Existence of solutions to a one-dimensional Hamilton–Jacobi equation with a degenerate Hamiltonian

We study a one-dimensional Hamilton–Jacobi initial value problem for a degenerate Hamiltonian occurring in multipeakon dynamics. Such a degenerate problem does not obey the usual viscosity solutions theory; viability type extensions do not seem to cover it either. Thanks to a particular change of variables, we reduce the problem to the case where Hopf theory is sufficient. Moreover, we show that our solutions are solutions in the viscosity sense.