Extinction and non‐extinction for viscous Hamilton–Jacobi equations in $\mathbb{R}$ N

Extinction in finite time and non‐compactness of the support are investigated for non‐negative classical solutions to the Cauchy problem u_t−Δu+|∇u|^p=0 when p∈(0,1). The occurrence of these phenomena is shown to depend on the behaviour of the initial data u₀ for large values of x. In particular, we obtain the optimal decay rate of u₀ at infinity to ensure finite‐time extinction of u. The case of periodic boundary conditions is also considered. Finally the optimality of temporal L^∞‐decay estimates obtained previously is discussed.