﻿ 一类带时滞的分数阶脉冲偏微分方程解的振动性质 Oscillation of Certain Impulsive Partial Fractional Differential Equations with Several Delays

Pure Mathematics
Vol. 09  No. 03 ( 2019 ), Article ID: 30471 , 8 pages
10.12677/PM.2019.93063

Oscillation of Certain Impulsive Partial Fractional Differential Equations with Several Delays

Zhuo Qu, Weijie Xu, Anping Liu*

School of Mathematics and Physics, China University of Geosciences, Wuhan Hubei

Received: May 6th, 2019; accepted: May 16th, 2019; published: May 28th, 2019

ABSTRACT

In this paper, the oscillatory properties of a class of impulsive partial fractional differential equations with several delays subject to a class of boundary conditions are investigated. By using differential inequality method, some sufficient conditions for oscillation of the solutions are obtained and an example is given to illustrate the main results.

Keywords:Oscillation, Impulsive, Partial Fractional Differential Equations, Delays

1. 引言

$\left\{\begin{array}{l}{D}_{+,t}^{\alpha }\left[u\left(t,x\right)-u\left(t-\tau ,x\right)\right]\\ =a\left(t\right)h\left(u\left(t,x\right)\right)\Delta u\left(t,x\right)+\underset{i=1}{\overset{m}{\sum }}{a}_{i}\left(t\right){h}_{i}\left(u\left(t-{\tau }_{i}\left(t\right),x\right)\right)\Delta u\left(t-{\tau }_{i}\left(t\right),x\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\underset{j=1}{\overset{n}{\sum }}{q}_{j}\left(t,x\right)\cdot {f}_{j}\left({\int }_{0}^{t}{\left(t-v\right)}^{-\alpha }u\left(v,x\right)\text{d}v\right)-g\left(t,x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ne {t}_{k},\left(t,x\right)\in {R}_{+}×\Omega \equiv G,\\ {D}_{+,t}^{\alpha }u\left({t}_{k}^{+},x\right)-{D}_{+,t}^{\alpha }u\left({t}_{k}^{-},x\right)=\sigma \left({t}_{k},x\right){D}_{+,t}^{\alpha }u\left({t}_{k},x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t={t}_{k},k=1,2,\cdots ,\end{array}$ (1)

$u\left(t,x\right)=\text{0},\left(t,x\right)\in {R}_{+}×\partial \Omega ,t\ne {t}_{k}.$ (2)

(H1)： 为连续函数，且对常数 ${k}_{j}$ ，有 ${f}_{i}\left(u\right)/u\ge {k}_{i}>0$ ，且 $u\ne 0$

(H2)： ${q}_{j}\left(t,x\right)\in PC\left[{R}_{+}×\stackrel{¯}{\Omega },{R}_{+}\right]$ ，且满足

(H3)： $\sigma :{R}_{+}×\stackrel{¯}{\Omega }\to {R}_{+}$ ，且满足 $\sigma \left({t}_{k},x\right)\le {\alpha }_{k}$

(H4)： $h\left(u\right)\in C\left(R,R\right);u{h}^{\prime }\left(u\right)\ge 0$

$\begin{array}{l}y\left(t\right)={\int }_{\Omega }u\left(t,x\right)\text{d}x,\\ V\left(t\right)={\int }_{0}^{t}{\left(t-v\right)}^{-\alpha }y\left(v\right)\text{d}v,\\ G\left(t\right)={\int }_{\Omega }g\left(t,x\right)\text{d}x.\end{array}$ (3)

2. 预备知识

(4)

$V\left(t\right)={\int }_{0}^{t}{\left(t-v\right)}^{-\alpha }y\left(v\right)\text{d}v,t>0$

$\begin{array}{l}\Delta w\left(x\right)+\lambda w\left(x\right)=0,x\in \Omega \\ w\left(x\right)=0,或者\frac{\partial w}{\partial x}+cw\left(x\right)=0,x\in \partial \Omega \end{array}$

$\begin{array}{l}{\omega }^{\prime }\left(t\right)\le {g}_{1}\left(t\right)\omega \left(t\right)+{g}_{2}\left(t\right),\text{\hspace{0.17em}}t\ne {t}_{k},\text{\hspace{0.17em}}t\ge \mu ,\\ \omega \left({t}_{k}^{+}\right)\le \left(1+{a}_{k}\right)\omega \left({t}_{k}\right),\text{\hspace{0.17em}}k=1,2,\cdots ,\end{array}$

