## Blow-up of Solutions to the Euler-Poisson-Darboux-Tricomi Equation with a Nonlinear Memory Term

Ouyang Baiping,

Department of Apllied Mathematics, Guangzhou Huashang College, Guangzhou 511300

 基金资助: 广东省普通高校自然科学重点项目.  2019KZDXM042广东省普通高校创新团队项目.  2020WCXTD008广州华商学院校内项目.  2020HSDS01广州华商学院校内项目.  2021HSKT01

 Fund supported: Key Projects of Universities in Guangdong Province (NATURAL SCIENCE).  2019KZDXM042Innovation Team Project in Colleges and Universities of Guangdong Province.  2020WCXTD008Science Foundation of Guangzhou Huashang College.  2020HSDS01Science Foundation of Guangzhou Huashang College.  2021HSKT01

Abstract

Blow-up phenomenon of solutions to the Euler-Poisson-Darboux-Tricomi equation with a nonlinear memory term in the subcritical case is studied. By using functional methods associated with a modified Bessel equation, an iteration frame and the first lower bound are derived. Then, nonexistence of global solutions to the Cauchy problem for the Euler-Poisson-Darboux-Tricomi equation and an upper bound estimate of solutions for the lifespan are obtained via the iteration technique.

Keywords： Nonlinear memory term ; Euler-Poisson-Darboux-Tricomi equation ; Blow-up

Ouyang Baiping. Blow-up of Solutions to the Euler-Poisson-Darboux-Tricomi Equation with a Nonlinear Memory Term. Acta Mathematica Scientia[J], 2023, 43(1): 169-180 doi:

## 1 引言

$\begin{matrix}\label{1} \left\{\begin{array}{ll} u_{tt}-t^{2l}\Delta u+\mu t^{-1}u_t+\nu^2t^{-2}u=C_{\gamma}\int_1^t (t-s)^{-\gamma} |u(s,x)|^p {\rm d}s,& (x,t)\in \Bbb R^n \times (1,T), \\ (u,u_t)(1,x)= \varepsilon (u_0,u_1)(x),& x\in \Bbb R^n, \end{array}\right. \end{matrix}$

$\left\{\begin{array}{ll} u_{tt}-t^{2l}\Delta u+\mu t^{-1}u_t+\nu^2t^{-2}u=|u|^p,& (x,t)\in \Bbb R^n \times (1,T), \\ (u,u_t)(1,x)= \varepsilon (u_0,u_1)(x),& x\in \Bbb R^n, \end{array}\right.$

