论文标题
粘弹性波方程的溶液的能量衰减速率，具有可变指数和弱阻尼的速率
Energy decay rates of solutions to a viscoelastic wave equation with variable exponents and weak damping
论文作者
论文摘要
本文的目的是研究具有可变指数\ [U_ {u_ {tt}-ΔU+\ int_0^tg（t-s-s）ΔU（s）ds+a | u_t | u_t | u_t |^{m（x）-2} -2} u_t = b | u | u | u | u | u | u |^u |^u |^u_t（p） - 指数$ p（x）$和$ m（x）$具有功能，$ a，〜b> 0 $是常数。更准确地说，在条件下，$ g'（t）\ le -ξ（t）g（t）$，此处$ξ（t）：\ Mathbb {r}^+\ to \ to \ mathbb {r}^+$是具有$ξ（0）> 0，〜\ fint_0^^\ ferty Every Essef $ noind increas nistriage nestiand = ferty necive $ nestive。此外，当$ g $衰减在多个方面时，也分别获得了指数和多项式衰减率。这项工作概括并改善了文献中的早期结果。
The goal of the present paper is to study the asymptotic behavior of solutions for the viscoelastic wave equation with variable exponents \[ u_{tt}-Δu+\int_0^tg(t-s)Δu(s)ds+a|u_t|^{m(x)-2}u_t=b|u|^{p(x)-2}u\] under initial-boundary condition, where the exponents $p(x)$ and $m(x)$ are given functions, and $a,~b>0$ are constants. More precisely, under the condition $g'(t)\le -ξ(t)g(t)$, here $ξ(t):\mathbb{R}^+\to\mathbb{R}^+$ is a non-increasing differential function with $ξ(0)>0,~\int_0^\inftyξ(s)ds=+\infty$, general decay results are derived. In addition, when $g$ decays polynomially, the exponential and polynomial decay rates are obtained as well, respectively. This work generalizes and improves earlier results in the literature.
