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We present some results on the formation of singularities
for C^1 - solutions of the quasi-linear N × N strictly hyperbolic system Ut +
A(U )Ux = 0 in [0, +∞) × Rx . Under certain weak non-linearity conditions
(weaker than genuine non-linearity), we prove that the first order derivative
of the solution blows-up in finite time.
We consider the (characteristic and non-characteristic) Cauchy problem for a system of constant coefficients partial differential equations with initial data on an affine subspace of arbitrary codimension. We show that evolution is equivalent to the validity of a principle on the complex characteristic variety and we study the relationship of this condition with the one introduced by Hörmander in the case of scalar operators and initial data on a hypersurface.
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