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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
##### Keywords:
Stefan problem; Hopf bifurcation
##### Software:
Algorithm 731; CWRESU; CWRESX
Full Text:
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