﻿ 轴向力作用下具有记忆项的热弹耦合梁方程组的初边值问题

# 轴向力作用下具有记忆项的热弹耦合梁方程组的初边值问题The Initial Boundary ValueProblem of the Thermoelastic Coupled Beam Equations with Memory Term under Axial Force

Abstract: In this paper, we study the initial boundary value of a thermally coupled beam equation with a memory term under the action of an axial force.

1. 引言

1968年，Gurtin Monton E等 [1] 提出了具有记忆项的热传导理论。之后关于含记忆性的各类非线性梁方程的初边值问题的研究有若干的进展 [2] - [8]，比如2009年，Wang Junmin等 [2] 研究了如下具有记忆项的热传导方程

${\theta }_{t}+{\int }_{0}^{t}g\left(t-\tau \right){\theta }_{xx}\left(x,\tau \right)\text{d}\tau =0$

${u}_{tt}-M\left({|{u}_{x}|}^{2}\right){u}_{xx}+{\int }_{0}^{t}g\left(t-\tau \right){u}_{xx}\left(x,\tau \right)\text{d}\tau =0$

$\left\{\begin{array}{l}{u}_{tt}-{u}_{xx}+\alpha {\theta }_{x}=0\\ {\theta }_{t}-{\theta }_{xx}-{\int }_{0}^{t}g\left(t-\tau \right){\theta }_{xx}\left(x,\tau \right)\text{d}\tau +\beta {u}_{tx}=0\end{array}$

$\left\{\begin{array}{l}{u}_{tt}+{u}_{xxxx}-{\int }_{0}^{t}{k}_{1}\left(t-\tau \right){u}_{xxxx}\left(x,\tau \right)\text{d}\tau -M\left({\int }_{0}^{L}{|{u}_{x}|}^{2}\text{d}x\right){u}_{xx}+\alpha {\theta }_{xx}=q\left(x\right)\\ {\theta }_{t}-{\theta }_{xx}-{\int }_{0}^{t}{k}_{2}\left(t-\tau \right){\theta }_{xx}\left(x,\tau \right)\text{d}\tau -\alpha {u}_{xxt}=0\end{array}$ (1)

$u\left(0,t\right)=u\left(L,t\right)={u}_{xx}\left(0,t\right)={u}_{xx}\left(L,t\right)=0$, $\theta \left(0,t\right)=\theta \left(L,t\right)=0$ (2)

, ${u}_{t}\left(x,0\right)={u}_{1}\left(x\right)$, $\theta \left(x,0\right)={\theta }_{0}\left(x\right)$ (3)

2. 预备知识

${S}_{1}=\left\{u\in {H}^{4}\left(0,L\right)|u,{u}_{xx}\in {H}_{0}^{1}\left(0,L\right)\right\}$

${S}_{2}={H}_{0}^{1}\left(0,L\right)\cap {H}^{2}\left(0,L\right)$

${S}_{1}\subset {S}_{2}\subset {L}^{2}$

(h1) 对函数M，假设 $M\in {C}^{1}\left(\left[0,\infty \right)\right)$，且M为非负函数，有

$M\left(s\right)\ge a+bs\left(a,b>0\right)$, $|{M}^{\prime }\left(s\right)|<\sigma$ (4)

(h2) 假设 ${k}_{1}\left(t\right)\in {C}^{1}\left[\left[0,\infty \right),\left[0,\infty \right)\right]$, ${k}_{1}\left(t\right)\ge 0$, ${{k}^{\prime }}_{1}\left(t\right)\le 0$, ${\int }_{0}^{\infty }{k}_{1}\left(t\right)\text{d}t

(h3) 假设 ${k}_{2}\left(t\right)\in {C}^{1}\left[\left[0,\infty \right),\left[0,\infty \right)\right]$, ${k}_{2}\left(0\right)=0$，且 $1-{\int }_{0}^{\infty }{k}_{2}\left(t\right)\text{d}t=l>0$, $\exists \xi >0$，使得 ${{k}^{\prime }}_{2}\left(t\right)\le -\xi {k}_{2}\left(t\right)$

