## 具有不一定递减核的线性粘弹性波动方程振动传递问题的一般衰减估计

1 青岛科技大学数理学院 山东青岛 266061

2 中国海洋大学数学科学学院 山东青岛 266100

## General Decay for the Transmission Problem of Viscoelastic Waves with not Necessarily Decreasing Kernel

Liu Zhiqing,1, Fang Zhongbo,2

1 School of Mathematics and Physics, Qingdao University of Science and Technology, Shandong Qingdao 266061

2 School of Mathematical Sciences, Ocean University of China, Shandong Qingdao 266100

 基金资助: 山东省自然科学基金.  ZR2019MA072中央高校基本科研基金.  201964008

 Fund supported: the NSF of Shandong Province.  ZR2019MA072the Fundamental Research Funds for the Central Universities.  201964008

Abstract

In this paper, we are concerned with the asymptotic behavior for a transmission problem of viscoelastic waves with not necessarily decreasing kernel. We construct a new Lyapunov functional to derive the general decay estimate. Meanwhile, we present an example to illustrate that the decay rate we obtained includes exponential, algebraic and logarithmic decay etc.

Keywords： Transmission problem ; Not necessarily decreasing kernel ; General decay

Liu Zhiqing, Fang Zhongbo. General Decay for the Transmission Problem of Viscoelastic Waves with not Necessarily Decreasing Kernel. Acta Mathematica Scientia[J], 2021, 41(5): 1428-1444 doi:

## 1 引言

$$$u_{tt}-\Delta u+\int_{0}^{t}g(t-s)\Delta u(s){\rm d}s+au_{t} = 0, \;(x, t)\in\Omega_{1}\times(0, +\infty),$$$

$$$v_{tt}-\Delta v = 0, \;(x, t)\in\Omega_{2}\times(0, +\infty),$$$

$$$\frac{\partial u}{\partial\nu}-\int_{0}^{t}g(t-s)\frac{\partial u(s)}{\partial\nu}{\rm d}s+u_{t} = 0, \;(x, t)\in\Gamma_{2}\times(0, +\infty),$$$

$$$v(x, t) = 0, \;(x, t)\in\Gamma_{0}\times(0, +\infty),$$$

$$$u = v, \;\frac{\partial u}{\partial\nu}-\int_{0}^{t}g(t-s)\frac{\partial u(s)}{\partial\nu}{\rm d}s = \frac{\partial v}{\partial\nu}, \;(x, t)\in\Gamma_{1}\times(0, +\infty)$$$

$$$u(x, 0) = u_{0}(x), \;u_{t}(x, 0) = u_{1}(x), \;x\in\Omega_{1},$$$

$$$v(x, 0) = v_{0}(x), \;v_{t}(x, 0) = v_{1}(x), \;x\in\Omega_{2},$$$

## 2 预备知识及主要结论

(H2)    存在可微正函数$\xi(t)$及常数$\alpha$使得

$g(t)$非负可积且振荡. 同时, 取$\xi(t) = \frac{\rho}{\rho-1}\left(1+\frac{1}{t}\right)$可满足条件(H2).

$\begin{eqnarray} E(t): = \frac{1}{2}\left(\|u_{t}\|_{\Omega_{1}}^{2}+\|v_{t}\|_{\Omega_{2}}^{2}+\|\nabla u\|_{\Omega_{1}}^{2}+\|\nabla v\|_{\Omega_{2}}^{2}\right). \end{eqnarray}$

$\begin{eqnarray} \frac{{\rm d}E(t)}{{\rm d}t} = -a\|u_{t}\|_{\Omega_{1}}^{2}-\|u_{t}\|_{\Gamma_{2}}^{2} +\int_{\Omega_{1}}\nabla u_{t}\cdot\int_{0}^{t}g(t-s)\nabla u(s){\rm d}s{\rm d}x. \end{eqnarray}$

$$${\cal E}(t): = \frac{1}{2}\|u_{t}\|_{\Omega_{1}}^{2}+\frac{1}{2}\|v_{t}\|_{\Omega_{2}}^{2}+\frac{1}{2}\left(1-\int_{0}^{t}g(s){\rm d}s\right)\|\nabla u\|_{\Omega_{1}}^{2}+\frac{1}{2}\|\nabla v\|_{\Omega_{2}}^{2}+\frac{1}{2}(g\diamond\nabla u)(t),$$$

