Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure
The author considers the nonlinear parabolic system of the form \[ u_t=-\nabla V\left( u\right) +u_{xx} \] where \(u=\left( u_1,\dots,u_n\right) \in \mathbb{R}^n,n\geq 1,x\in \mathbb{R},\) the potential \(V:\mathbb{R}^n\rightarrow \mathbb{R}\) is of class \(C^k\), \(k\geq 3,V\left( u\right) \rightarrow +\infty ,\) when \(\left\| u\right\| \rightarrow +\infty .\) The author proves a global convergence toward bistable fronts which states the invasion of a stable homogeneous equilibrium (a local minimum of the potential) necessarily occurs via traveling front connecting to another equilibrium. The proof is based only on energy methods and it doesn't use any comparison principle.
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