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.