Gradient blowup behavior for a viscous Hamilton-Jacobi equation with degenerate gradient nonlinearity

Abstract

The paper is concerned with gradient blowup behavior for a semilinear parabolic equationut−Δu=δm(x)|∇u|p+ΛinΩ×(0,T) with the zero Dirichlet condition. Two results with p>m+2 and m⩾0 are established. One of which is that any gradient blowup solution follows a global ODE type behavior, with domination of normal derivatives over the tangential derivatives. The other is about time-increasing solutions. Zhang and Hu (2010) [50] obtained the precise gradient blowup rate in one-dimensional case, but the higher dimensional case was left as an open problem. Here we solve this problem by establishing the gradient blowup rate, for any small γ>0,C(T−t)−m+1p−m−2⩽‖∇u‖∞⩽Cγ(T−t)−m+1p−m−2−γ for suitable ranges of p and m, which extends the result of (Attouchi and Souplet (2020) [3]) to the case m≠0. As an important by-product which is of independent interests itself, the gradient estimate near boundary for the corresponding elliptic equation is derived under weaker assumptions on the inhomogeneous term.