论文标题
$ \ mathbb {r}^n $的粘性汉密尔顿 - 雅各布方程无限解决方案的大型行为
Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi Equations in $\mathbb{R}^N$
论文作者
论文摘要
我们研究了整个空间中的抛物线粘性汉密尔顿 - 雅各比方程的限制的大型行为。这种解决方案的存在和独特性在一个非常一般的框架中显示，即源项和初始数据仅在无限时的任意增长下从下面界定。我们的主要结果是，当$ t \ to + \ infty $，即它们的行为就像$λ^*t + ϕ（x）$，其中$λ^*$是最大的ergodic常数，而$ ϕ $是相关的细能性问题的解决方案。该结果的主要独创性来自数据的普遍性：尤其是，初始数据在无穷大时的生长可能与Ergodic问题的解决方案具有完全不同的增长。
We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space $\mathbb{R}^N$ in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are shown in a very general framework, namely when the source term and the initial data are only bounded from below with an arbitrary growth at infinity. Our main result is that these solutions have an ergodic behavior when $t\to +\infty$, i.e., they behave like $λ^*t + ϕ(x)$ where $λ^*$ is the maximal ergodic constant and $ϕ$ is a solution of the associated ergodic problem. The main originality of this result comes from the generality of the data: in particular, the initial data may have a completely different growth at infinity from those of the solution of the ergodic problem.
