This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation \(u_t-\Delta u=u^p+M|\nabla u|^q\) in \(\Omega \times I\subset \mathbb {R}^N\times \mathbb {R}\) , where \(M>0\) , and \(p,q>1\) . We first establish the local pointwise gradient estimates when q is subcritical, critical and supercritical with respect to p. With these estimates, we can prove the parabolic Liouville-type theorems for time-decreasing ancient solutions. Next, we use Gidas–Spruck type integral methods to prove the Liouville-type theorem for the entire solutions when q is critical. Finally, as an application of the Liouville-type theorem, we use the doubling lemma to derive universal priori estimates for local solutions of parabolic equations with general nonlinearities. Our approach relies on a parabolic differential inequality containing a suitable auxiliary function rather than Keller–Osserman type inequality, which allows us to generalize and extend the partial results of the elliptic equation (Bidaut-Véron et al. in Math. Ann. 378(1–2):13–56, 2020) to the parabolic case.