$(u_0,u_1)\in H^1(\Bbb R^n)\times L^2(\Bbb R^n)$ 是非负的紧支集函数, 其支集包含于$B_R(R>0)$.$u\in {\cal C}([1,T),W_{\rm loc}^{1,1}(\Bbb R^n))\cap {\cal C}^1([1,T),L_{\rm loc}^1(\Bbb R^n))\cap L_{\rm loc}^p((1,T)\times \Bbb R^n)$ 为问题(1.1)的能量解,满足$u_1(x)+(\frac{\mu-1}{2}-\delta)u_0(x)\geq 0,\delta^2=\frac{(\mu-1)^2-4\nu^2}{4}, \delta\geq 0.$ 其生命跨度记作$T(\varepsilon)$, 则存在一个正常数$\varepsilon_0$, 使得当$\varepsilon\in (0,\varepsilon_0]$$u 在有限时间内爆破, 其生命跨度上界估计为 T(\varepsilon)\leq \check{C}\varepsilon^{-\frac{2p(p-1)}{\Theta(p,n,l,\gamma,\mu)}}, 其中 \breve{C} 为不依赖于 \varepsilon 的正常数, 且 \Theta(p,n,l,\gamma,\mu)=2(l+1)+(5-2\gamma+\mu+(n-2)(l+1))p-p^2(\mu+l+(n-1)(l+1)). ## 3 定理的证明 研究如下方程的正解 \begin{matrix}\label{4} \Phi_{ss}+(\mu+\nu^2)s^{-2}\Phi=s^{2l}\Delta \Phi+\mu s^{-1}\Phi_s. \end{matrix} 设函数\phi(x) (参见文献[21])定义如下 \phi(x) \doteq \left\{\begin{array}{ll} {\rm e}^{x}+{\rm e}^{-x},&\ \, n=1,\\ \int_{{\Bbb S}^{n-1}} {\rm e}^{x\cdot \omega}{\rm d} \sigma_\omega,& \ \, n\geq 2, \end{array}\right. 其中 {\Bbb S}^{n-1}表示n-1维球面, \phi(x) 是正光滑函数, 有如下性质 \Delta \phi(x) =\phi(x), \phi (x) \sim |x|^{-\frac{n-1}{2}} {\rm e}^{|x|}, \qquad \mbox{当} \,\, |x|\to \infty. \Phi=\Phi(s,x)=\lambda(s)\phi(x), 结合(3.1)式, 可推得 \begin{matrix}\label{5} \lambda''+(\mu+\nu^2)s^{-2}\lambda=s^{2l}\lambda+\mu s^{-1}\lambda'. \end{matrix} 作变量代换, 设\eta=w_l(s), 有 \begin{matrix}\label{6} \frac{{\rm d}\lambda}{{\rm d}s}s^{-l}=\frac{{\rm d}\lambda}{{\rm d}\eta}, \frac{{\rm d}^2\lambda}{{\rm d}s^2}=\frac{{\rm d}^2\lambda}{{\rm d}\eta^2}s^{2l}+\frac{{\rm d}\lambda}{{\rm d}\eta}ls^{l-1}. \end{matrix} 联立(3.2)-(3.3)式, 整理可得 \begin{matrix}\label{7} \eta^2\frac{{\rm d}^2\lambda}{{\rm d}\eta^2}-\frac{(l-\mu)\eta {\rm d}\lambda}{(l+1){\rm d}\eta}+(\frac{\mu+\nu^2}{(l+1)^2}-\eta^2)\lambda=0. \end{matrix} \lambda(\eta)=\eta^m \xi(\eta), 结合(3.4)式, 得到 \begin{matrix}\label{8} \eta^l\frac{{\rm d}^2\xi}{{\rm d}\eta^2}+(2m+\frac{l-\mu}{l+1})\eta\frac{{\rm d}\xi}{{\rm d}\eta}+(m(m-1)+\frac{(l-\mu)m}{l+1}+(\frac{\mu+\nu^2}{(l+1)^2}-\eta^2))\xi=0. \end{matrix} (3.5)式中, 取 m=\frac{1+\mu}{2(l+1)}, 得 \begin{matrix}\label{9} \eta^2\frac{{\rm d}^2\xi}{{\rm d}\eta^2}+\eta\frac{{\rm d}\xi}{{\rm d}\eta}-(\frac{(\mu-1)^2-4\nu^2}{4(l+1)^2}+\eta^2)\xi=0. \end{matrix} (3.2)-(3.6)式表明,(3.2)式的解 \lambda亦为(3.6)式的解. \delta^2=\frac{(\mu-1)^2-4\nu^2}{4}, \delta\geq 0, 则(3.6)式化为 \begin{matrix}\label{10} \eta^2\frac{{\rm d}^2\xi}{{\rm d}\eta^2}+\eta\frac{{\rm d}\xi}{{\rm d}\eta}-(\frac{\delta^2}{(l+1)^2}+\eta^2)\xi=0. \end{matrix} 选取第二类Bessel函数K_{\frac{\delta}{l+1}}(\eta)作为(3.7)式的解. 于是, 如果忽略一个正常数因子, 有 \begin{matrix}\label{11} \lambda(s)=s^{\frac{\mu+1}{2}}K_{\frac{\delta}{l+1}}(w_l(s)). \end{matrix} 由此, 可得 \begin{matrix}\label{12} \Phi(s,x)=\lambda(s)\phi(x)=s^{\frac{\mu+1}{2}}K_{\frac{\delta}{l+1}}(w_l(s))\phi(x). \end{matrix} 接下来, 将通过引入辅助泛函得到需要的迭代框架. 为此, 设 \begin{matrix}\label{13} F(t)=\int_{\Bbb R^n} u(t,x){\rm d}x. \end{matrix} (2.2)式中, 取 \varphi=\varphi(s,x)\equiv 1,其中 (s,x)\in [t]\times \Bbb R^n$$ |x|\leq R+A_l(s)$, 得到

