^{m}U_k(t)(I-P)||\\ &\leq &\frac{\eta}{2\pi |{\rm Im}\lambda_{n+1}|^{r_0} }{\rm {e}}^{\beta_{n, \varepsilon} t} + \sum _{k=0}^m {\rm {e}}^{(\omega(U)+\varepsilon)t} {M_1}^{k+1}\frac{|| B ||^k t^k}{k!} \\ &\leq &M {\rm {e}}^{\beta_{n, \varepsilon} t}=M {\rm {e}}^{({\rm Re}\lambda_{n+1}+\varepsilon)t}, \end{eqnarray}$

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