## Geometric explosion for quasilinear systems. (Explosion géométrique pour des systèmes quasi-linéaires.)(French)Zbl 0840.35060

A local geometric theory for blow-up in general quasilinear hyperbolic systems is sketched. The main idea of this theory is to correspond to a given system $$L$$ another “blow-up” system $$Le$$ and to correspond to each convenient solution of $$Le$$ one blow-up solution of $$L$$. This solution has the form $$u(\phi(x))= v(x)$$, where $$\phi(x)= (\varphi(x), x_2,\dots, x_n)$$ and $$u(\varphi, v)$$ is a smooth solution of $$Le$$. Of course, $$\partial\varphi(0)/\partial x_1 = 0$$ and each singularities of $$\phi$$ on $$x= 0$$ (fold, crease etc…) corresponds to a solution with the same name. It is proved that there is no obstruction in $$Le$$ to find solutions $$(\varphi, v)$$ the jet of which has the desired properties. In fact, this theory is microlocal in sense that it involves only on real eigenvalue of the system $$L$$ in a characteristic point $$(x^0, u^0, \varsigma^0)$$. In more simple cases (folds, creases) it is analyzed the characteristics linearized on the constructed singular solution. That makes obvious the essential difference between one singular solution with the other blow-up solution. The blow-up solution is a solution which becomes singular in a point of a influence zone where it is regular: the observed singularity is therefore created, but not propagated. The link between the blow-up solutions and genuine nonlinearity is studied.
Reviewer: L.G.Vulkov (Russe)

### MSC:

 35L60 First-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs

### Keywords:

blow-up; quasilinear hyperbolic systems
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