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Remarks on characteristics of partial differential equations of first order. (English) Zbl 0694.35026
The authors consider the Cauchy problem for general nonlinear partial differential equations of first order. Let \(x=x(t,y)\) be a characteristic curve which combines a point (t,x) with an initial point (0,y). For the formation of singularities of generalized solutions, one must know the behavior of the family of characteristic curves \(x=x(t,y)\) in neighbourhoods of points where the Jacobian (Dx/Dy)(t,y) vanishes. The first aim of this note is to show examples in which, though the Jacobian (Dx/Dy)(t,y) vanishes at a point \((t^ 0,y^ 0)\), the characteristic curves \(x=x(t,y)\) do not meet in a neighbourhood of \((t^ 0,y^ 0)\). The second is to give sufficient conditions which guarantee the collision of characteristic curves after the time where the Jacobian vanishes.
Reviewer: M.Tsuji

35F25 Initial value problems for nonlinear first-order PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs