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Globally classical solutions for nonlinear equations of first order. (English) Zbl 0594.35052
Etant donné le problème différentiel: $(1)\quad \partial u/\partial t+f(t,x,u,Du)=0$ $$(t,x)\in D=\{t>0$$, $$x\in {\mathbb{R}}^ N\}$$, $$u(0,x)=\phi (u)$$, f et $$\phi$$ étant de classe $$C^ 2$$. Si les solutions du problème: $dx_ i/dt=\partial f(t,x,v,p)/\partial p_ i,\quad dv/dt=\sum^{N}_{i=1}p_ i \partial f/\partial p_ i-f,\quad dp_ i/dt=-\partial f/\partial x_ i-p_ i \partial f/\partial v$ possèdent le bonnes propriétés. L’A. montre (Théorème 5) que (1) admet une unique solution de classe $$C^ 2$$ dans $$\bar D$$ si et seulement si (Dx/Dy)(t,y)$$\neq 0$$ pour tout (t,y)$$\in \bar D$$.
Reviewer: M.-T.Lacroix

##### MSC:
 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 35K55 Nonlinear parabolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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##### References:
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