$\begin{matrix}\label{14} &&\int_{\Bbb R^n} u_{t}(t,x){\rm d}x+\mu\int^t_1\int_{\Bbb R^n}s^{-1}u_t(s,x){\rm d}x{\rm d}s+\int^t_1\int_{\Bbb R^n}\nu^{2}s^{-2}u(s,x){\rm d}x{\rm d}s \\ &= &C_{\gamma}\int_1^t \int_{\Bbb R^n}\int_1^s(s-\tau)^{-\gamma} |u(\tau,x)|^p {\rm d}\tau {\rm d}x{\rm d}s+\varepsilon \int_{\Bbb R^n}u_1(x){\rm d}x. \end{matrix}$

$\begin{matrix}\label{15} &&F'(t)+\mu\int^t_1\int_{\Bbb R^n}s^{-1}u_t(s,x){\rm d}x{\rm d}s+\int^t_1\int_{\Bbb R^n}\nu^{2}s^{-2}u(s,x){\rm d}x{\rm d}s \\ &=& C_{\gamma}\int_1^t \int_{\Bbb R^n}\int_1^s(s-\tau)^{-\gamma} |u(\tau,x)|^p {\rm d}\tau {\rm d}x{\rm d}s+F'(1). \end{matrix}$

(3.12)式对$t$求导, 可推出

$\begin{matrix}\label{16} F''(t)+\mu t^{-1}F'(t)+\nu^{2}t^{-2}F(t)= C_{\gamma} \int_{\Bbb R^n}\int_1^t(t-s)^{-\gamma} |u(s,x)|^p {\rm d}s{\rm d}x. \end{matrix}$

$\begin{matrix}\label{17} t^{-(q_2+1)}\frac{{\rm d}}{{\rm d}t}(t^{q_2+1-q_1}\frac{{\rm d}(t^{q_1}F(t))}{{\rm d}t})= C_{\gamma} \int_{\Bbb R^n}\int_1^t(t-s)^{-\gamma} |u(s,x)|^p {\rm d}s{\rm d}x, \end{matrix}$

$\begin{matrix}\label{18} F(t)&=&t^{-q_1}F(1)+t^{-q_1}\int_1^t F'(1)s^{-(q_2+1-q_1)}{\rm d}s \\ &&+ C_{\gamma}t^{-q_1} \int_1^t s^{-(q_2+1-q_1)}\int_1^s \tau^{q_2+1}\int_{\Bbb R^n}\int_1^\tau(\tau-\sigma)^{-\gamma} |u(\sigma,x)|^p {\rm d}\sigma {\rm d}x {\rm d}\tau {\rm d}s \\ &\geq & C_{\gamma}t^{-q_1} \int_1^t s^{-(q_2+1-q_1)}\int_1^s \tau^{q_2+1}\int_{\Bbb R^n}\int_1^\tau(\tau-\sigma)^{-\gamma} |u(\sigma,x)|^p {\rm d}\sigma {\rm d}x {\rm d}\tau {\rm d}s. \end{matrix}$

$\begin{matrix}\label{19} \int_{\Bbb R^n} |u(\sigma,x)|^p {\rm d}x\geq c_0(R+A_l(\sigma))^{-n(p-1)}(F(\sigma))^p, \end{matrix}$

$\begin{matrix}\label{20} F(t)&\geq & C_{\gamma}c_0t^{-q_1} \int_1^t s^{-(q_2+1-q_1)}\int_1^s \tau^{q_2+1}\int_1^\tau(\tau-\sigma)^{-\gamma} (R+A_l(\sigma))^{-n(p-1)}(F(\sigma))^p {\rm d}\sigma {\rm d}\tau {\rm d}s \\ &\geq & C_{\gamma}\widetilde{c}_0t^{-q_1} \int_1^t s^{-(q_2+1-q_1)}\int_1^s \tau^{q_2+1}\int_1^\tau(\tau-\sigma)^{-\gamma} (\sigma+1)^{-n(l+1)(p-1)}(F(\sigma))^p {\rm d}\sigma {\rm d}\tau {\rm d}s \\ &\geq & C_{\gamma}\widetilde{c}t^{-(q_2+1+\gamma)-n(l+1)(p-1)} \int_1^t \int_1^s \tau^{q_2+1}\int_1^\tau(F(\sigma))^p {\rm d}\sigma {\rm d}\tau {\rm d}s, \end{matrix}$

