Unified stability analysis for a Volterra integro-differential equation under creation time perspective

Abstract

Many real-world applications are modeled by Volterra integral–differential equations of the form

$$\begin{aligned} u_{tt}-\Delta u +\int \limits _{\alpha }^{t}g(t-s)\Delta u(s)\, \mathrm{d}s = 0 \;\; \text{ in } \;\; \Omega \times (0,\infty ), \end{aligned}$$

where $\Omega$ is a bounded domain of ${\mathbb {R}}^N$ and g is a memory kernel. Our main concern is with the concept of so-called creation time, the time $\alpha$ where past history begins. Separately, the cases $\alpha =-\infty$ (history) and $\alpha =0$ (null history) were extensively studied in the literature. However, as far as we know, there is no unified approach with respect to the intermediate case $-\infty< \alpha <0$. Therefore we provide new stability results featuring (i) uniform and general stability when the creation time $\alpha$ varies over full range $(-\infty ,0)$ and (ii) connection between the history and the null history cases by means of a rigorous backward ($\alpha \rightarrow -\infty$) and forward ($\alpha \rightarrow 0^-$) limit analysis.