On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton–Jacobi equations
On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton–Jacobi equations
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次二次粘性 Hamilton-Jacobi 方程 Dirichlet 问题解的大时间行为
DOI：
10.1016/j.matpur.2010.03.006
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
Thierry Tabet Tchamba
中科院分区：
文献类型：
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作者：
G. Barles;A. Porretta;Thierry Tabet Tchamba
在本文中，我们感兴趣的是次二次粘性 Hamilton-Jacobi 方程的狄利克雷问题解的大时间行为。在超二次案例中，第三位作者证明了这些解只能有两种不同的行为：要么演化方程的解收敛到相关的平稳广义狄利克雷问题的解（假设它存在），要么表现得像 −ct+φ(x)，其中 c⩾0 是一个常数，通常称为“遍历常数”，而 φ 是所谓“遍历问题”的解。在当前的次二次情况下，我们表明情况稍微复杂一些：如果方程中的梯度增长类似于 |Du|m 并且 m>3/2，那么类似的结果在超二次情况下成立，至少在 c>0 的情况下。但是，相反，如果 m⩽3/2 或 c=0，则会出现另一种不同的行为，因为 u(x,t)+ct 可以从下面无界，其中 u 是次二次粘性 Hamilton-Jacobi 方程的解。
In this article, we are interested in the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton–Jacobi equations. In the superquadratic case, the third author has proved that these solutions can have only two different behaviors: either the solution of the evolution equation converges to the solution of the associated stationary generalized Dirichlet problem (provided that it exists) or it behaves like −ct+φ(x) where c⩾0 is a constant, often called the “ergodic constant” and φ is a solution of the so-called “ergodic problem”. In the present subquadratic case, we show that the situation is slightly more complicated: if the gradient-growth in the equation is like |Du|mwith m>3/2, then analogous results hold as in the superquadratic case, at least if c>0. But, on the contrary, if m⩽3/2 or c=0, then another different behavior appears since u(x,t)+ct can be unbounded from below where u is the solution of the subquadratic viscous Hamilton–Jacobi equation.