General energy decay for wave equation with space-time potential and time delay in ${\mathbb {R}}^n$

Abstract

In this paper, we consider the following wave equation

$$\begin{aligned} \left\{ \begin{aligned} u&_{tt} - \varDelta u + a_0 b(t, x)u_t +a_1b(t, x) u_t(t-\tau ) +\vert u\vert ^{p-1}u=0, \quad t>0, \;\; x\in {\mathbb {R}}^n\\ u&(0, x ) = u_0(x), \qquad u_t(0, x) = u_1(x) \qquad x\in {\mathbb {R}}^n\qquad \\ u&_t( t-\tau , x)=h_0(t-\tau , x),\qquad x\in {\mathbb {R}}^n, \quad 0<t<\tau ,\quad \end{aligned} \right. \end{aligned}$$

with space time dependent potential and a time delay in the internal feedback. Under appropriate conditions on the damping coefficients b and the constants $a_0$, $a_1$, we establish a general energy decay result of the solution where the initial data have compact support.