$\begin{matrix}\label{21} U(t)=\int_{\Bbb R^n}u(t,x)\Phi(t,x){\rm d}x. \end{matrix}$

(2.2)式中, 取$\Phi=\varphi$, $\Phi$的定义见(3.9)式, 由波方程的有限传播速度可知, $u$ 具有紧支集, 因此 $\Phi$ 的支集条件可以去掉, 有

$\begin{matrix}\label{22} &&\int_{\Bbb R^n} u_{t}(t,x)\Phi(t,x) {\rm d}x-\int_{\Bbb R^n} u(t,x)\Phi_t(t,x) {\rm d}x+\int^t_1\int_{\Bbb R^n}u(s,x)\Phi_{ss}(s,x){\rm d}x{\rm d}s \\ &&-\int^t_1\int_{\Bbb R^n}s^{2l} u(s,x)\Delta\Phi(s,x) {\rm d}x{\rm d}s+\mu\int_{\Bbb R^n}t^{-1}u(t,x)\Phi(t,x){\rm d}x \\ &&-\mu\int^t_1\int_{\Bbb R^n}s^{-1}u(s,x)\Phi_s(s,x){\rm d}x{\rm d}s+\int^t_1\int_{\Bbb R^n}(\mu+\nu^{2})s^{-2}u(s,x)\Phi(s,x){\rm d}x{\rm d}s \\ &=& C_{\gamma}\int_1^t \int_{\Bbb R^n}\Phi(s,x)\int_1^s(s-\tau)^{-\gamma} |u(\tau,x)|^p {\rm d}\tau {\rm d}x{\rm d}s+\varepsilon \int_{\Bbb R^n}u_1(x)\Phi(1,x){\rm d}x \\ &&-\varepsilon \int_{\Bbb R^n}u_0(x)\Phi_t(1,x){\rm d}x+\varepsilon \mu\int_{\Bbb R^n}u_0(x)\Phi(1,x){\rm d}x. \end{matrix}$

$\begin{matrix}\label{23} &&\int_{\Bbb R^n} u_{t}(t,x)\Phi(t,x) {\rm d}x-\int_{\Bbb R^n} u(t,x)\Phi_t(t,x) {\rm d}x+\mu\int_{\Bbb R^n}t^{-1}u(t,x)\Phi(t,x){\rm d}x \\ &=& C_{\gamma}\int_1^t \int_{\Bbb R^n}\Phi(s,x)\int_1^s(s-\tau)^{-\gamma} |u(\tau,x)|^p {\rm d}\tau {\rm d}x{\rm d}s+\varepsilon \int_{\Bbb R^n}u_1(x)\Phi(1,x){\rm d}x \\ &&-\varepsilon \int_{\Bbb R^n}u_0(x)\Phi_t(1,x){\rm d}x+\varepsilon \mu\int_{\Bbb R^n}u_0(x)\Phi(1,x){\rm d}x. \end{matrix}$

$\begin{matrix}\label{24} U'(t)=\int_{\Bbb R^n}u_t(t,x)\Phi(t,x){\rm d}x+\frac{\lambda'(t)}{\lambda(t)}U(t). \end{matrix}$

$\begin{matrix}\label{25} &&U'(t)-\frac{2\lambda'(t)}{\lambda(t)}U(t)+\mu t^{-1}U(t) \\ &=& C_{\gamma}\int_1^t \int_{\Bbb R^n}\Phi(s,x)\int_1^s(s-\tau)^{-\gamma} |u(\tau,x)|^p {\rm d}\tau {\rm d}x{\rm d}s +\varepsilon \int_{\Bbb R^n}u_1(x)\lambda(1)\phi(x){\rm d}x \\ &&+\varepsilon \int_{\Bbb R^n}(\mu \lambda(1)-\lambda'(1))u_0(x)\phi(x){\rm d}x. \end{matrix}$

