带非局部阻尼项的粘弹性方程解的衰减估计

In this paper,we consider the following viscoelastic equation u tt- △u +∫t 0 g(t-s)△u(s)ds + a(x)u t + u |u|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 → +∞.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.

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

In this paper, we consider the following viscoelastic equation $$u_{tt} -Delta u+int_0^t {g(t-s)Delta u(s)d s} +a(x)u_t +uleft| uright|^r=0$$ 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 $trightarrow +infty$. 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.