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

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.