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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

##### MSC:
 35F25 Initial value problems for nonlinear first-order PDEs 35A30 Geometric theory, characteristics, transformations in context of PDEs
##### Keywords:
characteristic curve; singularities