Affiliation:
1. Institut für Mathematik , Technische Universität Berlin , Straße des 17. Juni 136, 10623 Berlin , Germany
Abstract
Abstract
The initial value problem for an evolution equation of type
v
′
+
A
v
+
B
K
v
=
f
{v^{\prime}+Av+BKv=f}
is studied, where
A
:
V
A
→
V
A
′
{A:V_{A}\to V_{A}^{\prime}}
is a monotone, coercive operator and where
B
:
V
B
→
V
B
′
{B:V_{B}\to V_{B}^{\prime}}
induces an inner product. The Banach space
V
A
{V_{A}}
is not required to be embedded in
V
B
{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.
Subject
Applied Mathematics,Computational Mathematics,Numerical Analysis
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