Global Estimates and Regularity of Retarded Parabolic Equations with Fast-growing Nonlinearities

Abstract

This paper is concerned with global estimates and regularity of solutions for the initial value problem of the retarded parabolic equation $$\frac{\patial u}{\patial t}-\Delta u=f(x,u)+g(u(x,t-r_1(t)),\cdots,u(x,t-r_m(t)))+h(x,t)$$ in a bounded domain $\Omega\subset R^n$ with fast-growing nonlinearities and a dissipative structure, which is associated with the homogeneous Dirichlet boundary condition. Our results reveal some deeper inherent connections between dissipative structures and the regularity of solutions for such problems.