$\begin{array}{l}\omega \left(t\right)\le \omega \left({t}_{0}\right)\underset{{t}_{0}<{t}_{l}

3. 主要结果及证明

${D}_{+,t}^{\alpha }y\left(t\right)-{D}_{+,t}^{\alpha }y\left(t-\tau \right)+\underset{j=1}{\overset{n}{\sum }}{k}_{j}{q}_{j}\left(t\right)V\left(t\right)\le -G\left(t\right),$ (7)

$V\left({t}_{k}^{+}\right)-V\left({t}_{k}^{+}-\tau \right)\le \left(1+{\alpha }_{k}\right)\left(V\left({t}_{k}\right)-V\left({t}_{k}-\tau \right)\right),k=1,2,\cdots ,$ (8)

${D}_{+,t}^{\alpha }y\left(t\right)-{D}_{+,t}^{\alpha }y\left(t-\tau \right)+\underset{j=1}{\overset{n}{\sum }}{k}_{j}{q}_{j}\left(t\right)V\left(t\right)\ge -G\left(t\right),$ (9)

$V\left({t}_{k}^{+}\right)-V\left({t}_{k}^{+}-\tau \right)\ge \left(1+{\alpha }_{k}\right)\left(V\left({t}_{k}\right)-V\left({t}_{k}-\tau \right)\right),k=1,2,\cdots ,$ (10)

(I) 当 $t\ne {t}_{k}$ ，对(1)的第一个方程，两边同时对x在有界域 $\Omega$ 上积分得到：

 (11)

$\begin{array}{l}{\int }_{\Omega }h\left(u\right)\Delta u\left(t,x\right)\text{d}x\\ ={\int }_{\partial \Omega }h\left(u\right)\frac{\partial u\left(t,x\right)}{\partial n}\text{d}x-{\int }_{\Omega }{h}^{\prime }\left(u\right){|gradu|}^{2}\text{d}x\\ ={\int }_{\partial \Omega }h\left(u\right)w\left(t,x,u\right)\text{d}x-{\int }_{\Omega }{h}^{\prime }\left(u\right){|gradu|}^{2}\text{d}x\le 0\end{array}$ (12)

${h}_{i}\left(u\left(t-{\tau }_{i}\left(t\right),x\right)\right)\Delta u\left(t-{\tau }_{i}\left(t\right),x\right)\text{d}x\le \text{0}\text{.}$ (13)

$\begin{array}{l}{\int }_{\Omega }\underset{j=1}{\overset{n}{\sum }}{q}_{j}\left(t,x\right)\cdot {f}_{j}\left({\int }_{0}^{t}\left(t-v\right)u\left(v,x\right)\text{d}v\right)\text{d}x\\ \ge \underset{j=1}{\overset{n}{\sum }}{k}_{j}{q}_{j}\left(t\right){\int }_{0}^{t}{\left(t-v\right)}^{-\alpha }y\left(v\right)\text{d}v\\ =\underset{j=1}{\overset{n}{\sum }}{k}_{j}{q}_{j}\left(t\right)V\left(t\right),t\ge \mu .\end{array}$ (14)

 (15)

(II) 当 $t={t}_{k}$ ，对(1)的第二个方程，两边同时对x在有界域 $\Omega$ 上积分

$\underset{\Omega }{\int }u\left({t}_{k}^{+},x\right)\text{d}x-\underset{\Omega }{\int }u\left({t}_{k}^{-},x\right)\text{d}x=\underset{\Omega }{\int }\sigma \left({t}_{k},x,u\left({t}_{k},x\right)\right)\text{d}x$ (16)

$y\left({t}_{k}^{+}\right)=y\left({t}_{k}\right)\underset{\Omega }{\int }\left(1+\sigma \left({t}_{k},x,u\left({t}_{k},x\right)\right)\right)\text{d}x$ (17)

 (18)

$V\left({t}_{k}^{+}-\tau \right)={\int }_{0}^{{t}_{k}^{+}}{\left({t}_{k}^{+}-{v}_{2}\right)}^{-\alpha }\underset{\Omega }{\int }\left(1+\sigma \left(v,x,u\left(v,x\right)\right)\right)\text{d}x\text{d}vV\left({t}_{k}-\tau \right)$ (19)

$V\left({t}_{k}^{+}\right)-V\left({t}_{k}^{+}-\tau \right)\le \left(1+{\alpha }_{k}\right)\left(V\left({t}_{k}\right)-V\left({t}_{k}-\tau \right)\right)$ (20)

