General Decay of Solutions in a Viscoelastic Equation with Nonlinear Localized Damping

DOI：10.3770/j.issn:2095-2651.2012.01.007

 作者 单位 宋小军 西华师范大学数学与信息学院, 四川 南充 637002 曾嵘 重庆大学数学与统计学院, 重庆 401331 穆春来 重庆大学数学与统计学院, 重庆 401331

在本文中我们研究了在 Dirichlet 边界条件下的一类带阻尼项的粘弹性方程的边值问题$$u_{tt} -\Delta u+\int_0^t {g(t-s)\Delta u(s)ds} +a(x)u_t +u\left| u\right|^r=0.$$ 方程的能量函数的的衰减性质与松弛函数在无穷远处的性质密切相关。在文献 [3,7,11] 中，要求松弛函数 $g(t)$ 在无穷远处满足指数或多项式形式的衰减。最近作者 Messaoudi 在文献 [12,13] 中证明了对更广泛的一类松弛函数,能量函数的衰减率相同于松弛函数的衰减率.受他工作的启发,通过引入一个新的摄动能量函数对上面的方程我们也得到了相同也文献 [12,13] 中的结果.

In this paper, we consider the following viscoelastic equation $$u_{tt} -\Delta u+\int_0^t {g(t-s)\Delta u(s)\d s} +a(x)u_t +u\left| u\right|^r=0$$ with initial condition and Dirichlet boundary condition.\,The decay property of the energy function closely depends on the properties of the relaxation function $g(t)$ at infinity. In the previous works of [3,7,11], it was required that the relaxation function $g(t)$ decay exponentially or polynomially as $t\rightarrow +\infty$. In the recent work of Messaoudi [12,13], it was shown that the energy decays at a similar rate of decay of the relaxation function, which is not necessarily dacaying in a polynomial or exponential fashion. Motivated by [12,13], under some assumptions on $g(x)$, $a(x)$ and $r$, and by introducing a new perturbed energy, we also prove the similar results for the above equation.