Travelling fronts for monostable reaction-diffusion systems with gradient-dependence

Abstract

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 $E^- < E^+ , ~E^-$ ``stable'' and $E^+$ ``unstable'', that depends not only on $u$ but also on $u_{x}$ and satisfies both conditions sufficient for a comparison principle to hold for (0.1) (so is ``weakly-coupled''), and also either a subquadratic growth condition in $u_x$ or alternatively a mild monotonicity property in $u_x$. It is shown that there exist ${c^{*}}$ and $c_{0}$ with $c^{*} \geq c_{0}$ such that there is a ``stable-monotone direction'' at ${E^{+}}$ for all $c \geq c_{0}$, and solutions $w_{c} (x-ct)$ of (0.1) that connect $E^-$ to ${E^{+}}$ exist for all $c \geq c^{*}$ (${c^{*}} \neq c_{0}$ in general). We then discuss the stability of fronts of speed $c > {c_{0}}$ 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 $w_{c}' \not\in X_{c}$, 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 $c=c^{*}$.