We investigate the long-term dynamical behavior of the partially viscoelastic wave equation subject to a localized frictional damping, given by$u_{tt}-\Delta u+{\int }_0^{t}g(t-s)\mathrm{div}\left(a\left(x\right)\nabla u\left(s\right)\right)ds+b\left(x\right)g_1\left(u_t\right)=0,\mathrm{in}\Omega \times {\mathbb{R}}^{+},$where g denotes the memory kernel, $a\left(x\right)\in C^1\left(\overline{\Omega }\right),b\left(x\right)\in L^{\infty }\left(\Omega \right)$ are nonnegative functions satisfying the assumption$a\left(x\right)+b\left(x\right)\ge 2\delta >0,\forall x\in {\omega }_0,$ω₀ is a subset of Ω, and b(x)g₁(u_t) denotes the frictional damping. Under as less as possible restrictions imposed on memory kernel g(·) and some geometric condition on the subset ω₀, we show that there does not exist bifurcation and chaos for this physical model and actually the energy of the solution for the equation above decays definitely to zero with uniform decay rate as the time goes to infinity. Moreover, such a uniform decay rate is determined by the solution of an ordinary differential equation.

我们研究了局部粘弹性阻尼作用下的部分粘弹性波动方程的长期动力学行为，其公式为${ü}_{ŤŤ}-\Delta ü+{\int }_0^{Ť}G（Ť-s）\mathrm{div}（\mathrm{一种}（X）\nabla ü（s））ds+b（X）G_{\mathrm{1个}}（{ü}_{Ť}）=0，\mathrm{在}\Omega \times {\mathbb{[R}}^{+}，$其中g表示内存内核，$\mathrm{一种}（X）\in C^{\mathrm{1个}}（\overline{\Omega }），b（X）\in {\mathrm{大号}}^{\infty }（\Omega ）$ 是满足假设的非负函数$\mathrm{一种}（X）+b（X）\ge 2\delta >0，\forall X\in {\omega }_0，$ω ₀是Ω的子集，和b（X）克₁（Ú_吨）表示摩擦阻尼。下尽可能少地限制施加在存储器内核克（·），并且在子集某些几何条件ω ₀，我们表明，不存在对于该物理模型的分岔与混沌和实际解决方案的能量为上述衰变方程时间到无穷远，绝对会以零衰减率归零。而且，这种均匀的衰减率是由一个常微分方程的解确定的。