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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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