Large time behavior for a viscous Hamilton–Jacobi equation with Neumann boundary condition

We prove the existence and the uniqueness of strong solutions for the viscous Hamilton–Jacobi equation: with Neumann boundary condition, and initial data μ0, a continuous function. The domain Ω is a bounded and convex open set with smooth boundary, and p>0. Then, we study the large time behavior of the solution and we show that for p (0,1), the extinction in finite time of the gradient of the solution occurs, while for p 1 the solution converges uniformly to a constant, as t→∞.