Search

The paper is concerned with gradient blowup behavior for a semilinear parabolic equation u t − Δ u = δ m (x) | ∇ u | p + Λ in Ω × (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 + 1 p − m − 2 ⩽ ‖ ∇ u ‖ ∞ ⩽ C γ (T − t) − m + 1 p − 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. [ABSTRACT FROM AUTHOR]