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Nonanalyticity of solutions to t u = ² x u + u ²

Grzegorz Łysik (2003)

Colloquium Mathematicae

It is proved that the solution to the initial value problem t u = ² x u + u ² , u(0,x) = 1/(1+x²), does not belong to the Gevrey class G s in time for 0 ≤ s < 1. The proof is based on an estimation of a double sum of products of binomial coefficients.

The Cauchy problem for systems through the normal form of systems and theory of weighted determinant

Waichiro Matsumoto (1998/1999)

Séminaire Équations aux dérivées partielles

The author propose what is the principal part of linear systems of partial differential equations in the Cauchy problem through the normal form of systems in the meromorphic formal symbol class and the theory of weighted determinant. As applications, he choose the necessary and sufficient conditions for the analytic well-posedness ( Cauchy-Kowalevskaya theorem ) and C well-posedness (Levi condition).

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