Verification of asymptotic solutions for one-dimensional nonlinear parabolic equations (Q1316893)

The one-dimensional semilinear diffusion equation of the form \[ u_ t = u_{xx} + f(u,\varepsilon x), \quad x \in \mathbb{R},\quad u(x,0) = \varphi(x) \] is studied where \(\varepsilon\) is a small parameter. The author proves local asymptotic stability of a kink solution, i.e. a monotone stationary solution of the problem with different limits for \(x \to \pm \infty\).