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A REACTION–DIFFUSION–ADVECTION EQUATION WITH COMBUSTION NONLINEARITY ON THE HALF-LINE
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 98 / Issue 2 / October 2018
- Published online by Cambridge University Press:
- 02 July 2018, pp. 277-285
- Print publication:
- October 2018
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- Article
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We study the dynamics of a reaction–diffusion–advection equation
$u_{t}=u_{xx}-au_{x}+f(u)$ on the right half-line with Robin boundary condition 
$u_{x}=au$ at 
$x=0$, where 
$f(u)$ is a combustion nonlinearity. We show that, when 
$0<a<c$ (where 
$c$ is the travelling wave speed of 
$u_{t}=u_{xx}+f(u)$), 
$u$ converges in the 
$L_{loc}^{\infty }([0,\infty ))$ topology either to 
$0$ or to a positive steady state; when 
$a\geq c$, a solution 
$u$ starting from a small initial datum tends to 
$0$ in the 
$L^{\infty }([0,\infty ))$ topology, but this is not true for a solution starting from a large initial datum; when 
$a>c$, such a solution converges to 
$0$ in 
$L_{loc}^{\infty }([0,\infty ))$ but not in 
$L^{\infty }([0,\infty ))$ topology.
