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

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Tahara H.

Banach Center Publications

2012

tom 97

nr 1

125-140

We will consider the nonlinear partial differential equation
$t^{γ}(∂/∂t)^{m}u = F(t,x,{(∂/∂t)^{j}(∂/∂x)^{α}u}_{j+|α|≤L,j(with γ ≥ 0 and 1 ≤ m ≤ 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 $u(t,x) ∈ C^{∞}([0,T],𝓔^{σ}(V))$ 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.

Removable singularities of solutions of nonlinear singular partial differential equations

100%

Tahara H.

Banach Center Publications

1996

tom 33

nr 1

395-399

1. Introduction. The study of singularities has been one of the main subjects of research in partial differential equations. In the case of linear equations the singularities are now pretty well understood; but in the nonlinear case there seems to be still very few studies. In this paper I want to discuss the singularities of solutions of a class of nonlinear singular partial differential equations in the complex domain. The class is only a model, but it helps one understand that the situation in the nonlinear case is more complicated than in the linear case.

Ograniczanie wyników

2 Banach Center Publications

2 Tahara H.

1 2012

1 1996

Wyniki wyszukiwania

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

Removable singularities of solutions of nonlinear singular partial differential equations