Global existence and asymptotic behavior for diffusive Hamilton–Jacobi equations with Neumann boundary conditions

We investigate the diffusive Hamilton–Jacobi equation $$\begin{aligned} u_t-\Delta u = |\nabla u|^p \end{aligned}$$ with $$p>1$$ , in a smooth bounded domain of $${\mathbb {R}^n}$$ 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 $$t\rightarrow \infty $$ , 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  $$|\nabla u|^p$$ .