Exponential decay of a first order linear Volterra equation

Monica Conti; Filippo Dell'Oro; Vittorino Pata; Monica Conti; Filippo Dell'Oro; Vittorino Pata

doi:10.3934/mine.2020021

Mathematics in Engineering

2020, Volume 2, Issue 3: 459-471. doi: 10.3934/mine.2020021

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Exponential decay of a first order linear Volterra equation

Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy

Received: 10 February 2020 Accepted: 04 March 2020 Published: 09 March 2020

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.
- Volterra equation,
- memory kernel,
- exponential decay,
- heat equation
Citation: Monica Conti, Filippo Dell'Oro, Vittorino Pata. Exponential decay of a first order linear Volterra equation[J]. Mathematics in Engineering, 2020, 2(3): 459-471. doi: 10.3934/mine.2020021

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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.

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