Weighted well-posedness and kernel stability for coercive evolution equations with measure-valued delays

Abstract

We consider coercive evolution equations with measure-valued delay on a Gelfand triple. The delayed feedback is induced by a bounded form on $V$ and may contain principal spatial derivatives, so it is naturally $V^*$-valued rather than bounded on the pivot space. For every finite signed Borel kernel with no atom at the origin, an exponential weight makes the causal history operator contractive relative to the coercive parabolic solution operator. This yields finite-time well-posedness without a smallness condition on the total variation and without any positivity assumption on the delayed form. An asymmetric residual identity gives a total-variation Lipschitz estimate for signed kernels and strong stability for non-negative retarded kernels under narrow, equivalently weak, convergence of finite measures. A delayed-diffusion realisation shows that the framework genuinely covers principal-part delays, and narrow stability yields qualitative distributed-to-discrete convergence at every fixed positive lag.