Geometric analysis of a truncated Galerkin discretization of fast-slow PDEs with transcritical singularities

Abstract

We consider a fast-slow partial differential equation (PDE) with reaction-diffusion dynamics in the fast variable and the slow variable driven by a differential operator on a bounded domain. Assuming a transcritical normal form for the reaction term and viewing the slow variable as a dynamic bifurcation parameter, we analyze the passage through the fast subsystem bifurcation point for the spectral Galerkin approximation of the PDE. We characterize the invariant manifolds for the finite-dimensional Galerkin ODEs using geometric desingularization via a blow-up analysis. In addition to the crucial approximation procedure, we also make the domain dynamic during the blow-up analysis. Finally, we elaborate in which sense our results approximate the infinite-dimensional problem. Within our analysis, we find that the PDEs appearing in entry and exit blow-up charts are quasi-linear free boundary value problems, while in the central/scaling chart we obtain a PDE, which is often encountered in classical reaction-diffusion problems exhibiting solutions with finite-time singularities.

Figures (4)

• Figure 1: Depiction of the slow Galerkin manifolds close to $S_{{\textnormal{a}}}^{-,1}$ in the $(u_1, v_1)$-plane for $k_0 =1$, anchored at a fixed $u_1 = v_1 = c$ and parameterized by $u_1 = h_1(v_1, \varepsilon)$\ref{['eq:cm_k01']}, i.e. $G_{\varepsilon, \zeta}$ for $- \frac{9\pi^2}{4a} < \zeta^{-1} \omega_A < -\frac{\pi^2}{a}$ and $\varepsilon$ sufficiently small, when (a) $\mu < 1$ and (b) $\mu > 1$. The curves with arrows indicate the behaviour around the singularity at the origin as known from the classical planar ODE treatment, see e.g. KruSzm4, where the blow-up method is used for the extension of the slow manifolds.
• Figure 1: Behavior of solutions $x_1= v_{1,1} e_1 + \sum_{k=2}^{k_0} u_{k,1}^{k_0} e_k$ for the Galerkin problem in chart $K_1$ along $M_{{\textnormal{a}},1}^{-, k_0}$ (a), with decreasing $a_1$ and $r_1$, and $M_{{\textnormal{a}},1}^{+, k_0}$ (b), with increasing $a_1$ and $r_1$. Note that replacing $v_{1,1}$ by $u_{1,1} =-1$ means replacing $x_1$ by $u_1$, approixmating the solution of the corresponding PDE in \ref{['eq:mainPDE_K1_fullsystem']}. The behavior around $r_1=0$, i.e. in between the two illustrated regions, is captured by the analysis in chart $K_2$.
• Figure 1: Sketch of the dynamics of a typical solution of the $k_0$-Galerkin truncation of the PDE system \ref{['eq:mainPDE_fasttime']} along a slow attracting manifold close to the critical manifold as described in Theorem A for $\mu < 1$ (a) and $\mu > 1$ (b).
• Figure 2: Depiction of the slow Galerkin manifolds close to $S_{{\textnormal{a}}}^{-,2}$ projected to the $(u_1, v_1)$-plane for $k_0 =2$, anchored at a fixed $u_1 = v_1 = c, u_2 = v_2 =0$ and parameterized by $u_1 = h_1(v_1, v_2, \varepsilon)$\ref{['eq:cm_k02_1']}, i.e. $G_{\varepsilon, \zeta}$ for $- \frac{9\pi^2}{4a} < \zeta^{-1} \omega_A < -\frac{\pi^2}{a}$ and $\varepsilon$ sufficiently small, when (a) $\mu < 1$ and (b) $\mu > 1$. The curves with arrows indicate the flow close to the manifolds. We note that the perturbation depending on $v_2^2$ tends to the left and, hence, may lead to $G_{\varepsilon, \zeta}$ lying to the left of $S_{{\textnormal{a}}}^{-,2}$ also for $\mu > 1$ (b), depending on the sizes of $\varepsilon$ and $v_2^2$.

Theorems & Definitions (45)

• Theorem 1.1
• Proposition 2.1
• Proof 1
• Proposition 2.2
• Proof 2
• Proposition 2.3
• Proof 3
• Remark 2.4
• Lemma 3.1
• Proof 4