Global Existence for Reaction-diffusion Equations with State-Dependent Delay and Fast-growing Nonlinearities

This work aims to study the initial-boundary value problem of the reaction-diffusion equation with state-dependent delay ${\text{pa}}_{t}u-\mathrm{\Delta }u=f(u)+g(u,u(t-\tau (t,u_t)))+h(t,x)$ in a bounded domain. We establish the global existence of the problem under suitable dissipative-type structural conditions, allowing both nonlinear terms $f$ and $g$ to have arbitrary polynomial growth rates. Another highlight in this work is that, we significantly relax the continuity assumptions imposed on the delay functions.