$K'_{\chi}(z)=-K_{\chi+1}(z)+\frac{\chi}{z}K_{\chi}(z),$

$\begin{matrix}\label{26} \lambda'(s)&=&\frac{\mu+1}{2}s^{\frac{\mu-1}{2}}K_{\frac{\delta}{l+1}}(w_l(s)) +s^{\frac{\mu-1}{2}}K'_{\frac{\delta}{l+1}}(w_l(s))w'_l(s) \\ &=&(\frac{\mu+1}{2}+\delta)s^{\frac{\mu-1}{2}}K_{\frac{\delta}{l+1}}(w_l(s)) -s^{\frac{\mu+1}{2}+l}K_{\frac{\delta}{l+1}+1}(w_l(s)). \end{matrix}$

$\begin{matrix}\label{27} && \int_{\Bbb R^n}u_1(x)\lambda(1)\phi(x){\rm d}x+ \int_{\Bbb R^n}(\mu \lambda(1)-\lambda'(1))u_0(x)\phi(x){\rm d}x \\ &=&\int_{\Bbb R^n}K_{\frac{\delta}{l+1}+1}(1)u_0(x)\phi(x){\rm d}x+ \int_{\Bbb R^n}K_{\frac{\delta}{l+1}}(1)(u_1(x) +(\frac{\mu-1}{2}-\delta)u_0(x))\phi(x){\rm d}x \\ &>&0. \end{matrix}$

$I=I[u_0,u_1]= \int_{\Bbb R^n}u_1(x)\lambda(1)\phi(x){\rm d}x+ \int_{\Bbb R^n}(\mu \lambda(1)-\lambda'(1))u_0(x)\phi(x){\rm d}x,$

$\begin{matrix}\label{28} U'(t)-\frac{2\lambda'(t)}{\lambda(t)}U(t)+\mu t^{-1}U(t)\geq \varepsilon I. \end{matrix}$

$\begin{matrix}\label{29} \frac{\lambda^2(t)}{t^\mu}\frac{{\rm d}}{{\rm d}t}(\frac{t^\mu}{\lambda^2(t)}U(t)) \geq \varepsilon I. \end{matrix}$

$\begin{matrix}\label{30} U(t)&\geq&\frac{U(1)}{\lambda^2(1)}t^{-\mu}\lambda^2(t)+ \varepsilon I t^{-\mu}\lambda^2(t)\int_1^t \frac{s^\mu}{\lambda^2(s)}{\rm d}s \geq \varepsilon I t^{-\mu}\lambda^2(t)\int_1^t \frac{s^\mu}{\lambda^2(s)}{\rm d}s. \end{matrix}$

$\begin{matrix}\label{31} C_1{\rm e}^{-2w_l(s)}s^{\mu-l}\leq\lambda^2(s)\leq C_2{\rm e}^{-2w_l(s)}s^{\mu-l}. \end{matrix}$

$\begin{matrix}\label{32} U(t)&\geq &\frac{\varepsilon IC_1}{C_2}{\rm e}^{-2w_l(t)} t^{-l}\int_{t_0}^t {\rm e}^{2w_l(s)}s^l{\rm d}s \geq \frac{\varepsilon IC_1}{C_2}{\rm e}^{-2w_l(t)} t^{-l}\int_{\frac{t}{2}}^t {\rm e}^{2w_l(s)}s^l{\rm d}s \\ &\geq&\frac{\varepsilon IC_1}{2C_2} t^{-l}(1- {\rm e}^{-\frac{2}{l+1}(1-2^{\frac{1}{l+1}})(2t_0)^{l+1}})=\widetilde{C}\varepsilon t^{-l}, \end{matrix}$

$\begin{matrix}\label{33} U(t)\leq \left(\int_{\Bbb R^n} |u(t,x)|^p {\rm d}x\right)^{\frac{1}{p}} \bigg(\int_{B_{R+A_l(t)}} \Phi(t,x)^{p'} {\rm d}x\bigg)^{\frac{1}{p'}}, \end{matrix}$

$\begin{matrix}\label{39} F(t)\geq K_0t^{-\alpha_0}(t-t_1)^{\beta_0},\quad t\geq t_1. \end{matrix}$

$\begin{matrix}\label{40} F(t)\geq K_jt^{-\alpha_j}(t-L_jt_1)^{\beta_j}, \end{matrix}$

$L_j=\prod\limits_{k=0}^j l_k, l_k=1+p^{-k}, k,j\in {\Bbb N}.$

$\{L_j\}_{j\in {\Bbb N}}$ 的无限积定义为$\prod\limits_{k=0}^\infty l_k, l_k=1+p^{-k}, k\in {\Bbb N}$.