(h4) 假设 $q\left(x\right)\in {L}^{2}\left(0,L\right)$

3. 主要结论

3.1. 弱解的存在性

$\left\{\begin{array}{l}\frac{\text{d}}{\text{d}t}\left({u}_{t},\omega \right)+\left({u}_{xx},{\omega }_{xx}\right)-{\int }_{0}^{t}{k}_{1}\left(t-\tau \right)\left({u}_{xx},{\omega }_{xx}\right)\text{d}\tau +M\left({‖{u}_{x}\left(t\right)‖}_{2}^{2}\right)\left({u}_{x},{\omega }_{x}\right)+\alpha \left({\theta }_{xx},\omega \right)=\left(q\left(x\right),\omega \right)\\ \frac{\text{d}}{\text{d}t}\left(\theta ,\stackrel{˜}{\omega }\right)+\left({\theta }_{x},{\stackrel{˜}{\omega }}_{x}\right)+{\int }_{0}^{t}{k}_{2}\left(t-\tau \right)\left({\theta }_{x},{\stackrel{˜}{\omega }}_{x}\right)\text{d}\tau -\alpha \left({u}_{xxt},\stackrel{˜}{\omega }\right)=0\end{array}$, $\forall \omega \in {S}_{2}$, $\stackrel{˜}{\omega }\in {H}_{0}^{1}$ (5)

${u}^{m}\left(x,t\right)=\underset{j=1}{\overset{m}{\sum }}{r}_{j}\left(t\right){\omega }_{j}\left(x\right)$, ${\theta }^{m}\left(t\right)=\underset{j=1}{\overset{m}{\sum }}{h}_{j}\left(t\right){\stackrel{˜}{\omega }}_{j}\left( x \right)$

$\left\{\begin{array}{l}\left({u}_{tt}^{m},{\omega }_{j}\right)+\left({u}_{xx}^{m},{\omega }_{jxx}\right)-{\int }_{0}^{t}{k}_{1}\left(t-\tau \right)\left({u}_{xx}^{m},{\omega }_{jxx}\right)\text{d}\tau +M\left({‖{u}_{x}^{m}\left(t\right)‖}_{2}^{2}\right)\left({u}_{x}^{m},{\omega }_{jx}\right)+\alpha \left({\theta }_{xx}^{m},{\omega }_{j}\right)=\left(q\left(x\right),{\omega }_{j}\right)\\ \left({\theta }_{t}^{m},{\stackrel{˜}{\omega }}_{j}\right)+\left({\theta }_{x}^{m},{\stackrel{˜}{\omega }}_{jx}\right)+{\int }_{0}^{t}{k}_{2}\left(t-\tau \right)\left({\theta }_{x}^{m},{\stackrel{˜}{\omega }}_{jx}\right)\text{d}\tau -\alpha \left({u}_{xxt}^{m},{\stackrel{˜}{\omega }}_{j}\right)=0\end{array}$ (6)

$\left\{\begin{array}{l}{u}^{m}\left(x,0\right)={u}_{0}^{m}\to {u}_{0}\in {S}_{2}\\ {u}_{t}^{m}\left(x,0\right)={u}_{1}^{m}\to {u}_{1}\in {L}^{2}\\ {\theta }^{m}\left(x,0\right)={\theta }_{0}^{m}\to {\theta }_{0}\in {L}^{2}\end{array}$ (7)

2) 先验估计

$\left(k\circ \phi \right)\left(t\right)={\int }_{0}^{t}k\left(t-\tau \right){\int }_{0}^{L}|\phi \left(t\right)-\phi \left(\tau \right)|\text{d}x\text{d}\tau$ (8)

$\begin{array}{l}\left({u}_{tt}^{m},{u}_{t}^{m}\right)+\left({u}_{xx}^{m},{u}_{txx}^{m}\right)-{\int }_{0}^{t}{k}_{1}\left(t-\tau \right)\left({u}_{xx}^{m}\left(\tau \right),{u}_{txx}^{m}\left(t\right)\right)\text{d}\tau \\ \text{ }+M\left({‖{u}_{x}^{m}\left(t\right)‖}_{2}^{2}\right)\left({u}_{x}^{m},{u}_{tx}^{m}\right)+\alpha \left({\theta }_{xx}^{m},{u}_{t}^{m}\right)=\left(q\left(x\right),{u}_{t}^{m}\right)\end{array}$ (9)

$\left({\theta }_{t}^{m},{\theta }^{m}\right)+\left({\theta }_{x}^{m},{\theta }_{x}^{m}\right)+{\int }_{0}^{t}{k}_{2}\left(t-\tau \right)\left({\theta }_{x}^{m}\left(\tau \right),{\theta }_{x}^{m}\left(t\right)\right)\text{d}\tau -\alpha \left({u}_{xxt}^{m},{\theta }_{x}^{m}\right)=0$ (10)

