Classification of entire and ancient solutions of the diffusive Hamilton-Jacobi equation

TL;DR

This work provides a comprehensive Liouville-type and rigidity analysis for the diffusive Hamilton–Jacobi equation $u_t-\Delta u=|\nabla u|^p$ under Dirichlet conditions, addressing both the whole space and the half-space. The authors establish optimal Liouville-type theorems for ancient solutions in $\mathbb{R}^n$ and classify entire and ancient solutions in the half-space, showing stationarity and one-dimensionality under sharp growth restrictions, while revealing nonstationary ancient half-space solutions for all $p>1$ without such constraints. A new pair of local estimates—Li–Yau type additive bounds and Bernstein gradient bounds—drive the analysis, enabling integral and pointwise control, positivity results, and a translation-compactness framework that yields full classification results and the construction of backward and forward self-similar solutions. The combination of delicate integral estimates, translation-compactness arguments, and optimal local bounds sharpens our understanding of GBU-type phenomena, asymptotic profiles, and rigidity in nonlinear parabolic problems with gradient nonlinearity. Overall, the paper advances both the qualitative theory and the toolkit for analyzing Liouville-type properties in nonlinear diffusion equations with first-order terms.

Abstract

Consider the diffusive HJ eq. with Dirichlet conditions, which arises in stochastic control as well as in KPZ type models of surface growth. It is known that, for $p>2$ and suitably large, smooth initial data, the sol. undergoes finite time gradient blowup on the boundary. On the other hand, Liouville type rigidity or classif. ppties play a central role in the study of qualitative behavior in nonlinear elliptic and parabolic problems, and notably appear in the famous BCN conjecture about one-dimensionality of solutions in a half-space. With this motivation, we study the Liouville type classif. and symmetry ppties for entire and ancient sol. in $\R^n$ and in a half-space with Dirichlet B.C. - First, we show that any ancient sol. in $\R^n$ with sublinear upper growth at infinity is necessarily constant. This result is {\it optimal}, in view of explicit examples and solves a long standing open problem. - Next we turn to the half-space problem for $p>2$ and we completely classify entire solutions: any entire sol. is stationary and one-dimensional. The assumption is sharp in view of explicit examples for $p=2$. - Then we show that the situation is also completely different for ancient sol. in a half-space: there exist nonstationary ancient sol. for all $p>1$. Nevertheless, we show that any ancient sol. is necessarily positive, and that stationarity and one-dimensionality are recovered provided a -- close to optimal -- polynomial growth restriction is imposed on the sol. - In addition we establish new and optimal, local estimates of Bernstein and Li-Yau type. The proofs of the Liouville and classif. results are delicate, based on integral estimates, a translation-compactness procedure and comparison arguments, combined with our Bernstein and Li-Yau type estimates.

Theorems & Definitions (59)

• Conjecture 1.1
• Theorem 1
• Theorem 2
• Remark 2.1
• Definition 2.1
• Theorem 3
• Remark 2.2
• Theorem 4
• Theorem 5
• Theorem 6