Maillet type theorem and Gevrey regularity in time of solutions to nonlinear partial differential equations

Tom 97 / 2012

Hidetoshi Tahara Banach Center Publications 97 (2012), 125-140 MSC: Primary 35B65; Secondary 35G20. DOI: 10.4064/bc97-0-9


We will consider the nonlinear partial differential equation \[ t^{\gamma}(\partial /\partial t)^mu =F(t,x,\{(\partial /\partial t)^j (\partial /\partial x)^{\alpha}u \}_{j+|\alpha| \leq L, j < m}) \tag*{(E)} \] (with $\gamma \geq 0$ and $1 \leq m \leq L$) and show the following two results: (1) (Maillet type theorem) if (E) has a formal solution it is in some formal Gevrey class, and (2) (Gevrey regularity in time) if (E) has a solution $u(t,x) \in C^{\infty}([0,T], {\mathcal E}^{\{\sigma \}}(V))$ it is in some Gevrey class also with respect to the time variable $t$. It will be explained that the mechanism of these two results are quite similar, but still there appears some difference between them which is very interesting to the author.


  • Hidetoshi TaharaDepartment of Information and Communication Sciences
    Sophia University
    Kioicho, Chiyoda-ku
    Tokyo, 102-8554 Japan

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