$\begin{array}{l}{F}_{m}\left(t\right)+\left({k}_{1}\circ {u}_{xx}^{m}\right)\left(t\right)+{\int }_{0}^{t}\left[{k}_{1}\left(s\right){‖{u}_{xx}^{m}‖}_{2}^{2}-\left({{k}^{\prime }}_{1}\circ {u}_{xx}^{m}\right)\left(s\right)\right]\text{d}s\\ \text{ }+2{\int }_{0}^{t}\left[{\int }_{0}^{s}{k}_{2}\left(s-\tau \right)\left({\theta }_{x}^{m}\left(\tau \right),{\theta }_{x}^{m}\left(s\right)\right)\text{d}\tau \right]\text{d}s+2{\int }_{0}^{t}{‖{\theta }_{x}^{m}\left(s\right)‖}_{2}^{2}\text{d}s\\ ={F}_{m}\left(0\right)+{\int }_{0}^{t}\left(q\left(x\right),{u}_{t}^{m}\right)\text{d}s\end{array}$ (11)

${F}_{m}\left(t\right)={‖{u}_{t}^{m}‖}_{2}^{2}+{‖{u}_{xx}^{m}‖}_{2}^{2}+\stackrel{^}{M}\left({‖{u}_{x}^{m}‖}_{2}^{2}\right)+{‖{\theta }^{m}‖}_{2}^{2}-{\int }_{0}^{t}{k}_{1}\left(s\right)\text{d}s{‖{u}_{xx}^{m}‖}_{2}^{2}$

${F}_{m}\left(0\right)={‖{u}_{1}^{m}‖}_{2}^{2}+{‖{u}_{0xx}^{m}‖}_{2}^{2}+\stackrel{^}{M}\left({‖{u}_{0x}^{m}‖}_{2}^{2}\right)+{‖{\theta }_{0}^{m}‖}_{2}^{2}-{\int }_{0}^{t}{k}_{1}\left(s\right)\text{d}s{‖{u}_{0xx}^{m}‖}_{2}^{2}$

$\begin{array}{l}|{\int }_{0}^{s}{k}_{2}\left(s-\tau \right)\left({\theta }_{x}^{m}\left(\tau \right),{\theta }_{x}^{m}\left(s\right)\right)\text{d}\tau |\\ =|{\int }_{0}^{L}{\int }_{0}^{s}{k}_{2}\left(s-\tau \right){\theta }_{x}^{m}\left(\tau \right){\theta }_{x}^{m}\left(s\right)\text{d}\tau \text{d}x|\\ \le \eta {‖{\theta }_{x}^{m}\left(s\right)‖}_{\text{2}}^{2}+\frac{1}{4\eta }{\int }_{0}^{L}{\left[{\int }_{0}^{s}{k}_{2}\left(s-\tau \right){\theta }_{x}^{m}\left(\tau \right)\text{d}\tau \right]}^{2}\text{d}x\\ \le \eta {‖{\theta }_{x}^{m}\left(s\right)‖}_{\text{2}}^{2}+\frac{1}{4\eta }{\int }_{0}^{s}{k}_{2}\left(\tau \right)\text{d}\tau {\int }_{0}^{s}{k}_{2}\left(s-\tau \right){\int }_{0}^{L}{|{\theta }_{x}^{m}\left(\tau \right)|}^{2}\text{d}x\text{d}\tau \\ \le \eta {‖{\theta }_{x}^{m}\left(s\right)‖}_{\text{2}}^{2}+\frac{\left(1-l\right){k}_{2}\left(0\right)}{4\eta }{\int }_{0}^{s}{‖{\theta }_{x}^{m}\left(\tau \right)‖}_{2}^{2}\text{d}\tau \end{array}$ (12)

$|2{\int }_{0}^{t}\left[{\int }_{0}^{s}{k}_{2}\left(s-\tau \right)\left({\theta }_{x}^{m}\left(\tau \right),{\theta }_{x}^{m}\left(s\right)\right)\text{d}\tau \right]\text{d}s|\le \text{2}\eta {\int }_{0}^{t}{‖{\theta }_{x}^{m}\left(s\right)‖}_{2}^{2}ds$ (13)

