Asymptotics of periodic solutions of autonomous parabolic equations with small diffusion
Assume that the system \(\dot v=F(v,x)\) admits an exponentially orbitally stable periodic solution \(v_ 0(t,x)\) which depends smoothly on the parameter \(x\in [0,1]\). It is shown that close to \(v_ 0\) there exists an exponentially orbitally stable periodic solution of the system \[ u_ t=\mu u_{xx}+F(u,x),\quad 0<x<1, \] with Neumann or Dirichlet boundary conditions if \(\mu\) is small enough. The asymptotics (as \(\mu\to 0)\) of such solutions are derived.
scientific article; zbMATH DE number 4017473
scientific article