Local existence and blow-up behavior for the diffusive Hamilton-Jacobi equation in a half-space with unbounded initial data

Abstract

We study the local existence and blow-up behavior of solutions, with possible growth at space infinity, for the diffusive Hamilton-Jacobi equation $u_t-Δu=|\nabla u|^p$ ($p>1$), in a half-space with Dirichlet boundary conditions. Under optimal, polynomial growth assumptions on the initial data, characterized by the critical exponent $p/(p-1)$, we first prove local existence and uniqueness of a maximal classical solution and establish a blow-up alternative. This is complemented by a nonexistence result showing that $p/(p-1)$ is the sharp threshold for admissible growth at infinity. Next, for initial data with subcritical growth, we show global-in-time existence for $1<p\le2$ and that finite-time blow-up can occur only through gradient blow-up for $p>2$, and we derive sharp upper and lower estimates on the gradient, including a precise type~II boundary blow-up rate and a dichotomy in its behavior near the singular time. Finally, in the case $p>2$, for any type~II blow-up solution, we prove convergence after rescaling) to an explicit one-dimensional profile and providing a refined description of the asymptotic singularity formation.