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Minimax Theory and its Applications 06 (2021), No. 1, 145--154
Copyright Heldermann Verlag 2021

On the Uniqueness of Solutions to One-Dimensional Constrained Hamilton-Jacobi Equations

Yeoneung Kim
Department of Mathematics, University of Wisconsin, Madison, WI 53706, U.S.A.


\def\R{\mathbb{R}} The goal of this paper is to study the uniqueness of solutions to a constrained Hamilton-Jacobi equation \begin{equation*} \begin{cases} u_t=u_x^2+R(x,I(t)) &\text{in }\R \times (0,\infty), \\ \max_{\R} u(\cdot,t)=0 &\text{on }[0,\infty), \end{cases} \end{equation*} with an initial condition $u(x,0)=u_0(x)$ on $\R$. A reaction term $R(x,I(t))$ is given while $I(t)$ is an unknown constraint (Lagrange multiplier) that forces maximum of $u$ to be always zero. In the paper, we prove uniqueness of a pair of unknowns $(u,I)$ using the dynamic programming principle for a particular class of non-separable reaction $R(x,I(t))$ when the space is one-dimensional.

Keywords: Hamilton-Jacobi equation with constraint, selection-mutation model.

MSC: 35A02, 35F21, 35Q92.

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