$\begin{eqnarray} \frac{{\rm d}{\cal E}(t)}{{\rm d}t} = -a\|u_{t}\|_{\Omega_{1}}^{2}-\|u_{t}\|_{\Gamma_{2}}^{2} +\frac{1}{2}(g'\diamond \nabla u)(t)-\frac{g(t)}{2}\|\nabla u\|_{\Omega_{1}}^{2}. \end{eqnarray}$

(i) 当$a>0$时, 若$\overline{l}_{\alpha}$充分小, 则存在常数$C, r_{1}>0$使得$E(t)$的衰减速率为

(ii) 当$a = 0$时, 若$g(0)$, $\overline{g} $$\overline{l}_{\alpha} 充分小, 则存在常数 C, r_{2}>0 使得 E(t) 的衰减速率为 ## 3 一般衰减估计 本节中, 我们讨论问题(1.1)–(1.7)能量的一般衰减估计值. 为了证明主要结论, 我们先引入几个引理. 其中 w(t): = u(t)-\int_{0}^{t}g(t-s)u(s){\rm d}s$$ 0<\theta<1$为待定常数.

(i) 当$a>0$时, 有如下微分不等式成立

$\begin{eqnarray} \frac{{\rm d}\Phi(t)}{{\rm d}t}&\leq&\frac{1}{(1-\theta)(1-\overline{g})}\bigg\{4R^{2}\left[2\xi^{2}(0)\overline{g}+2\beta^{2}\overline{l}+g^{2}(0)\right] \\ &&+a^{2}\left[8R^{2}+\lambda_{1}^{2}(N-2\theta)\right]\bigg\}\|u_{t}\|_{\Omega_{1}}^{2}-\frac{(1-\theta)(1-\overline{g})}{2}\|\nabla u\|_{\Omega_{1}}^{2}\\ &&+\left[R+\left(\frac{N}{2}-\theta\right)^{2}\frac{4\lambda^{2}}{(1-\theta)(1-\overline{g})}\right]\|u_{t}\|_{\Gamma_{2}}^{2}\\ &&+\left[\left(\frac{N}{2}-1\right)+\frac{(1-\theta)(1-\overline{g})}{8}+\frac{20}{(1-\theta)(1-\overline{g})}\right](g\diamond \nabla u)(t)\\ & &+\frac{(1-\theta)(1-\overline{g})}{16}(l\diamond \nabla u)(t)-\theta\|v_{t}\|_{\Omega_{2}}^{2}-(1-\theta)\|\nabla v\|_{\Omega_{2}}^{2}, \end{eqnarray}$

(ii) 当$a = 0$时, 有如下微分不等式成立

$\begin{eqnarray} \frac{{\rm d}\Phi(t)}{{\rm d}t}&\leq&-\left\{\theta-\frac{3R^{2}\left[2\xi^{2}(0)\overline{g}+2\beta^{2}\overline{l}+g^{2}(0)\right]}{(1-\theta)(1-\overline{g})}\right\} \|u_{t}\|_{\Omega_{1}}^{2}\\ &&-\frac{(1-\theta)(1-\overline{g})}{2}\|\nabla u\|_{\Omega_{1}}^{2}+\frac{(1-\theta)(1-\overline{g})}{12}(l\diamond \nabla u)(t)\\ &&+\left[R+\left(\frac{N}{2}-\theta\right)^{2}\frac{3\lambda^{2}}{(1-\theta)(1-\overline{g})}\right]\|u_{t}\|_{\Gamma_{2}}^{2}\\ &&+\left[\left(\frac{N}{2}-1\right)+\frac{(1-\theta)(1-\overline{g})}{12}+\frac{15}{(1-\theta)(1-\overline{g})}\right](g\diamond \nabla u)(t)\\ &&-\theta\|v_{t}\|_{\Omega_{2}}^{2}-(1-\theta)\|\nabla v\|_{\Omega_{2}}^{2}, \end{eqnarray}$

