Description:
<jats:p>We study the dynamics of a reaction–diffusion–advection equation <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline1" /><jats:tex-math>$u_{t}=u_{xx}-au_{x}+f(u)$</jats:tex-math></jats:alternatives></jats:inline-formula> on the right half-line with Robin boundary condition <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline2" /><jats:tex-math>$u_{x}=au$</jats:tex-math></jats:alternatives></jats:inline-formula> at <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline3" /><jats:tex-math>$x=0$</jats:tex-math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline4" /><jats:tex-math>$f(u)$</jats:tex-math></jats:alternatives></jats:inline-formula> is a combustion nonlinearity. We show that, when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline5" /><jats:tex-math>$0<a<c$</jats:tex-math></jats:alternatives></jats:inline-formula> (where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline6" /><jats:tex-math>$c$</jats:tex-math></jats:alternatives></jats:inline-formula> is the travelling wave speed of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline7" /><jats:tex-math>$u_{t}=u_{xx}+f(u)$</jats:tex-math></jats:alternatives></jats:inline-formula>), <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline8" /><jats:tex-math>$u$</jats:tex-math></jats:alternatives></jats:inline-formula> converges in the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline9" /><jats:tex-math>$L_{loc}^{\infty }([0,\infty ))$</jats:tex-math></jats:alternatives></jats:inline-formula> topology either to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline10" /><jats:tex-math>$0$</jats:tex-math></jats:alternatives></jats:inline-formula> or to a positive steady state; when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline11" /><jats:tex-math>$a\geq c$</jats:tex-math></jats:alternatives></jats:inline-formula>, a solution <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline12" /><jats:tex-math>$u$</jats:tex-math></jats:alternatives></jats:inline-formula> starting from a small initial datum tends to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline13" /><jats:tex-math>$0$</jats:tex-math></jats:alternatives></jats:inline-formula> in the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline14" /><jats:tex-math>$L^{\infty }([0,\infty ))$</jats:tex-math></jats:alternatives></jats:inline-formula> topology, but this is not true for a solution starting from a large initial datum; when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline15" /><jats:tex-math>$a>c$</jats:tex-math></jats:alternatives></jats:inline-formula>, such a solution converges to <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline16" /><jats:tex-math>$0$</jats:tex-math></jats:alternatives></jats:inline-formula> in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline17" /><jats:tex-math>$L_{loc}^{\infty }([0,\infty ))$</jats:tex-math></jats:alternatives></jats:inline-formula> but not in <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0004972718000370_inline18" /><jats:tex-math>$L^{\infty }([0,\infty ))$</jats:tex-math></jats:alternatives></jats:inline-formula> topology.</jats:p>