A priori estimates and Liouville-type theorems for the semilinear parabolic equations involving the nonlinear gradient source

This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation u(t)-Delta u=u(p)+M|del u|(q) in Omega xI subset of R(N)xR, 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-V & eacute;ron et al. in Math. Ann. 378(1-2):13-56, 2020) to the parabolic case.