Kernel-Robust Dynamics for Reaction-Diffusion Equations with Measure-Valued Delay
Abstract
We study reaction-diffusion equations with finite signed measure delays and nonlinear, possibly nonlocal feedback. We treat abstract reactions that are Lipschitz on bounded L2-balls with quadratic coercivity, and pointwise cubic polynomials with positive leading coefficient in spatial dimensions at most three. For affine-growth feedback and time-independent forcing in H−1(Ω), both classes generate global weak solution semiflows on X=C([−r,0];L2(Ω)). Besides total-variation Lipschitz stability, we obtain quantitative weak-star stability with modulus ϖ(d)=dlog(e+d−1) for the abstract class and ϖ(d)1/2 for cubic reactions, where d is the bounded-Lipschitz distance between delay measures. The estimates are uniform on bounded sets of continuous histories and yield concentration and atomic-quadrature rates. A damping condition gives common compact absorption for the abstract class. Cubic coercivity removes that smallness condition on every fixed total-variation-bounded kernel class. In both cases the global attractors are upper semicontinuous in X and in C([−r,0];H01(Ω)).
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BibTeX
@misc{shikhman2026kernelrobust,
title = {{Kernel-Robust Dynamics for Reaction-Diffusion Equations with Measure-Valued Delay}},
author = {Lennon Shikhman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.04195},
eprint = {2606.04195},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.04195},
}
RIS
TY - UNPB
TI - Kernel-Robust Dynamics for Reaction-Diffusion Equations with Measure-Valued Delay
AU - Lennon Shikhman
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2606.04195
UR - https://arxiv.org/abs/2606.04195
AB - We study reaction-diffusion equations with finite signed measure delays and nonlinear, possibly nonlocal feedback. We treat abstract reactions that are Lipschitz on bounded L^2-balls with quadratic coercivity, and pointwise cubic polynomials with positive leading coefficient in spatial dimensions at most three. For affine-growth feedback and time-independent forcing in H^−1(Ω), both classes generate global weak solution semiflows on X=C([−r,0];L^2(Ω)). Besides total-variation Lipschitz stability, we obtain quantitative weak-star stability with modulus ϖ(d)=dlog(e+d^−1) for the abstract class and ϖ(d)^1/2 for cubic reactions, where d is the bounded-Lipschitz distance between delay measures. The estimates are uniform on bounded sets of continuous histories and yield concentration and atomic-quadrature rates. A damping condition gives common compact absorption for the abstract class. Cubic coercivity removes that smallness condition on every fixed total-variation-bounded kernel class. In both cases the global attractors are upper semicontinuous in X and in C([−r,0];H_0^1(Ω)).
ER -