The goal of this paper is to study a Hamilton-Jacobi equation$$ \textstyle\begin{cases} u_{t}=H(Du)+R(x,I(t)) &\text{in }\mathbb{R}^{n} \times (0,\infty ), \\ \sup_{\mathbb{R}^{n}} u(\cdot ,t)=0 &\text{on }[0,\infty ), \end{cases} $$with initial conditions \(I(0)=I_{0}>0\), \(u(x,0)=u_{0}(x)\) on \(\mathbb{R}^{n}\). Here \((u,I)\) is a pair of unknowns and the Hamiltonian \(H\) and the reaction term \(R\) are given. Moreover, \(I(t)\) is an unknown constraint (Lagrange multiplier) that constrains the supremum of \(u\) to be always zero. We construct a solution in the viscosity setting using a fixed point argument when the reaction term \(R(x,I)\) is strictly decreasing in \(I\). We also discuss both uniqueness and nonuniqueness. For uniqueness, a certain structural assumption on \(R(x,I)\) is needed. We also provide an example with infinitely many solutions when the reaction term is not strictly decreasing in \(I\).

中文翻译：

---

约束的Hamilton-Jacobi方程的适定性

本文的目的是研究Hamilton-Jacobi方程$$ \ textstyle \ begin {cases} u_ {t} = H（Du）+ R（x，I（t））＆\ text {in} \ mathbb { R} ^ {n} \ times（0，\ infty），\\ \ sup _ {\ mathbb {R} ^ {n}} u（\ cdot，t）= 0＆\ text {on} [0，\ infty ），\ end {cases} $$具有初始条件\（I（0）= I_ {0}> 0 \），\（u（x，0）= u_ {0}（x）\）在\（\ mathbb {R} ^ {n} \）。这里\（（u，I）\）是一对未知数，并给出了哈密顿量\（H \）和反应项\（R \）。此外，\（I（t）\）是一个未知约束（拉格朗日乘数），它约束\（u \）的最大值始终为零。当反应项\（R（x，I）\）在\（I \）中严格减小时，我们使用定点参数构造粘度设置的解决方案。我们还将讨论唯一性和非唯一性。为了唯一，需要对\（R（x，I）\）进行一定的结构假设。当反应项未严格按\（I \）减小时，我们还提供了具有无限多个解的示例。