GLOBAL PROPAGATION OF SINGULARITIES FOR TIME DEPENDENT HAMILTON-JACOBI EQUATIONS
GLOBAL PROPAGATION OF SINGULARITIES FOR TIME DEPENDENT HAMILTON-JACOBI EQUATIONS
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DOI：
10.3934/dcds.2015.35.4225
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发表时间：
2015-09-01
影响因子：
1.1
通讯作者：
Sinestrari, Carlo
中科院分区：
文献类型：
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作者：
Cannarsa, Piermarco;Mazzola, Marco;Sinestrari, Carlo
本文研究了方程u（t）（t，x）+ H（delu（t，x））- 0，a. e.（t，x）是Rn+1的（0，+无穷大）x Ω子集的元素。(1)It众所周知，这种解的奇异性是局部沿着广义特征传播的。在对哈密顿量H的结构作一些假设的情况下，可以构造出满足能量条件的特殊广义特征。在本文中，我们提供了估计的耗散行为的能量沿着这样的曲线。作为应用，我们证明了（1）的任何粘性解的奇性在有限时间内不会消失。
We investigate the properties of the set of singularities of semiconcave solutions of Hamilton-Jacobi equations of the formu(t)(t, x) + H(del u(t, x)) - 0, a.e. (t, x) is an element of (0, +infinity) x Omega subset of Rn+1. (1)It is well known that the singularities of such solutions propagate locally along generalized characteristics. Special generalized characteristics, satisfying an energy condition, can be constructed, under some assumptions on the structure of the Hamiltonian H. In this paper, we provide estimates of the dissipative behavior of the energy along such curves. As an application, we prove that the singularities of any viscosity solution of (1) cannot vanish in a finite time.