Global Existence, Regularity, and Dissipativity of Reaction-diffusion Equations with State-dependent Delay and Supercritical Nonlinearities

This work aims to study the initial-boundary value problem of the reaction-diffusion equation ${\text{pa}}_{t}u-\mathrm{\Delta }u=f(u)+g(u(t-\tau (t,u_t)))+h(t,x)$ in a bounded domain with state-dependent delay and supercritical nonlinearities. We establish the global existence and discuss the regularity and dissipativity of the problem under weaker assumptions. In particular, the existence of a global pullback attractor is proved regardless of uniqueness.