$\begin{matrix}\label{41} F(t)&\geq& C_{\gamma}\widetilde{c}K_j^pt^{-(q_2+1+\gamma)-n(l+1)(p-1)} \int_{L_jt_1}^t \int_{L_jt_1}^s \tau^{q_2+1}\int_{L_jt_1}^\tau \sigma^{-p\alpha_j}(\sigma-L_jt_1)^{p\beta_j} {\rm d}\sigma {\rm d}\tau {\rm d}s \\ &\geq & C_{\gamma}\widetilde{c}K_j^pt^{-(q_2+1+\gamma)-n(l+1)(p-1)-p\alpha_j} \int_{L_jt_1}^t \int_{L_jt_1}^s \tau^{q_2+1}\int_{L_jt_1}^\tau (\sigma-L_jt_1)^{p\beta_j} {\rm d}\sigma {\rm d}\tau {\rm d}s \\ &\geq &\frac{C_{\gamma}\widetilde{c}K_j^pt^{-(q_2+1+\gamma)-n(l+1)(p-1)-p\alpha_j}} {(p\beta_j+1)(p\beta_j+q_2+3)} \int_{\frac{t}{l_{j+1}}}^t(s-L_jt_1)^{p\beta_j+q_2+3} {\rm d}s \\ &\geq & \frac{C_{\gamma}\widetilde{c}K_j^p(l_{j+1}-1)t^{-(q_2+1+\gamma)-n(l+1)(p-1)-p\alpha_j} (t-L_{j+1}t_1)^{p\beta_j+q_2+4}}{(p\beta_j+1)(p\beta_j+q_2+3)l_{j+1}^{p\beta_j+q_2+4}}, \end{matrix}$

$$$\label{42} K_{j+1}=\frac{C_{\gamma}\widetilde{c}K_j^p(l_{j+1}-1)} {(p\beta_j+1)(p\beta_j+q_2+3)l_{j+1}^{p\beta_j+q_2+4}},$$$
$$$\label{43} \alpha_{j+1}=(q_2+1+\gamma)+n(l+1)(p-1)+p\alpha_j=p\alpha_j+r,$$$
$$$\label{44} \beta_{j+1}=p\beta_j+q_2+4.$$$

$\begin{matrix}\label{45} F(t)\geq K_{j+1}t^{-\alpha_{j+1}}(t-L_{j+1}t_1)^{\beta_{j+1}}. \end{matrix}$

(3.42)式说明(3.37)式对于 $j+1$成立. 以下将对$\alpha_j,\beta_j,K_j$ 进行估计.

$\begin{matrix}\label{46} \alpha_j&=&p\alpha_{j-1}+r=\cdots=r(1+p+p^2+\cdots+p^{j-1})+\alpha_0p^j \\ &=&(\frac{r}{p-1}+\alpha_0)p^j-\frac{r}{p-1}, \end{matrix}$
$\begin{matrix} \label{47} \beta_{j}&=&p\beta_{j-1}+q_2+4=\cdots=(q_2+4)(1+p+p^2+\cdots+p^{j-1})+\beta_0p^j \\ &=&(\frac{q_2+4}{p-1}+\beta_0)p^j-\frac{q_2+4}{p-1}. \end{matrix}$

$$$\label{48} \lim\limits_{j\rightarrow\infty}l_{j}^{\beta_j}=\lim\limits_{j\rightarrow\infty}(1+p^{-j})^{(\frac{q_2+4}{p-1}+\beta_0)p^j-\frac{q_2+4}{p-1}}={\rm e}^{\frac{q_2+4}{p-1}+\beta_0}.$$$

$\begin{matrix}\label{49} K_{j}&=&\frac{C_{\gamma}\widetilde{c}K_{j-1}^p(l_{j}-1)} {(p\beta_{j-1}+1)(p\beta_{j-1}+q_2+3)l_{j}^{p\beta_{j-1}+q_2+4}} \\ &\geq &C_{\gamma}\widetilde{c}{\rm e}^{-(\frac{q_2+4}{p-1}+\beta_0)}(\frac{q_2+4}{p-1}+\beta_0)^{-2}p^{-3j}K_{j-1}^p=Dp^{-3j}K_{j-1}^p. \end{matrix}$