$2{\int }_{0}^{t}\left(q\left(x\right),{u}_{t}^{m}\left(s\right)\right)\text{d}s\le 2{\int }_{0}^{t}\left[‖q\left(x\right)‖\cdot ‖{u}_{t}^{m}\left(s\right)‖\right]\text{d}s\le {‖q\left(x\right)‖}_{2}^{2}+{\int }_{0}^{t}{F}_{m}\left(s\right)\text{d}s$ (14)

${F}_{m}\left(t\right)+2\left(1-\eta \right){\int }_{0}^{t}{‖{\theta }_{x}^{m}\left(s\right)‖}_{2}^{\text{2}}\text{d}s\le {F}_{m}\left(0\right)+{‖q\left(x\right)‖}_{2}^{2}+C{\int }_{0}^{t}{F}_{m}\left(s\right)\text{d}s$ (15)

${F}_{m}\left(t\right)+2\left(1-\eta \right){\int }_{0}^{t}{‖{\theta }_{x}^{m}\left(s\right)‖}^{2}\text{d}s\le \left({F}_{m}\left(0\right)+{‖q\left(x\right)‖}_{2}^{2}\right){\text{e}}^{Ct}$

$‖{u}_{t}^{m}‖\le C$, $‖{u}_{xx}^{m}‖\le C$, $‖{\theta }^{m}‖\le C$, ${\int }_{0}^{t}{‖{\theta }_{x}^{m}\left(s\right)‖}_{2}^{\text{2}}\text{d}s.

3) 收敛：

${u}^{m}\to u$${L}^{\infty }\left(0,T;{S}_{2}\right)$ 弱*收敛

${u}_{t}^{m}\to {u}_{t}$${L}^{\infty }\left(0,T;{L}^{2}\right)$ 弱*收敛

${\theta }^{m}\to \theta$${L}^{\infty }\left(0,T;{L}^{2}\right)$ 弱*收敛

${\int }_{0}^{t}{‖{\theta }_{x}^{m}\left(s\right)‖}_{2}^{\text{2}}\text{d}s$${L}^{\infty }\left(0,T;{L}^{2}\right)$ 弱*收敛

${u}^{m}\to u$${L}^{2}\left(0,T\right)$ 中强收敛且几乎处处收敛

$\forall \phi \in {L}^{1}\left(0,T\right)$，有 $\omega \phi \in {L}^{1}\left(0,T;{L}^{2}\right)$，那么有

${\int }_{0}^{T}\left({u}_{xx}^{m},{\omega }_{xx}\right)\phi \text{d}t={\int }_{0}^{T}{\int }_{0}^{L}{u}_{xx}^{m}{\omega }_{xx}\phi \text{d}x\text{d}t={\int }_{0}^{T}\left({u}^{m},{\omega }_{xxxx}\phi \right)\text{d}t\to {\int }_{0}^{T}\left(u,{\omega }_{xxxx}\phi \right)\text{d}t={\int }_{0}^{T}\left({u}_{xx},{\omega }_{xx}\right)\phi \text{d}t$

$\left({u}_{xx}^{m},{\omega }_{xx}\right)\to \left({u}_{xx},{\omega }_{xx}\right)$${L}^{\infty }\left(0,T\right)$ 弱*收敛。

${\int }_{0}^{T}\left({\theta }_{x}^{m},{\stackrel{˜}{\omega }}_{x}\right)\phi \text{d}t={\int }_{0}^{T}{\int }_{0}^{L}{\theta }_{x}^{m}{\stackrel{˜}{\omega }}_{x}\phi \text{d}x\text{d}t={\int }_{0}^{T}\left({\theta }^{m},{\stackrel{˜}{\omega }}_{xx}\phi \right)\text{d}t\to {\int }_{0}^{T}\left(\theta ,{\stackrel{˜}{\omega }}_{xx}\phi \right)\text{d}t={\int }_{0}^{T}\left({\theta }_{x},{\stackrel{˜}{\omega }}_{x}\right)\phi \text{d}t$

$\left({\theta }_{x}^{m},{\omega }_{x}\right)\to \left({\theta }_{x},{\omega }_{x}\right)$${L}^{\infty }\left(0,T\right)$ 弱*收敛。

${\int }_{0}^{t}{k}_{1}\left(t-\tau \right)\left({u}_{xx}^{m},{\omega }_{xx}\right)\text{d}\tau \to {\int }_{0}^{t}{k}_{1}\left(t-\tau \right)\left({u}_{xx},{\omega }_{xx}\right)\text{d}\tau$${L}^{\infty }\left(0,T\right)$ 弱*收敛。

