| Abstract: |
| Consider the degenerate Fisher equation in a cylinder $R\times \Omega$, with bounded $\Omega \subset R^n$,
\begin{equation}
\left\{\begin{array}{ll}
u_t=\triangle u +\beta (y) u_x+ g(u), &t>0,\; (x,y)\in R\times \Omega,\
\frac{\partial u}{\partial n}=0, &t>0, \;(x,y)\in R\times \partial\Omega,
\end{array}\right. \label{1}
\end{equation}
with $g(0)=g(1)=g`(0)=0$ and $g(u)>0$ for $u\in (0,1)$.
In this talk we shall talk about our recent work on the spatial decay and stability of multidimensional cylinder fronts $\phi_c(x-ct,y)$ for the degenerate Fisher type equation. By applying the generalized center manifold theorem and asymptotic expansions we obtain the spatial decay of traveling fronts with all speeds, especially for the $p$-degree Fisher type equation we get the precise algebraic decaying rates and the higher order expansion of traveling fronts with non-critical speeds. By applying spectral analysis and sub-super solution method we prove the exponential stability of all traveling fronts in some exponentially weighted spaces and the Lyapunov stability of traveling fronts with noncritical speeds in some polynomially weighted spaces. We also investigate the asymptotic behavior and spreading speed of the solution with more general initial data, which are proved to be solely determined by the spatial decay of the initial data at one end. |
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