LAUSR

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.