Asymptotic stability of solutions to the Hamilton–Jacobi equation.

Abstract This paper is concerned with the asymptotic behavior of the solution for the generalized Hamilton–Jacobi equation u t + f (D u) = Δ u. It is known that one dimensional equation v t + C 0 v x 2 = v x x has a self-similar solution v (x 1 + t ) if v + ≠ v − where v + = v (t , + ∞) , v − = v (t , − ∞). This kind of solution is called planar diffusion wave in multi-dimension (M-D). In this paper, it is shown that under some smallness conditions, the solutions to the generalized Hamilton–Jacobi equation converge to the above diffusion wave v (x 1 + t ) with C 0 = 1 2 ∂ 2 f (ξ 1 , ξ 2 , ⋯ , ξ n) ∂ ξ 1 2 | ξ 1 = 0 , as time tends to infinity. The convergence rate in time is also obtained. [ABSTRACT FROM AUTHOR]