Radiation fields. (English) Zbl 1085.35015
This paper is concerned with the question whether polyhomogeneity of initial data is preserved under evolution dictated by semilinear wave equations; and for the wave map equation, on Minkowski space-time. For such equations, local existence of solutions in weighted Sobolev spaces is established, and it is shown that polyhomogeneity is preserved under evolution when appropriate corner conditions are satisfied by the initial data, which could be more singular than allowed in the existing related results. Estimates on the time derivatives are shown to be uniform in time in a neighbourhood of the initial data that satisfy certain compatibility conditions.
Reviewer: V. D. Sharma (Mumbai)
MSC:
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
| 35L40 | First-order hyperbolic systems |
| 58J47 | Propagation of singularities; initial value problems on manifolds |
| 35L67 | Shocks and singularities for hyperbolic equations |
| 76L05 | Shock waves and blast waves in fluid mechanics |
Keywords:
hyperbolical Cauchy problem; linear and semilinear wave equations; symmetric hyperbolic systems; asymptotic behavior; preservation of polyhomogeneity; weighted Sobolev spacesReferences:
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