Heteroclinic Travelling Waves of Gradient Diffusion Systems

We establish existence of travelling waves to the gradient system $u_t = u_{zz} - \nabla W(u)$ connecting two minima of $W$ when $u : \R \times (0,\infty) \larrow \R^N$, that is, we establish existence of a pair $(U,c) \in [C^2(\R)]^N \by (0,\infty)$, satisfying $$\{{array}{l} U_{xx} - \nabla W (U) = - c U_x U(\pm \infty) = a^{\pm}, {array}.$$ where $a^{\pm}$ are local minima of the potential $W \in C_{\textrm{loc}}^2(\R^N)$ with $W(a^-)< W(a^+)=0$ and $N \geq 1$. Our method is variational and based on the minimization of the functional $E_c (U) = \int_{\R}\Big\{{1/2}|U_x|^2 + W(U) \Big\}e^{cx} dx$ in the appropriate space setup. Following Alikakos-Fusco \cite{A-F}, we introduce an artificial constraint to restore compactness and force the desired asymptotic behavior, which we later remove. We provide variational characterizations of the travelling wave and the speed. In particular, we show that $E_c(U)=0$.