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

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.