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

TL;DR

This paper establishes existence, uniqueness, and stability of solutions for a class of nonlinear evolution equations with exponentially decaying memory, using time discretisation and integral operator techniques.

Contribution

It introduces a novel approach to prove existence and stability for equations with memory without requiring space embedding conditions.

Findings

Global-in-time solutions are proven to exist.

Uniqueness of solutions is established.

Stability results are demonstrated.

Abstract

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