$\begin{matrix}\label{50} \log K_j&\geq& \log D-3j\log p+p\log K_{j-1}\geq\cdots \\ &\geq&(1+p+p^2+\cdots+p^{j-1})\log D+p^j\log K_0-3(p^{j-1}+2p^{j-2}+\cdots+j)\log p \\ &=&p^j(\frac{\log D}{p-1}+\log K_0-\frac{3p\log p}{(p-1)^2})+\frac{3(p+(p-1)j)\log p}{(p-1)^2}-\frac{\log D}{p-1}, \ \ \forall j\in {\Bbb N}. \end{matrix}$

$j_0=j_0(n,p,l,\mu,\gamma)\in {\Bbb N}$, 满足

$j_0\geq \frac{\log D}{3\log p}-\frac{p}{p-1}.$

$\begin{matrix}\label{51} \log K_j\geq p^j(\frac{\log D}{p-1}+\log K_0-\frac{3p\log p}{(p-1)^2})= p^j\log(E_0\varepsilon^p), \end{matrix}$

$L_j$定义以及数学分析相关理论, 设$L=\lim\limits_{j\rightarrow\infty}L_j=\prod\limits_{j=0}^\infty l_j\in \Bbb R.$$l_j>1, 故j\rightarrow\infty时, 得到L_j\rightarrow L. 另外, 当j\in \Bbb R$$t\geq Lt_1$ 时,(3.37)式成立. 也就是,

$\begin{matrix}\label{52} F(t)\geq K_jt^{-\alpha_j}(t-Lt_1)^{\beta_j}. \end{matrix}$

$\begin{matrix}\label{53} F(t)&\geq& \exp(p^j\log(E_0\varepsilon^p))t^{-((\frac{r}{p-1}+\alpha_0)p^j-\frac{r}{p-1})} (t-Lt_1)^{(\frac{q_2+4}{p-1}+\beta_0)p^j-\frac{q_2+4}{p-1}} \\ &=&\exp(p^j(\log(E_0\varepsilon^p)-(\frac{r}{p-1}+\alpha_0)\log t+(\frac{q_2+4}{p-1}+\beta_0)\log (t-Lt_1))) \\ &&\times t^{\frac{r}{p-1}}(t-Lt_1)^{-\frac{q_2+4}{p-1}}. \end{matrix}$

$t\geq 2Lt_1$时, 可得

$\begin{matrix}\label{54} F(t)&\geq& \exp p^j(\log(E_02^{-(\frac{q_2+4}{p-1}+\beta_0)}\varepsilon^pt^{\frac{q_2+4}{p-1}+\beta_0-(\frac{r}{p-1}+\alpha_0)})) t^{\frac{r}{p-1}}(t-Lt_1)^{-\frac{q_2+4}{p-1}} \\ & =& \exp p^j(\log(E_1\varepsilon^pt^{\frac{q_2+4}{p-1}+\beta_0-(\frac{r}{p-1}+\alpha_0)})) t^{\frac{r}{p-1}}(t-Lt_1)^{-\frac{q_2+4}{p-1}}, \end{matrix}$

(3.51)式指数函数$t$的指数如下

$\begin{matrix}\label{55} & &\frac{q_2+4}{p-1}+\beta_0-(\frac{r}{p-1}+\alpha_0)=\frac{q_2+4-r}{p-1}+\beta_0-\alpha_0 \\ &=&\frac{2(l+1)+(5-2\gamma+\mu+(n-2)(l+1))p-p^2(\mu+l+(n-1)(l+1))}{2(p-1)} \\ &=&\frac{\Theta(n,p,l,\gamma,\mu)}{2(p-1)}. \end{matrix}$

$\varepsilon_0^{-\frac{2p(p-1)}{\Theta(n,p,l,\gamma,\mu)}}\geq 2Lt_1E_1^{\frac{2(p-1)}{\Theta(n,p,l,\gamma,\mu)}}.$

$T(\varepsilon)\leq \check{C}\varepsilon^{-\frac{2p(p-1)}{\Theta(p,n,l,\gamma,\mu)}},$

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