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

The paper is concerned with gradient blowup behavior for a semilinear parabolic equation u(t) - Delta u = delta(m)(x)vertical bar del u vertical bar(p) + Lambda in Omega x (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 gamma > 0, C(T - t) (-m+1/p-m-2) <= parallel to del u parallel to(infinity) <= C-gamma (T - t) (-m+1/p-m-2-gamma) for suitable ranges of p and m, which extends the result of (Attouchi and Souplet (2020) [3]) to the case m not equal 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. (c) 2023 Elsevier Inc. All rights reserved.