We prove that the solution operators e t 0Y y for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space H 1 L &1 Â L 2 to H s q H for t j 0, and 0 % s % 1Y n 1a1a2 À 1aq H 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here e t 0Y y uÁY t, where u is a solution ofwhere n^4Y m^0 and & b & Ã n 2an À 2 in the supercritical case.
The wave and scattering operators for the equation with m > 0 and λ > 0 on four-dimensional Minkowski space are analytic on the space of finite-energy Cauchy data, i.e. L 2 (R 3 )®L 2 (R 3 ).
We prove a regularity result for a Fredholm integral equation with weakly singular kernel, arising in connection with the neutron transport equation in an infinite cylindrical domain. The theorem states that the solution has almost two derivatives in L1, and is proved using Besov space techniques. This result is applied in the error analysis of the discrete ordinates method for the numerical solution of the neutron transport equation. We derive an error estimate in the L1-norm for the scalar flux, and as a consequence, we obtain an error bound for the critical eigenvalue.
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