Abstract
The initial – boundary value problem is considered for the Hamilton-Jacobi of evolutionary type in the case when the state space is one-dimensional. The Hamiltonian depends on the state and momentum variables, and the dependence on the momentum variable is exponential. The problem is considered on fixed bounded time interval, and the state variable changes from a given fixed value to infinity. The initial and boundary functions are subdifferentiable. It is proved that such a problem has a continuous generalized viscosity) solution. The representative formula is given for this solution. Sufficient conditions are indicated under which the generalized solution is unique. Hamilton-Jacobi equations with an exponential dependence on the momentum variable are atypical for theory, but such equations arise in practical problems, for example, in molecular genetics.
Highlights
Some practical problems and applied research, in particular, in molecular genetics [Saakian, Rozanova and Akmetzhanov, 2008], lead to the need to study Hamilton – Jacobi equations with an exponential dependence of the Hamiltonian on the momentum variable
Exact formulas for solutions can be obtained in very few special cases
It is required to construct a viscosity solution [Crandall and Lions, 1983] u(·, ·) of the equation (1) which is continuous on the set clG+ – the closure of the domain G+, and such that the following initial and boundary conditions are satisfied u(0, x) = u0(x), x ∈ R, x ≥ x∗; (7)
Summary
Some practical problems and applied research, in particular, in molecular genetics [Saakian, Rozanova and Akmetzhanov, 2008], lead to the need to study Hamilton – Jacobi equations with an exponential dependence of the Hamiltonian on the momentum (impulse) variable. Hamilton-Jacobi equation, initial-boundary value problem, viscosity solution, characteristics, subdifferentials The initial and boundary function are supposed to be subdifferentiable and such that their Dini derivatives are finite.
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