Delayed diffusion with measure-valued kernels in nonlinear parabolic equations

We study nonlinear parabolic equations with delayed diffusion terms governed by finite signed measure kernels. The atom of the kernel at the origin is absorbed into the present-time operator, while the remaining part is treated as a residual delay kernel. Under structural assumptions on the effective present-time operators and a pathwise coercivity condition for the total memory operator, we prove the existence and uniqueness of weak solutions and their stability under weak-star convergence of the kernels. The stability result covers collapsing delayed atoms, whose mass is transferred to the present-time diffusion coefficient in the limit. We verify the assumptions for p-Laplacian type examples, including separated kernels and a regularized class of kernels reaching the origin.