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 ut−Δ u=|∇ 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≤2 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.
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BibTeX
@misc{chabi2026local,
title = {{Local existence and blow-up behavior for the diffusive Hamilton-Jacobi equation in a half-space with unbounded initial data}},
author = {Loth Damagui Chabi},
howpublished = {HAL},
year = {2026},
month = jun,
eprint = {2606.25624},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05801886v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local existence and blow-up behavior for the diffusive Hamilton-Jacobi equation in a half-space with unbounded initial data
AU - Loth Damagui Chabi
PY - 2026
DA - 2026/06/24
PB - HAL
UR - https://hal.science/hal-05801886v1
AB - 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−Δ u=|∇ 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≤2 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.
ER -