Optimized Uniform Decay Estimate of the Solution to Petrovsky Equation with Memory

Abstract

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

$$\begin{aligned} u_{tt}+\Delta ^2u-\int _ 0^t g(t-s)\Delta ^2u(s)ds=0 \end{aligned}$$

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 $H(\cdot )$.