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Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model. (English) Zbl 1197.35021
The paper deals with the one-dimensional, singularly perturbed Gray-Scott model:
\[ \begin{cases} v_t=\varepsilon^2v_{xx}-v+{\mathcal A}uv^2,\;\tau u_t=Du_{xx}+(1-u)-\varepsilon^{-1}uv^2, &|x|<1, \\ u_x=v_x=0, &x=\pm 1, \end{cases}\tag{1} \] where \(\varepsilon >0\) is a small parameter which goes to 0, \({\mathcal A}>0\) is the feed- rate parameter, \(\tau>0\) is the reaction-time parameter, and \(D>0\) with \(D=O(1)\). The aim of the paper is to study \(k\)-spike solutions to (1), in the intermediate regime \(O(1)\ll {\mathcal A}\ll O(\varepsilon^{-\frac12})\). A new scaling of \(u,v\) and a matched method between the inner and the outer solutions for \(u\) yields, when \(\tau\varepsilon^3{\mathcal A}^2\ll 1\), to the Stefan-type problem:
\[ \begin{cases} \tau\varepsilon^2{\mathcal A}^2u_\sigma= Du_{xx}+(1-u)- \sum_{j=1}^k 6\gamma_j\delta(x-x_j), &|x|\leq 1,\\ \frac{dx_j}{d\sigma}=\gamma_j[u_x(x_j^+,\sigma)+u_x(x_j^-,\sigma)], &j=1,\dots,k,\\ u(x_j(\sigma),\sigma)= \frac{1}{\gamma_j{\mathcal A}^2}, &j=1,\dots,k, \end{cases}\tag{2} \] with \( u_x(\pm 1,\sigma)=0.\) Here, \(\delta\) is the Dirac distribution, \(\sigma =\varepsilon^2{\mathcal A}^2t \) is the slow time scale, \(x_j(\sigma)\) the center of the \(j\)th spike and \(\gamma_j(\sigma)\),\( j=1,\dots,k\) the spike amplitudes. Next, it is supposed that \(\tau\varepsilon^2{\mathcal A}^2 \ll 1\). In the subregime \(1\ll{\mathcal A}\ll \varepsilon ^{-\frac13}\), a \(k\)-spike quasi-equilibrium solution to (2) and the stability of the equilibrium are studied, while in the subregime \(\varepsilon^{-\frac13}\ll{\mathcal A}\ll 1\), when \(\tau\varepsilon^2{\mathcal A}^2=O(1)\), the spike motion is not quasi-steady and the Stefan problem (2) is solved numerically.

MSC:
35B25 Singular perturbations in context of PDEs
35K57 Reaction-diffusion equations
80A22 Stefan problems, phase changes, etc.
35B32 Bifurcations in context of PDEs
35Q79 PDEs in connection with classical thermodynamics and heat transfer
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