Travelling fronts for monostable reaction-diffusion systems with gradient-dependence
Elaine Crooks · Advances in Differential Equations · 2003
We study the existence and stability of travelling-front solutions for parabolic systems of the form \begin{equation} u_{t} = A u_{xx} + f(u, u_{x}),~~x \in \mathbb R,~ t>0,~ u(x,t) \in \mathbb R^N, \tag*{(0.1)} \end{equation} where $A$ is a positive-definite diagonal matrix. The nonlinearity $f$ is a ``monostable'' function with equilibria $E^- {c_{0}}$ in the framework of exponentially weighted spaces, using Sattinger's approach [36] that was also exploited by the Volperts in [41]. The stability of the travelling front $w_{c}$ depends on whether $w_{c}'$ belongs to a $c-$dependent space, $X_{c}$ say. If $w_{c}' \in X_{c}$, the solution $u^{{E^{+}}hi}$ of (0.1) with initial data $\phi \in {BUC}^{1}$ converges in $X_{c}$ to a {\em translate} of $w_{c}$ if $\phi(x) - w_{c}(x)$ is small when $|x|$ is large and $\phi$ behaves like $w_{c}$ in a neighbourhood of the ``unstable'' equilibrium ${E^{+}}$. If $w_{c}' ot\in X_{c}$, the theorem is local --- $u^{\phi}$ converges in $X_{c}$ to $w_{c}$ if $\phi$ is close to $w_{c}$ in $X_{c}$. We also show that, as for scalar equations, at most one front solution of the parabolic system (0.1) can have ``fast-decay'', and if such a front exists, its velocity must be $c=c^{*}$.