Exponential decay of a first order linear Volterra equation

Abstract

We consider the linear Volterra equation of the first order in time $$\dot u(t)+\int_0^t g(s)A u(t-s) d s=0$$ where $A$ is a positive bounded operator on a Hilbert space $H$. The exponential decay of the related energy is shown to occur, provided that the kernel $g$ is controlled by a negative exponential.

Keywords

• volterra equation
• memory kernel
• exponential decay
• heat equation