Extinction and decay estimates for viscous Hamilton-Jacobi equations in ℝ^{ℕ}

Abstract

We consider non-negative and integrable classical solutions to the Cauchy problem u t − Δ u + | ∇ u | p = 0 u_t-\Delta u+\vert \nabla u\vert ^p=0 when p ∈ ( 0 , + ∞ ) p\in (0,+\infty ) . For p ∈ ( 0 , N / ( N + 1 ) ) p\in (0,N/(N+1)) we prove that any such solution vanishes identically after a finite time. For higher values of p p temporal decay estimates are obtained.