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Counterexample to the 1-summability of a divergent formal solution to some linear partial differential equations. (English) Zbl 1426.35071

The work under review provides a counterexample to the Borel-Laplace summability property of the formal solutions to certain family of linear partial differential equations of the form \[ \partial_t^m u+\sum_{(j,\alpha)\in\Lambda}a_{j,\alpha}(t)\partial_t^j\partial_x^{\alpha}u=f(t,x) \] under initial conditions \(\partial_{t}^{j}u|_{t=0}=\varphi_j(x)\), for \(j=0,\ldots,m-1\), where \(\Lambda\) is a finite subset of \(\mathbb{N}\times \mathbb{N}^{N}\), the coefficients \(a_{j,\alpha}(t)\) are holomorphic functions on some neighborhood of the origin, \(f\) is holomorphic on some neighborhood of the origin with respect to its first variables and an entire function with respect to \(x\), and the initial data are entire functions.
The author gives an example in which one of the conditions imposed in the previous work by H. Tahara and H. Yamazawa [J. Differ. Equations 255, No. 10, 3592–3637 (2013; Zbl 1320.35151)] on the 1-summability of the formal solution of the previous problem does not hold. In that example, the loss of such condition shows that the formal solution is not 1-summable along any direction.

MSC:

35C10 Series solutions to PDEs
35A20 Analyticity in context of PDEs

Citations:

Zbl 1320.35151
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References:

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[8] Tahara, H. and Yamazawa, H., Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations, J. Differential Equations, 255 (2013), 3592-3637. · Zbl 1320.35151
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