Nonlinear evolution equations with exponentially decaying memory: Existence via time discretisation, uniqueness, and stability

2018/06/17 by Eikmeier, André, Emmrich, Etienne, Kreusler, Hans-Christian
#34K30 #35K90 #35R09 #45K05 #47J35 #65J08 #65M12 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.06353

Abstract

The initial value problem for an evolution equation of type v' + Av + BKv = f is studied, where A:V_A → V_A' is a monotone, coercive operator and where B:V_B → V_B' 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.

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