On the Uniqueness of Solutions to One-Dimensional Constrained Hamilton-Jacobi Equations
Keywords:
Hamilton-Jacobi equation with constraint, selection-mutation model.Abstract
The goal of this paper is to study the uniqueness of solutions to a constrained Hamilton-Jacobi equation
{
ut = u
2
x + R(x, I(t)) in R × (0, ∞),
maxR u(·, t) = 0 on [0, ∞),
with an initial condition u(x, 0) = u0(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 dynamic programming principle for a particular class of non-separable reaction R(x, I(t)) when the space is one-dimensional.
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