由(1.1)式及Green公式, 我们得到

$\begin{eqnarray} & &\frac{{\rm d}}{{\rm d}t}\int_{\Omega_{1}}\left[(m\cdot\nabla w)+\left(\frac{N}{2}-\theta\right) u\right]u_{t}{\rm d}x\\ & = &\int_{\Omega_{1}}(m\cdot\nabla u_{t})u_{t}{\rm d}x-\int_{\Omega_{1}}u_{t}\int_{0}^{t}g'(t-s)\left(m\cdot\nabla u(s)\right){\rm d}s{\rm d}x-g(0)\int_{\Omega_{1}}(m\cdot\nabla u)u_{t}{\rm d}x\\ &&-\int_{\Omega_{1}}\nabla(m\cdot\nabla w)\cdot\nabla w{\rm d}x+\left(\frac{N}{2}-\theta\right)\|u_{t}\|_{\Omega_{1}}^{2}-\left(\frac{N}{2}-\theta\right)\|\nabla u\|_{\Omega_{1}}^{2}\\ &&+\left(\frac{N}{2}-\theta\right)\int_{\Omega_{1}}\nabla u(t)\cdot\int_{0}^{t}g(t-s)\nabla u(s){\rm d}s{\rm d}x\\ &&+\int_{\partial\Omega_{1}}\left[(m\cdot\nabla w)+\left(\frac{N}{2}-\theta\right) u\right]\frac{\partial w}{\partial\nu}{\rm d}\Gamma-a\int_{\Omega_{1}}\left[(m\cdot\nabla w)+\left(\frac{N}{2}-\theta\right) u\right]u_{t}{\rm d}x. \end{eqnarray}$

$\begin{eqnarray} \int_{\Omega_{1}}(m\cdot\nabla u_{t})u_{t}{\rm d}x = -\frac{N}{2}\|u_{t}\|_{\Omega_{1}}^{2}+\frac{1}{2}\int_{\partial\Omega_{1}}(m\cdot\nu)|u_{t}|^{2}{\rm d}\Gamma, \end{eqnarray}$

$\begin{eqnarray} & &-\int_{\Omega_{1}}\nabla(m\cdot\nabla w)\cdot\nabla w{\rm d}x\\ & = &-\int_{\Omega_{1}}\sum\limits_{i, j = 1}^{N}\left[\frac{\partial}{\partial x_{i}}\left(m_{j}\frac{\partial w}{\partial x_{j}}\right)\frac{\partial w}{\partial x_{i}}\right]{\rm d}x\\ & = &-\int_{\Omega_{1}}\sum\limits_{i, j = 1}^{N}\frac{\partial w}{\partial x_{i}}\frac{\partial w}{\partial x_{j}}\frac{\partial m_{j}}{\partial x_{i}}{\rm d}x-\frac{1}{2}\int_{\Omega_{1}}\sum\limits_{i, j = 1}^{N}\frac{\partial}{\partial x_{j}}\left(\frac{\partial w}{\partial x_{i}}\right)^{2}m_{j}{\rm d}x\\ & = &\left(\frac{N}{2}-1\right)\|\nabla w\|_{\Omega_{1}}^{2}-\frac{1}{2}\int_{\partial\Omega_{1}}(m\cdot\nu)|\nabla w|^{2}{\rm d}\Gamma. \end{eqnarray}$

