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Global version of the Cauchy-Kovalevskaia theorem for nonlinear PDEs. (English) Zbl 0733.35002

The main result of this paper is the proof of the following result. There exist generalized solutions to the n-th order, nonlinear, analytic differential equation \[ D_ t^ mU(t,y)=G(t,y,...,D_ t^ pD^ q_ yU,...),\quad y\in {\mathbb{R}}^{n-1}, \] \(0\leq p<m\), q a multi-index with \(p+| q| \leq m\), which satisfy the noncharacteristic Cauchy data \[ D_ t^ pU(t_ 0,y)=g_ p(y),\quad 0\leq p<m. \] The result is global and relies on previous work of the author and others on the nonlinear theory of generalized functions. The paper is very elegant but it is quite difficult as well.

MSC:

35A10 Cauchy-Kovalevskaya theorems
35D05 Existence of generalized solutions of PDE (MSC2000)
35G25 Initial value problems for nonlinear higher-order PDEs
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