$M\left({‖{u}_{t}^{m}\left(t\right)‖}_{2}^{2}\right)\left({u}_{x}^{m},{\omega }_{x}\right)\to M\left({‖{u}_{x}\left(t\right)‖}_{2}^{2}\right)\left({u}_{x},{\omega }_{x}\right)$${L}^{\infty }\left(0,T\right)$ 弱*收敛。

$\left\{\begin{array}{l}\left({u}_{tt},\omega \right)+\left({u}_{xx},{\omega }_{xx}\right)-{\int }_{0}^{t}{k}_{1}\left(t-\tau \right)\left({u}_{xx},{\omega }_{xx}\right)\text{d}\tau +M\left({‖{u}_{x}\left(t\right)‖}_{2}^{2}\right)\left({u}_{x},{\omega }_{x}\right)=q\left(L\right)\\ \left({\theta }_{t},\stackrel{˜}{\omega }\right)+\left({\theta }_{x},{\stackrel{˜}{\omega }}_{x}\right)-{\int }_{0}^{t}{k}_{2}\left(t-\tau \right)\left({\theta }_{x},{\stackrel{˜}{\omega }}_{x}\right)\text{d}\tau =0\end{array}$ (16)

$\left\{\begin{array}{l}\left({z}_{tt},{\omega }_{j}\right)+\left({z}_{xx},{\omega }_{jxx}\right)-{\int }_{0}^{t}{k}_{1}\left(t-\tau \right)\left({z}_{xx},{\omega }_{jxx}\right)\text{d}\tau +M\left({‖{z}_{x}\left(t\right)‖}_{2}^{2}\right)\left({z}_{x},{\omega }_{jx}\right)+\alpha \left({\gamma }_{xx},{\omega }_{j}\right)=0\\ \left({\gamma }_{t},{\stackrel{˜}{\omega }}_{j}\right)+\left({\gamma }_{x},{\stackrel{˜}{\omega }}_{jx}\right)+{\int }_{0}^{t}{k}_{2}\left(t-\tau \right)\left({\gamma }_{x},{\stackrel{˜}{\omega }}_{jx}\right)\text{d}\tau -\alpha \left({z}_{txx},{\stackrel{˜}{\omega }}_{j}\right)=0\end{array}$

${‖{z}_{t}‖}_{2}^{2}+{‖{z}_{xx}‖}_{2}^{2}+{‖\lambda ‖}_{2}^{2}\le \left\{{‖{u}_{1}-{v}_{1}‖}_{2}^{2}+{‖{u}_{0xx}-{v}_{0xx}‖}_{2}^{2}+{‖{\theta }_{0}-{\eta }_{0}‖}_{2}^{2}\right\}{\text{e}}^{Ct}$ (17)

3.2. 正则性

(h5) ${k}_{1}\left(0\right)=0$, ${{k}^{\prime }}_{1}\left(0\right)=0$, ${{k}^{\prime }}_{2}\left(t\right)\le 0$, ${{k}^{″}}_{2}\left(t\right)\ge 0$

$\left\{\begin{array}{l}\left({u}_{tt},\omega \right)+\left({u}_{xx},{\omega }_{xx}\right)-{\int }_{0}^{t}{k}_{1}\left(t-\tau \right)\left({u}_{xx},{\omega }_{xx}\right)\text{d}\tau +M\left({‖{u}_{x}\left(t\right)‖}_{2}^{2}\right)\left({u}_{x},{\omega }_{x}\right)+\alpha \left({\theta }_{xx},\omega \right)=\left(q\left(x\right),\omega \right)\\ \left({\theta }_{t},\stackrel{˜}{\omega }\right)+\left({\theta }_{x},{\stackrel{˜}{\omega }}_{x}\right)+{\int }_{0}^{t}{k}_{2}\left(t-\tau \right)\left({\theta }_{x},{\stackrel{˜}{\omega }}_{x}\right)\text{d}\tau -\alpha \left({u}_{xxt},\stackrel{˜}{\omega }\right)=0\end{array}$ $\forall \omega \in {S}_{1},\stackrel{˜}{\omega }\in {S}_{2}$ (18)

