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

Volume 47 / 2020

Tomasz Cieślak, Hiroshi Wakui Applicationes Mathematicae 47 (2020), 155-163 MSC: 35F21, 35C99, 35D40. DOI: 10.4064/am2407-7-2020 Published online: 19 October 2020

Abstract

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.

Authors

  • Tomasz CieślakInstitute of Mathematics
    Polish Academy of Sciences
    00-656 Warszawa, Poland
    ORCID: 0000-0003-4191-4231
    e-mail
  • Hiroshi WakuiMathematical Institute
    University of Wrocław
    pl. Grunwaldzki 2/4
    50-384 Wrocław, Poland
    and
    Faculty of Science Division I
    Tokyo University of Science
    Tokyo 162-8601, Japan
    e-mail

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