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Quenching of flames by fluid advection. (English) Zbl 1032.35087
This paper concerns the following nonlinear problem \[ T_t+Au(y)T_x=\kappa\Delta T +\frac{v^2_0}{\kappa}f(T), \quad T(0,x,y)=T_0(x,y), \] where \((x,y)\in D=\mathbb R\times[0,H]\), \(f(\cdot)\) is a normalized so-called “ignition-type” nonlinearity. The equation may be regarded as a simple model of flame propagation in a fluid advected by a shear flow. Consider boundary conditions periodic in \(y\) and decaying in \(x\), and assume that \(u(y)\) is periodic with zero mean, and \(T_0(x,y)\) is compactly supported. The authors prove that the flame becomes extinct (i.e. \(T(t,x,y)\to 0\) uniformly in \(D\) as \(t\to\infty\)) if the support of initial data is small compared to the scale of the laminar-flame width \(\kappa/v_0\), and the flame propagates (i.e. \(T(t,x,y)\to 1\) on the whole \(D\) as \(t\to\infty\)) if the support is large enough. Moreover, the influence of shear flow on the extinction-propagation is investigated, and some surprising results are obtained. Among them, the authors show that for certain profile \(u(y)\) the flame extinction will occur for sufficiently large \(A\), and for some other profile \(u(y)\) with certain (compactly supported) \(T_0(x,y)\) the flame will propagates for all \(A\).
Reviewer: Ning Su (Beijing)

MSC:
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
80A25 Combustion
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