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

TL;DR

This paper classifies entire and ancient solutions of the diffusive Hamilton-Jacobi equation, revealing conditions under which solutions are constant, stationary, or one-dimensional, and establishing new estimates for their behavior.

Contribution

It provides a complete classification of solutions in various settings and introduces optimal estimates, solving longstanding open problems in the field.

Findings

Ancient solutions with sublinear growth are necessarily constant.

Entire solutions in a half-space are stationary and one-dimensional.

Existence of nonstationary ancient solutions for all p > 1.

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.