Abstract
We consider local solutions to second order partial differential equations of the form Pu = f(x, u), for which u is smooth on the complement of a characteristic surface with a cusp singularity. If P is strictly hyperbolic and u is assumed to be regular in the past with respect to differentiation by a natural family of smooth vector fields, then u is regular in the future, and “conormal” with respect to a larger family of vector fields which are nonsmooth at the singularity of the cusp. If P is a Tricomi operator associated with the cusp, and the natural initial data (Dirichlet or Cauchy) are conormal with respect to a hyperplane, then u is again shown to be conormal with respect to the cusp.
Research partially supported by NSF Grant # DMS-8902136.
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© 1991 Springer-Verlag New York, Inc.
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Beals, M. (1991). Regularity of Nonlinear Waves Associated with a Cusp. In: Beals, M., Melrose, R.B., Rauch, J. (eds) Microlocal Analysis and Nonlinear Waves. The IMA Volumes in Mathematics and its Applications, vol 30. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9136-4_2
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DOI: https://doi.org/10.1007/978-1-4613-9136-4_2
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