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

TL;DR

This paper proves the global existence of solutions for reaction-diffusion equations with state-dependent delays and fast-growing nonlinearities, under relaxed continuity conditions and dissipative structural assumptions.

Contribution

It introduces new conditions that ensure global solutions for reaction-diffusion equations with complex nonlinearities and less restrictive delay function assumptions.

Findings

Established global existence under polynomial growth conditions.

Relaxed continuity assumptions on delay functions.

Applicable to equations with fast-growing nonlinearities.

Abstract

This work aims to study the initial-boundary value problem of the reaction-diffusion equation with state-dependent delay $\pa_{t}u-\Delta u=f(u)+g(u,u(t-\tau(t,u_t)))+h(t,x)$ 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.