${u}^{m}\left(x,t\right)=\underset{j=1}{\overset{m}{\sum }}{r}_{j}\left(t\right){\omega }_{j}\left(x\right)$, ${\theta }^{m}\left(t\right)=\underset{j=1}{\overset{m}{\sum }}{h}_{j}\left(t\right){\stackrel{˜}{\omega }}_{j}\left( x \right)$

$\left\{\begin{array}{l}\left({u}_{tt}^{m},{\omega }_{j}\right)+\left({u}_{xxxx}^{m},{\omega }_{j}\right)-{\int }_{0}^{t}{k}_{1}\left(t-\tau \right)\left({u}_{xxxx}^{m},{\omega }_{j}\right)\text{d}\tau -M\left({‖{u}_{x}^{m}\left(t\right)‖}_{2}^{2}\right)\left({u}_{xx}^{m},{\omega }_{j}\right)+\alpha \left({\theta }_{xx}^{m},{\omega }_{j}\right)=\left(q\left(x\right),{\omega }_{j}\right)\\ \left({\theta }_{t}^{m},{\stackrel{˜}{\omega }}_{j}\right)-\left({\theta }_{xx}^{m},{\stackrel{˜}{\omega }}_{j}\right)-{\int }_{0}^{t}{k}_{2}\left(t-\tau \right)\left({\theta }_{xx}^{m},{\stackrel{˜}{\omega }}_{j}\right)\text{d}\tau -\alpha \left({u}_{xxt}^{m},{\stackrel{˜}{\omega }}_{j}\right)=0\end{array}$ (19)

(20)

2) 先验估计

${r}_{j}={r}_{jtt}\left(0\right),{h}_{j}={h}_{jt}\left(0\right)$ 两式分别与方程组(19)中两式做积，对 $j=1,2,3,\cdots ,m$ 求和后，得

${‖{u}_{tt}^{m}\left(0\right)‖}_{2}^{2}+\left({u}_{xxxx}^{m}\left(0\right),{u}_{tt}^{m}\left(0\right)\right)-M\left({‖{u}_{x}^{m}\left(0\right)‖}_{2}^{2}\right)\left({u}_{xx}^{m}\left(0\right),{u}_{tt}^{m}\left(0\right)\right)+\alpha \left({\theta }_{xx}^{m}\left(0\right),{u}_{tt}^{m}\left(0\right)\right)=\left(q\left(x\right),{u}_{tt}^{m}\left(0\right)\right)$ (21)

${‖{\theta }_{t}^{m}\left(0\right)‖}_{2}^{2}+\left({\theta }_{xx}^{m}\left(0\right),{\theta }_{t}^{m}\left(0\right)\right)-\alpha \left({u}_{xxt}^{m}\left(0\right),{\theta }_{t}^{m}\left(0\right)\right)=0$ (22)

${‖{u}_{tt}^{m}\left(0\right)‖}_{2}^{2}\le \left[‖{u}_{0xxxx}^{m}‖+M\left({‖{u}_{0x}^{m}‖}_{2}^{2}\right)‖{u}_{0xx}^{m}‖+\alpha ‖{\theta }_{0xx}‖+‖q\left(x\right)‖\right]‖{u}_{tt}^{m}\left(0\right)‖$ (23)

${‖{\theta }_{t}^{m}\left(0\right)‖}_{2}^{2}\le \left[‖{\theta }_{0xx}^{m}‖+\alpha ‖{u}_{1xx}^{m}‖\right]‖{\theta }_{t}^{m}\left(0\right)‖$ (24)

$\begin{array}{l}{u}_{ttt}^{m}+{u}_{xxxxt}^{m}-\left[{M}^{\prime }\left({‖{u}_{x}^{m}\left(t\right)‖}_{2}^{2}\right)+2M\left({u}_{x}^{m},{u}_{xt}^{m}\right)\right]{u}_{xx}^{m}-M\left({‖{u}_{x}\left(t\right)‖}_{2}^{2}\right){u}_{xxt}^{m}\\ \text{ }-{\int }_{0}^{t}{{k}^{\prime }}_{1}\left(t-\tau \right){u}_{xxxx}^{m}\left(\tau \right)\text{d}\tau -{k}_{1}\left(0\right){u}_{xxxxt}^{m}+\alpha {\theta }_{xxt}^{m}=0\end{array}$ (25)

