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# Communications in Mathematical Sciences

## Volume 13 (2015)

### Number 4

### Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

### On the explosion problem in a ball

Pages: 1025 – 1032

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n4.a9

#### Author

#### Abstract

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