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**Asymptotic solutions with slow convergence rate of Hamilton-Jacobi equations in Euclidean \(n\) space.**
*(English)*
Zbl 1212.35208

Summary: In this paper, we study the long-time asymptotics of the Cauchy problem for the Hamilton-Jacobi equation \(u_t(x,t)+\alpha x\cdot Du(x,t)+H(Du(x,t))=f(x)\) in \(\mathbb R^n(0,\infty )\), where \(\alpha \) is a positive constant. In [Indiana Univ.Math.J.55, No. 5, 1671-1700 (2006; Zbl 1112.35041)], it was shown that there are a constant \(c\in \mathbb R\) and a viscosity solution \(v\) of \(c+\alpha x\cdot Dv(x)+H((Dv(x))=f(x)\) in \(\mathbb R^n\) such that \(u(\cdot ,t)-(v(\cdot )+ct)\to 0\) as \(t\to \infty \) locally uniformly in \(\mathbb R^n\). The function \(v(x)+ct\) is called the asymptotic solution. Our goal is to give a sufficient condition in order that the set of points where the rate of this convergence is slower than \(t^{-1}\) is non-empty. We also give several examples which show that we can not remove, in general, the assumptions in this sufficient condition in order that this set is nonempty. As a result, we clarify crucial factors which cause this slow rate of convergence. They are both a geometrical property of the set of equilibrium points and a lower bound of the initial data.

### MSC:

35K55 | Nonlinear parabolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |