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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

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)
Full Text: DOI
[1] DOI: 10.1007/BF00280178 · Zbl 0352.35029 · doi:10.1007/BF00280178
[2] Doubnov B., Theorie des perturbationset methodes asymptotiques” p
[3] D. Hoff, Locally Lipschitz solutions of a single conservation law in several space variables, preprint. · Zbl 0443.34005
[4] Li Ta-tsien and Chen Shu-xing, Regularity and singularity of solutions for nonlinear hyper bolicequtions , t o appear.
[5] Tie-hu Qin, Fudan Journal 22 pp 41– (1983)
[6] Ta-tsien Li, Fudan Journal 21 pp 361– (1982)
[7] DOI: 10.2307/2316266 · Zbl 0263.57015 · doi:10.2307/2316266
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