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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]

The paper is devoted to investigating a semilinear parabolic equation with a nonlinear gradient source term: ut = uxx + xm|ux|p, t > 0, 0 < x < 1,where p > m + 2, m ≥ 0. Zhang and Hu [Discrete Contin. Dyn. Syst. 26 (2010) 767-779] showed that finite time gradient blowup occurs at the boundary and the accurate blowup rate is also obtained for super-critical boundary value. Throughout this paper, we present a complete large time behavior of a classical solution : is global and converges to the unique stationary solution in norm for subcritical boundary value, and blows up in infinite time for critical boundary value. Gradient growup rate is also established by the method of matched asymptotic expansions. In addition, gradient estimate of solutions is obtained by the Bernstein-type arguments. [ABSTRACT FROM AUTHOR]