Global existence and asymptotic behavior for diffusive Hamilton-Jacobi equations with Neumann boundary conditions
Joaquin Dominguez-de-Tena, Philippe Souplet
TL;DR
The paper studies the Neumann problem for the diffusive Hamilton-Jacobi equation $u_t-\Delta u=|\nabla u|^p$ on a bounded domain with $p>1$, proving global existence and that all global solutions converge exponentially to a spatial constant at rate $e^{-\lambda t}$, where $\lambda>0$ is the second Neumann eigenvalue. A key technical advance is a Bernstein-type gradient estimate achieved via a multiplicative perturbation $h=\psi|\nabla u|^2$ and a Robin auxiliary function $\psi$, which controls boundary contributions without requiring convexity of the domain. The results extend to a broad class of gradient nonlinearities $F(\nabla u)$ with suitable growth, including $F(\nabla u)=\exp(|\nabla u|^q)-1$ for $q>1$, and include a uniformity statement: the exponential decay constant can be chosen uniformly for initial data bounded in $X$. The paper also establishes exponential decay of the gradient for small initial data using Neumann semigroup estimates and provides a Lyapunov-based convergence to a constant for global bounded solutions, enriching the long-time behavior theory for nonlinear parabolic equations with Neumann boundary conditions.
Abstract
We investigate the diffusive Hamilton-Jacobi equation $$u_t-\Lap u = |\nabla u|^p$$ with $p>1$, in a smooth bounded domain of $\RN$ with homogeneous Neumann boundary conditions and $W^{1,\infty}$ initial data. We show that all solutions exist globally, are bounded and converge in $W^{1,\infty}$ norm to a constant as $t\to\infty$, with a uniform exponential rate of convergence given by the second Neumann eigenvalue. This improves previously known results, which provided only an upper polynomial bound on the rate of convergence and required the convexity of the domain. Furthermore, we extend these results to a rather large class of nonlinearities $F(\nabla u)$ instead of~$|\nabla u|^p$.