$\begin{eqnarray} &&\frac{{\rm d}}{{\rm d}t}\int_{\Omega_{1}}\left[(m\cdot\nabla w)+\left(\frac{N}{2}-\theta\right) u\right]u_{t}{\rm d}x\\ & = &-\theta\|u_{t}\|_{\Omega_{1}}^{2}-\left(\frac{N}{2}-\theta\right)\|\nabla u\|_{\Omega_{1}}^{2}+\left(\frac{N}{2}-1\right)\|\nabla w\|_{\Omega_{1}}^{2}\\ &&+\left(\frac{N}{2}-\theta\right)\int_{\Omega_{1}}\nabla u(t)\cdot\int_{0}^{t}g(t-s)\nabla u(s){\rm d}s{\rm d}x\\ &&-\int_{\Omega_{1}}u_{t}\int_{0}^{t}g'(t-s)\left(m\cdot\nabla u(s)\right){\rm d}s{\rm d}x-g(0)\int_{\Omega_{1}}(m\cdot\nabla u)u_{t}{\rm d}x\\ &&+\frac{1}{2}\int_{\partial\Omega_{1}}2\left\{\left[(m\cdot\nabla w)+\left(\frac{N}{2}-\theta\right) u\right]\frac{\partial w}{\partial\nu}+(m\cdot\nu)\left(|u_{t}|^{2}-|\nabla w|^{2}\right)\right\}{\rm d}\Gamma\\ &&-a\int_{\Omega_{1}}\left[(m\cdot\nabla w)+\left(\frac{N}{2}-\theta\right) u\right]u_{t}{\rm d}x. \end{eqnarray}$

$\begin{eqnarray} &&\frac{{\rm d}}{{\rm d}t}\int_{\Omega_{2}}\left[(m\cdot\nabla v)+\left(\frac{N}{2}-\theta\right)v\right]v_{t}{\rm d}x\\ & = &-\theta\|v_{t}\|_{\Omega_{2}}^{2}-(1-\theta)\|\nabla v\|_{\Omega_{2}}^{2}+\frac{1}{2}\int_{\partial\Omega_{2}}(m\cdot\widetilde{\nu})\left(|v_{t}|^{2}-|\nabla v|^{2}\right){\rm d}\Gamma\\ &&+\int_{\partial\Omega_{2}}\left[(m\cdot\nabla v)+\left(\frac{N}{2}-\theta\right) v\right]\frac{\partial v}{\partial\widetilde{\nu}}{\rm d}\Gamma, \end{eqnarray}$

$\begin{eqnarray} \frac{{\rm d}\Phi(t)}{{\rm d}t}& = &-\theta\|u_{t}\|_{\Omega_{1}}^{2}-\left(\frac{N}{2}-\theta\right)\|\nabla u\|_{\Omega_{1}}^{2}+\left(\frac{N}{2}-1\right)\|\nabla w\|_{\Omega_{1}}^{2}\\ &&+\left(\frac{N}{2}-\theta\right)\int_{\Omega_{1}}\nabla u(t)\cdot\int_{0}^{t}g(t-s)\nabla u(s){\rm d}s{\rm d}x\\ &&-\int_{\Omega_{1}}u_{t}\int_{0}^{t}g'(t-s)\left(m\cdot\nabla u(s)\right){\rm d}s{\rm d}x-g(0)\int_{\Omega_{1}}(m\cdot\nabla u)u_{t}{\rm d}x\\ &&-\int_{\Gamma_{2}}\left[(m\cdot\nabla w)+\left(\frac{N}{2}-\theta\right) u\right]u_{t}{\rm d}\Gamma+\frac{1}{2}\int_{\Gamma_{2}}(m\cdot\nu)\left(|u_{t}|^{2}-|\nabla w|^{2}\right){\rm d}\Gamma\\ & &-a\int_{\Omega_{1}}\left[(m\cdot\nabla w)+\left(\frac{N}{2}-\theta\right) u\right]u_{t}{\rm d}x-\theta\|v_{t}\|_{\Omega_{2}}^{2}-(1-\theta)\|\nabla v\|_{\Omega_{2}}^{2}\\ &&-\frac{1}{2}\int_{\Gamma_{0}}(m\cdot\widetilde{\nu})|\nabla v|^{2}{\rm d}\Gamma+\int_{\Gamma_{0}}(m\cdot\nabla v)\frac{\partial v}{\partial\widetilde{\nu}}{\rm d}\Gamma. \end{eqnarray}$

$$$\frac{3R^{2}\left[2\xi^{2}(0)\overline{g}+2\beta^{2}\overline{l} +g^{2}(0)\right]}{(1-\theta)(1-\overline{g})}<\frac{\theta}{2},$$$

