Fast and heteroclinic solutions for a second order ODE related to Fisher-Kolmogorov's equation
It is well known that travelling wave solutions with speed \(c\) of the Fisher-Kolmogorov equation \(u_t= u_{xx}+ f(u)\), where \(f: [0,1]\to \mathbb{R}^+\) is Lipschitzian and satisfies \(f(0)= 0= f(1)\) and \(f(u)> 0\) for \(0< u< 1\), correspond to positive heteroclinic solutions of the second-order equation \(u''+ cu'+ f(u)= 0\). It is well known that such solutions exist for \(c\geq c^*> 0\). The authors consider the more general equation \[ u''+ cu'+\lambda f(u)= 0\tag{\(*\)} \] and call a heteroclinic solution \(u\) of \((*)\) for some \(\lambda> 0\) as fast heteroclinic solution if it belongs to the space \(H^1_{\text{loc}}(\mathbb{R}): e^{ct} u\in H^1(\mathbb{R})\) and satisfies \(u(-\infty)= 1\). They give a variational characterization of fast heteroclinic orbits as well as of \(c^*\).
scientific article; zbMATH DE number 2123515