Convergence to the travelling wave solution for a nonlinear reaction-diffusion equation (Q2504956)

The authors study the problem \[ u_t=(u^m)_{xx} +u(1-u^{m-1}), \;x \in \mathbb R, \;t > 0, \;\;u(0,x)=u_0(x) \geq 0, \;x \in \mathbb R \] with \(m > 1\). If the initial datum \(u_0(x)\) decays fast enough at infinity, it is proved that the solution of the Cauchy problem approaches the travelling wave solution spreading either to the right or to the left, or two travelling waves moving in opposite directions. The proofs of some auxiliary lemmas are not included in the paper.