General energy decay for a viscoelastic wave equation with space-time damping coefficient in $\mathbb{R}^n$

Document Type : Original Article

Authors

Paul Adewale Ogbiyele ¹

Salim A Messaoudi ²

¹ Department of Mathematics, University of Ibadan, Ibadan, Nigeria

² Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

10.22034/kjm.2023.350018.2587

Abstract

In this paper, we consider the following viscoelastic wave equation
\begin{equation}\small\nonumber
\left\{
\begin{split}
&u_{tt} -\bigl(\Delta u-\int^t_0 g(t-s)\Delta u(s) ds\bigr) + b(t , x) u_t +|u|^{p-1} u =0,\quad t >0, \; x\in \mathbb{R}^n\\
&u(0 , x ) = u_0(x), \qquad u_t(0 , x) = u_1(x), \qquad x\in \mathbb{R}^n,
\end{split}
\right.
\end{equation}
with space-time dependent potential and where the initial data $u_0(x)$, $u_1(x)$ have compact supports. Under suitable assumptions on the potential $b$ and for a relaxation function $g$ satisfying the condition $g^{\prime}(t) \leq -\mu(t) g^r(t),\quad t\geq 0,\; 1< r<\frac{3}{2}$, we obtain a general energy decay result that extends other results in the literature.

Keywords

Energy decay

damping potential

relaxation function

Subjects

• 35 Partial differential equations

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