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Computational Methods in Applied Mathematics

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

André EikmeierORCID iD: http://orcid.org/0000-0002-0270-6491 / Etienne EmmrichORCID iD: http://orcid.org/0000-0001-9869-0334 / Hans-Christian Kreusler
Published Online: 2018-11-21 | DOI: https://doi.org/10.1515/cmam-2018-0268


The initial value problem for an evolution equation of type v+Av+BKv=f is studied, where A:VAVA is a monotone, coercive operator and where B:VBVB induces an inner product. The Banach space VA is not required to be embedded in VB 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.

Keywords: Nonlinear Evolution Equation; Monotone Operator; Volterra Operator; Exponentially Decaying Memory; Existence; Uniqueness; Stability; Time Discretisation; Convergence

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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About the article

Received: 2018-06-20

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, DOI: https://doi.org/10.1515/cmam-2018-0268.

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