Gevrey versus \(q\)-Gevrey asymptotic expansions for some linear \(q\)-difference-differential Cauchy problem (Q6609576)

In this paper, the authors consider a family of singularly perturbed linear \(q\)-difference-differential equations of the form \N\[\NP(\varepsilon^k t^{k+1}\partial_t)\partial_z^Su(t,z,\varepsilon)=\mathcal{P}(t,z,\varepsilon,\partial_t,\partial_z,\sigma_q)u(t,z,\varepsilon)\N\]\Ntogether with the initial data \((\partial_z^ju)(t,0,\varepsilon)=\varphi_j(t,\varepsilon)\) for \(j=0,\dots,S-1\), and the following conditions:\N\begin{itemize} \N\item \(\varepsilon\) acts as a small complex perturbation parameter; \N\item \(\sigma_q\) stands for the \(q\)-difference operator acting on \(t\) for some \(q>1\); \N\item \(P(\tau)\) is a polynomial in \(\tau\) with coefficients in \(\mathcal{C}\); \N\item \(\mathcal{P}(t,z,\varepsilon,\tau_1,\tau_2,\tau_3)\) is a polynomial in \((t,z,\tau_1,\tau_2,\tau_3)\) with analytic coefficients on some neighborhood of the origin with respect to \(\varepsilon\); \N\item \(\varphi_j(t,\varepsilon)\) is a polynomial in \(t\) with analytic coefficients on some neighborhood of the origin with respect to \(\varepsilon\).\N\end{itemize}\NTheir aim is to study the asymptotic behavior of the analytic solutions of such a family. In particular, they provide two different asymptotic expansions with respect to the perturbation parameter \(\varepsilon\) and to the time variable \(t\): one of Gevrey nature, and another of mixed type Gevrey and \(q\)-Gevrey. Such asymptotic phenomena are observed due to the modification of the norm established on the space of coefficients of the formal solution.\N\NTo do that, the authors use techniques based on the adequate path deformation of the difference of two analytic solutions, and the application of several versions of Ramis-Sibuya Theorem.