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<p>The Hamilton-Jacobi equation of the evolutionary type is considered. The state space is one-dimensional, and the Hamiltonian depends on state and momentum variables, with the dependence on the momentum variable being exponential. The domain where the equation is considered is determined by zeros of the coefficient functions before the exponential terms in the Hamiltonian. The Cauchy initial-value problem with state constraints is investigated for special cases of Hamiltonians convex and concave in the momentum variable. It is proved that if the Hamiltonian is convex in momentum, there exists a&#xa0;unique global viscosity solution to this problem. In the case when the Hamiltonian is concave in momentum, a&#xa0;viscosity solution to the Cauchy problem, if it exists, has an empty subdifferential at all points on the boundary. We cannot construct such a&#xa0;solution in the general case, so we consider on bounded time interval the Cauchy-Dirichlet initial-boundary value problem obtained by adding boundary conditions defined by smooth functions to the initial condition. A&#xa0;generalized solution to this problem is determined, and sufficient conditions for its existence and uniqueness are obtained.</p>

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On generalized solutions for Hamilton-Jacobi equations with exponential dependence on momentum

  • 期刊论文

  • 发表日期：

    2026年4月17日

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  • Lyubov G. Shagalova

摘要

The Hamilton-Jacobi equation of the evolutionary type is considered. The state space is one-dimensional, and the Hamiltonian depends on state and momentum variables, with the dependence on the momentum variable being exponential. The domain where the equation is considered is determined by zeros of the coefficient functions before the exponential terms in the Hamiltonian. The Cauchy initial-value problem with state constraints is investigated for special cases of Hamiltonians convex and concave in the momentum variable. It is proved that if the Hamiltonian is convex in momentum, there exists a unique global viscosity solution to this problem. In the case when the Hamiltonian is concave in momentum, a viscosity solution to the Cauchy problem, if it exists, has an empty subdifferential at all points on the boundary. We cannot construct such a solution in the general case, so we consider on bounded time interval the Cauchy-Dirichlet initial-boundary value problem obtained by adding boundary conditions defined by smooth functions to the initial condition. A generalized solution to this problem is determined, and sufficient conditions for its existence and uniqueness are obtained.