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# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

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# Nonlinear Evolution Equations with Exponentially Decaying Memory: Existence via Time Discretisation, Uniqueness, and Stability

André Eikmeier
/ Etienne Emmrich
/ Hans-Christian Kreusler
Published Online: 2018-11-21 | DOI: https://doi.org/10.1515/cmam-2018-0268

## Abstract

The initial value problem for an evolution equation of type ${v}^{\prime }+Av+BKv=f$ is studied, where $A:{V}_{A}\to {V}_{A}^{\prime }$ is a monotone, coercive operator and where $B:{V}_{B}\to {V}_{B}^{\prime }$ induces an inner product. The Banach space ${V}_{A}$ is not required to be embedded in ${V}_{B}$ or vice versa. The operator K incorporates a Volterra integral operator in time of convolution type with an exponentially decaying kernel. Existence of a global-in-time solution is shown by proving convergence of a suitable time discretisation. Moreover, uniqueness as well as stability results are proved. Appropriate integration-by-parts formulae are a key ingredient for the analysis.

MSC 2010: 47J35; 45K05; 34K30; 35K90; 35R09; 65J08; 65M12

Dedicated to Professor Rolf D. Grigorieff on the occasion of his 80th birthday

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Accepted: 2018-10-20

Published Online: 2018-11-21

This work has been supported by the Deutsche Forschungsgemeinschaft through Collaborative Research Center 910 “Control of self-organizing nonlinear systems: Theoretical methods and concepts of application”.

Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

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