In this paper, we investigate uniform decay estimate of the solution to the Petrovsky equation with memory

with initial conditions and boundary conditions, where g is a memory kernel function. The related energy has been shown to decay exponentially or polynomially as \(t\rightarrow +\infty \) by the theorem established under the assumption \(g'(t)\leqslant -k g^{1+\frac{1}{p}}(t)\) with \(p\in (2,\infty )\) and \(k>0\) in the reference(J Funct Anal 254(5):1342–1372, 2008). Using the ideas introduced by by Lasiecka and Wang (Springer INdAM Series 10, Chapter, vol 14, pp 271–303, 2014), we prove the optimized uniform general decay result under the assumption \(g'(t)+H(g(t))\le 0\), where the function \(H(\cdot )\in C^1({\mathbb {R}}^1)\) is positive, increasing and convex with \(H(0)=0\), which is introduced for the first time by Alabau-Boussouira and Cannarsa (C R Acad Sci Paris Ser I 347:867–872, 2009) and studied systematically by Lasiecka and Wang (Springer INdAM Series 10, Chapter, vol 14, pp 271–303, 2014). The exponential decay result and polynomial decay result in the reference (J Funct Anal 254(5):1342–1372, 2008) are the special cases of this paper by choosing special \(H(\cdot )\).

在本文中，我们研究了具有记忆的 Petrovsky 方程解的均匀衰减估计

具有初始条件和边界条件，其中g是记忆核函数。根据假设\(g'(t)\leqslant -kg^{1+\frac{1}{1}{1}{1}{1}下建立的定理，相关能量已显示为指数或多项式衰减为\(t\rightarrow +\infty \) p}}(t)\)与参考文献中的\(p\in (2,\infty )\)和\(k>0\)（J Funct Anal 254(5):1342–1372, 2008）。使用 Lasiecka 和 Wang（Springer INdAM Series 10, Chapter, vol 14, pp 271–303, 2014）引入的思想，我们证明了在假设\(g'(t)+H(g (t))\le 0\)，其中函数\(H(\cdot )\in C^1({\mathbb {R}}^1)\)是正的，增加的和凸的\(H(0)=0\)，这是由 Alabau-Boussouira 和 Cannarsa 首次引入（CR Acad Sci Paris Ser I 347:867–872, 2009）并由Lasiecka 和 Wang（Springer INdAM 系列 10，第 14 卷，第 271–303 页，2014 年）。参考文献（J Funct Anal 254(5):1342–1372, 2008）中的指数衰减结果和多项式衰减结果是本文选择特殊\(H(\cdot )\)的特例。