| Abstract: |
| Consider the following general
autocatalytic reaction system without or with decay
\begin{equation}
\label{1.1} \left\{\begin{array}{l}
u_t=d\triangle u-uf(v)\
\noalign{\vskip8pt} v_t=\triangle v+uf(v)-Kv^q.
\end{array}\right.
\end{equation}
In this talk we shall talk about our recent progress on the asymptotic stability of traveling fronts for the system without decay ($K=0$) and the spectral stability of the wave solution ( Front, Pulse) for the system with strong decay ($K>0$ and $q>1$). Our arguments are based on Evan's function method, spectral analysis, semigroup estimates and numerical simulation on Evans function method. |
|