${\theta }_{tt}^{m}-{\theta }_{xxt}^{m}-{\int }_{0}^{t}{{k}^{\prime }}_{2}\left(t-\tau \right){\theta }_{xx}^{m}\left(\tau \right)\text{d}\tau -{k}_{2}\left(0\right){\theta }_{xx}^{m}-\alpha {u}_{xxt}^{m}=0$ (26)

(25) (26)两式分别与 ${u}_{tt}^{m}$${\theta }_{t}^{m}$ 做积，并从0到t积分，所得两式相加，结合假设(h1)及(8)式，得

$\begin{array}{l}{F}_{m}\left(t\right)+2{\int }_{0}^{t}{\int }_{0}^{s}{{k}^{\prime }}_{1}\left(s-\tau \right)\left({u}_{xxx}^{m}\left(\tau \right),{u}_{ttx}^{m}\left(s\right)\right)\text{d}\tau \text{d}s-\left({k}_{2}\circ {\theta }_{x}^{m}\right)\left(t\right)+{\int }_{0}^{t}\left[-{{k}^{\prime }}_{2}\left(s\right){‖{\theta }_{x}^{m}‖}_{2}^{2}+\left({{k}^{″}}_{2}\circ {\theta }_{x}^{m}\right)\left(s\right)\right]\text{d}s+2{\int }_{0}^{t}{‖{\theta }_{xt}^{m}‖}_{2}^{2}\text{d}s\\ ={F}_{m}\left(0\right)+{\int }_{0}^{t}M\left(\text{||}{u}_{x}\left(t\right){\text{||}}_{2}^{2}\right)\left({u}_{xxt}^{m},{u}_{tt}^{m}\right)+\left[{M}^{\prime }\left({‖{u}_{x}\left(t\right)‖}_{2}^{2}\right)+2M\left({u}_{x}^{m},{u}_{xt}^{m}\right)\right]\left({u}_{xx}^{m},{u}_{tt}^{m}\right)\text{d}s\end{array}$ (27)

$\begin{array}{l}M\left({‖{u}_{x}^{m}‖}_{2}^{2}\right)\left({u}_{xxt}^{m},{u}_{tt}^{m}\right)+\left[{M}^{\prime }\left({‖{u}_{x}^{m}‖}_{2}^{2}\right)+2M\left({u}_{x}^{m},{u}_{xt}^{m}\right)\right]\left({u}_{xx}^{m},{u}_{tt}^{m}\right)\\ \le \frac{1}{2}\sigma \left({‖{u}_{xx}^{m}‖}_{2}^{2}+{‖{u}_{tt}^{m}‖}_{2}^{2}\right)+2M{‖{u}_{x}^{m}‖}_{2}^{2}\cdot {‖{u}_{xt}^{m}‖}_{2}^{2}\cdot \frac{1}{2}\left({‖{u}_{xx}^{m}‖}_{2}^{2}+{‖{u}_{tt}^{m}‖}_{2}^{2}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+M\left({‖{u}_{x}^{m}‖}_{2}^{2}\right)\cdot \frac{1}{2}\left({‖{u}_{xxt}^{m}‖}_{2}^{2}+{‖{u}_{tt}^{m}‖}_{2}^{2}\right)\\ \le C\left({‖{u}_{xx}^{m}‖}_{2}^{2}+{‖{u}_{tt}^{m}‖}_{2}^{2}+{‖{u}_{x}^{m}‖}_{2}^{2}+{‖{u}_{xt}^{m}‖}_{2}^{2}+{‖{u}_{xxt}^{m}‖}_{2}^{2}\right)\\ \le C{F}_{m}\left(t\right)\end{array}$ (28)

