Abstract:

This paper investigates the asymptotic symmetry and monotonicity of solutions to a class of nonlocal first-order Hamilton-Jacobi equations. By extending the classical results on the nonlinear term $H(t,u)$ from the literature [Adv Math, 2021, 377: Art 107463] to the more general case of $H(t,x,u,\nabla u)$, we overcome the limitations of the original theoretical framework. The study employs the asymptotic moving plane method proposed in [Adv Math, 2021, 377: Art 107463] as the core tool. However, to address the new challenges posed by the gradient term $\nabla$u in the Hamiltonian, we make critical improvements to the construction of lower solution methods. This expands the applicability of the approach, enabling it to handle a broader range of nonlinear term types.

Key words: nonlocal Hamilton-Jacobi equation, asymptotic symmetry, asymptotic moving plane method.