Existence and Nonexistence of Solutions of the Reaction-Diffusion Equation with Robin Boundary Condition

2016 ◽  
Vol 68 (3) ◽  
pp. 475-484 ◽  
Author(s):  
E. Öztürk ◽  
K. Kallı
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Robin Boundary Condition ◽  
Reaction Diffusion Equation ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions ◽  
Robin Boundary

A REACTION–DIFFUSION–ADVECTION EQUATION WITH COMBUSTION NONLINEARITY ON THE HALF-LINE

2018 ◽  
Vol 98 (2) ◽  
pp. 277-285
Author(s):  
FANG LI ◽  
QI LI ◽  
YUFEI LIU
Keyword(s):  
Boundary Condition ◽  
Wave Speed ◽  
Reaction Diffusion ◽  
Robin Boundary Condition ◽  
Travelling Wave ◽  
Half Line ◽  
Advection Equation ◽  
The Right ◽  
Combustion Nonlinearity ◽  
Robin Boundary

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.

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