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Malek, Stéphane

On singular solutions to PDEs with turning point involving a quadratic nonlinearity. (English) Zbl 1470.35140

Abstr. Appl. Anal. 2017, Article ID 9405298, 32 p. (2017).
Summary: We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter \(\varepsilon\). The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in \(\mathbb{C}\). We construct a family of sectorial meromorphic solutions obtained as a small perturbation in \(\varepsilon\) of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in \(\varepsilon\) as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.

MSC:

35G20 Nonlinear higher-order PDEs
35B25 Singular perturbations in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs
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\textit{S. Malek}, Abstr. Appl. Anal. 2017, Article ID 9405298, 32 p. (2017; Zbl 1470.35140)
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