On the explosion problem in a ball

In $\Omega \subset \mathbb{R}^n$ we consider the explosion problem in an incompressible flow introduced in [L. Kagan, H. Berestycki, G. Joulin, and G. Sivashinsky, Comb. Theory Model., 1, 97–111, 1997]. If $\Omega$ is a ball, we show that the explosion threshold can only be increased by addition of an incompressible flow. Further, for any $\Omega$ we give a new proof of the $L^p - L^{\infty}$ estimate for elliptic advection-diffusion problems obtained in [H. Berestycki, A. Kiselev, A. Novikov, and L. Ryzhik, J. Anal. Math., 110, 31–65, 2010]. Our proof provides an optimal estimate when $\Omega$ is a ball.

Keywords

combustion, reaction-convection-diffusion equations

2010 Mathematics Subject Classification

35J05, 35J60

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Published 12 March 2015