$\begin{array}{l}|{\int }_{0}^{s}{{k}^{\prime }}_{1}\left(s-\tau \right)\left({u}_{xxx}^{m}\left(\tau \right),{u}_{xtt}^{m}\left(s\right)\right)\text{d}\tau |\\ \le |{\int }_{0}^{L}{\int }_{0}^{s}{{k}^{\prime }}_{1}\left(s-\tau \right){u}_{xxx}^{m}\left(\tau \right){u}_{xtt}^{m}\left(s\right)\text{d}\tau \text{d}x|\\ \le \eta {‖{u}_{xtt}^{m}\left(s\right)‖}_{2}^{2}+\frac{1}{4\eta }{\int }_{0}^{L}{\left({\int }_{0}^{s}{{k}^{\prime }}_{1}\left(s-\tau \right){u}_{xxx}^{m}\left(\tau \right)\text{d}\tau \right)}^{2}\text{d}x\\ \le \eta {‖{u}_{xtt}^{m}\left(s\right)‖}_{2}^{2}+\frac{1}{4\eta }{\int }_{0}^{s}{{k}^{\prime }}_{1}\left(\tau \right)\text{d}\tau {\int }_{0}^{s}{{k}^{\prime }}_{1}\left(s-\tau \right){\int }_{0}^{L}{|{u}_{xxx}^{m}\left(\tau \right)|}^{2}\text{d}x\\ \le \eta {‖{u}_{xtt}^{m}\left(s\right)‖}_{2}^{2}+\frac{\left(1-l\right){{k}^{\prime }}_{1}\left(0\right)}{4\eta }{\int }_{0}^{s}{‖{u}_{xxx}^{m}\left(\tau \right)‖}_{2}^{2}\text{d}\tau \end{array}$ (29)

$|{\int }_{0}^{t}{\int }_{0}^{s}{{k}^{\prime }}_{1}\left(s-\tau \right)\left({u}_{xxxx}^{m}\left(\tau \right),{u}_{xtt}^{m}\left(s\right)\right)\text{d}s\text{d}\tau |\le \text{2}\eta {\int }_{0}^{t}{‖{u}_{xtt}^{m}\left(s\right)‖}_{\text{2}}^{\text{2}}\text{d}s\le \text{2}\eta {\int }_{0}^{t}{‖{F}_{m}\left(s\right)‖}_{\text{2}}^{\text{2}}\text{d}s$ (30)

${F}_{m}\left(t\right)+2{\int }_{0}^{t}{‖{\theta }_{xt}^{m}‖}_{2}^{2}\text{d}s\le {F}_{m}\left(0\right)+C{\int }_{0}^{t}{F}_{m}\left(s\right)\text{d}s$ (31)

${F}_{m}\left(t\right)+2{\int }_{0}^{t}{‖{\theta }_{xt}^{m}‖}_{2}^{2}\text{d}s\le {F}_{m}\left(0\right){\text{e}}^{Ct}$

$‖{u}_{tt}^{m}‖\le C$, $‖{u}_{txx}^{m}‖\le C$, $‖{\theta }_{t}^{m}‖\le C$, ${\int }_{0}^{t}{‖{\theta }_{xt}^{m}‖}_{2}^{2}\text{d}s\le C$

$\exists {\xi }_{1}\in \left(0,L\right)$，使得 ${u}_{xxx}^{m}\left({\xi }_{1},t\right)=0$

$‖{u}_{xxx}^{m}‖\le \frac{1}{\sqrt{2}}‖{u}_{xxxx}^{m}‖

$\exists {\xi }_{2}\in \left(0,1\right)$，使得 ${\theta }_{x}^{m}\left({\xi }_{1},t\right)=0$

$‖{\theta }_{x}^{m}‖\le \frac{1}{\sqrt{2}}‖{\theta }_{xx}^{m}‖

3) 收敛性：

${u}^{m}\to u$${L}^{\infty }\left(0,T;{S}_{1}\right)$ 弱*收敛。

${u}_{t}^{m}\to {u}_{t}$${L}^{\infty }\left(0,T;{S}_{2}\right)$ 弱*收敛。

${\theta }^{m}\to \theta$${L}^{\infty }\left(0,T;{S}_{2}\right)$ 弱*收敛。

${u}^{m}\to u$${H}^{1}\left(0,T\right)$ 中强收敛且几乎处处收敛。

${u}_{tt}^{m}\to {u}_{tt}$${L}^{\infty }\left(0,T;{L}^{2}\right)$ 弱*收敛。

${\theta }_{t}^{m}\to {\theta }_{t}$${L}^{\infty }\left(0,T;{L}^{2}\right)$ 弱*收敛。

${\int }_{0}^{t}{\theta }_{xt}^{m}\text{d}s\to {\int }_{0}^{t}{\theta }_{xt}\text{d}s$${L}^{\infty }\left(0,T;{L}^{2}\right)$ 弱*收敛

${\theta }^{m}\to \theta$${L}^{2}\left(0,T\right)$ 中强收敛且几乎处处收敛。

4. 结论

1) 在满足初始条件的情况下，系统(1)~(3)弱解存在且唯一。

2) 在满足初始条件的情况下，系统(1) ~ (3)正则解存在且唯一。

NOTES

*通讯作者。

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