$$$\overline{l}_{\alpha}<\min\left\{1, \;\;\frac{(1-\theta)(1-\overline{g}){\alpha^{2}}}{4\xi(0)(12\alpha^{2}+7k\alpha+2k^{2})}\mu_{1}\right\},$$$

$$$\begin{array}{ll} \mu_{2} = 1, \\ { } \mu_{2}<\min\left\{\frac{1}{2\left[R+\frac{3}{4}\frac{\lambda^{2}(N-2\theta)^{2}}{(1-\theta)(1-\overline{g})}\right]}, \quad\frac{2\beta}{(1-\theta)(1-\overline{g})}, \frac{\beta}{4\left[\left(\frac{N}{2}-1\right)+\frac{(1-\theta)(1-\overline{g})}{12}+\frac{15}{(1-\theta)(1-\overline{g})} \right]}\right\}. \end{array}$$$

$\begin{eqnarray} \frac{{\rm d}{\cal L}(t)}{{\rm d}t}\leq-C_{2}\xi(t)\left[E(t)+(g\diamond \nabla u)(t)+(L_{\alpha}\diamond \nabla u)(t)\right], \end{eqnarray}$

$\begin{eqnarray} {\cal L}(t)\leq{\cal L}(0)\exp\left\{-r_{2}\int_{0}^{t}\xi(s){\rm d}s \right\}, \;t\geq0. \end{eqnarray}$

## 4 应用举例

(i) 当$a>0$时, 我们可以选取适当的$b, \alpha>0$使得

(ii) 当$a = 0$时, 我们可以选取适当的$b, \alpha>0$使得

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Marzocchi A , Mutõz Rivera J E , Grazia Naso M .

Asymptotic behaviour and exponential stability for a transmission problem in thermoelasticity

Math Method Appl Sci, 2002, 25 (11): 955- 980

Marzocchi A , Grazia Naso M .

Transmission problem in thermoelasticity with symmetry

IMA J Appl Math, 2003, 68 (1): 23- 46

Bastos W D , Raposo C A .

Transmission problem for waves with frictional damping

Electron J Differ Equa, 2007, 2007 (60): 1- 10

Mutõz Rivera J E , Oquendo H P .

The transmission problem of viscoelastic waves

Acta Appl Math, 2000, 62, 1- 21

Andrade D , Fatori L H , Mutõz Rivera J E .

Nonlinear transmission problem with a dissipative boundary condition of memory type

Electron J Differ Eq, 2006, 2006 (53): 1- 16

Alves M S , Raposo C A , Mutõz Rivera J E , Sepulveda M , Villagrán O V .

Uniform stabilization for the transmission problem of the Timoshenko system with memory

J Math Anal Appl, 2010, 369 (1): 323- 345

Li G , Wang D , Zhu B .

Well-posedness and decay of solutions for a transmission problem with history and delay

Electron J Differ Equa, 2016, 2016 (23): 1- 21

Zitouni S , Ardjouni A , Zennir K , Amiar R .

Well-posedness and decay of solution for a transmission problem in the presence of infinite history and varying delay

Nonlinear Studies, 2018, 25 (2): 445- 465

Medjden M , Tatar N E .

Asymptotic behavior for a viscoelastic problem with not necessarily decreasing kernel

Appl Math Comput, 2005, 167 (2): 1221- 1235

Kafini M , Tatar N E .

A decay result to a viscoelastic problem in with an oscillating kernel

J Math Phys, 2010, 51 (7): 073506

Djebabla A , Tatar N E .

Exponential stabilization of the Timoshenko system by a thermal effect with an oscillating kernel

Math Comput Model, 2011, 54 (1/2): 301- 314

Mesloub F , Boulaaras S .

General decay for a viscoelastic problem with not necessarily decreasing kernel

J Appl Math Comput, 2018, 58 (1/2): 647- 665

Ouchenane D , Boulaara S , Mesloub F .

General decay for a class of viscoelastic problem with not necessarily decreasing kernel

Appl Anal, 2019, 98 (9): 1677- 1693

/

 〈 〉