${\int }_{{\mu }_{2}}^{\infty }\mathrm{exp}\left(-{\int }_{{t}_{0}}^{t}r\left(\sigma \right)\text{d}\sigma \right)\text{d}s=\infty ,$ (21)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{{\int }_{{\mu }_{1}}^{t}\underset{s<{t}_{l} (22)

 (23)

${\left[V\left(t\right)-V\left(t-\tau \right)\right]}^{\prime }+\Gamma \left(1-\alpha \right)\underset{j=1}{\overset{n}{\sum }}{k}_{j}{q}_{j}V\left(t\right)\le -\Gamma \left(1-\alpha \right)G\left(t\right)\le 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge \mu .$ (24)

$\omega \left(t\right)=V\left(t\right)-V\left(t-\tau \right)$ (25)

${\omega }^{\prime }+\Gamma \left(1-\alpha \right)\underset{j=1}{\overset{n}{\sum }}{k}_{j}{q}_{j}V\left(t\right)\le -\Gamma \left(1-\alpha \right)G\left(t\right)\le 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge \mu .$ (26)

${\omega }^{\prime }+\Gamma \left(1-\alpha \right)\underset{j=1}{\overset{n}{\sum }}{k}_{j}{q}_{j}\left(V\left(t\right)-V\left(t-\tau \right)\right)\le -\Gamma \left(1-\alpha \right)G\left(t\right)\le 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge \mu .$

$\omega {\left(t\right)}^{\prime }+\Gamma \left(1-\alpha \right){k}_{j}{q}_{j}\omega \left(t\right)\le -\Gamma \left(1-\alpha \right)G\left(t\right)\le 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge \mu .$ (27)

$\begin{array}{l}\omega {\left(t\right)}^{\prime }\le -\Gamma \left(1-\alpha \right)\underset{j=1}{\overset{n}{\sum }}{k}_{j}{q}_{j}\omega \left(t\right)-\Gamma \left(1-\alpha \right)G\left(t\right)\le 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\ge \mu ,t\ne {t}_{k},\\ \omega \left({t}_{k}^{+}\right)\le \left(1+{\alpha }_{k}\right)\omega \left({t}_{k}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,2,\cdots \end{array}$

$\begin{array}{l}\omega \left(t\right)\le \omega \left({\mu }_{1}\right)\underset{{\mu }_{1}<{t}_{l} (28)

$\begin{array}{l}\frac{\omega \left(t\right)}{\underset{{\mu }_{1}<{t}_{l}

$t\to \infty$ ，根据条件(22)，由上式可得

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{\omega \left(t\right)}{\underset{{\mu }_{1}<{t}_{l}

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{\stackrel{˜}{\omega }\left(t\right)}{\underset{{\mu }_{1}<{t}_{l}

4. 举例

 (29)

$u\left(t,x\right)=\text{0},\left(t,x\right)\in {R}_{+}×\partial \Omega ,t\ne {t}_{k}.$ (30)

$\alpha =\frac{3}{4},\Omega =\left(0,\frac{\text{5π}}{2}\right),m=n=1,a\left(t\right)={\text{e}}^{-t},{a}_{1}\left(t\right)=\frac{{\text{e}}^{t}}{2},h\left(u\right)={h}_{i}\left(u\right)={u}^{2},$

${\delta }_{1}=\frac{2\text{π}}{3},\tau =\text{π},{\tau }_{1}=\frac{1}{2},{q}_{1}\left(t,x\right)=\frac{1}{\Gamma \left(1-\alpha \right)}\left({x}^{2}+\frac{1}{t}\right),{f}_{1}\left(v\right)=v,$

${\int }_{{\mu }_{2}}^{\infty }\mathrm{exp}\left(-{\int }_{{t}_{0}}^{t}r\left(\rho \right)\text{d}\rho \right)\text{d}s={\int }_{{\mu }_{2}}^{\infty }\mathrm{exp}\left(-{\int }_{{t}_{0}}^{t}\frac{1}{\rho }\text{d}\rho \right)\text{d}s={\int }_{{\mu }_{2}}^{\infty }\frac{{t}_{0}}{t}\text{d}t=\infty ,$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{{\int }_{{\mu }_{1}}^{t}\underset{s<{t}_{l}

Oscillation of Certain Impulsive Partial Fractional Differential Equations with Several Delays[J]. 理论数学, 2019, 09(03): 472-479. https://doi.org/10.12677/PM.2019.93063

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NOTES

*通讯作者。