The long-time behaviour of solutions to parabolic problems on unbounded intervals: the influence of boundary conditions

For a non-negative function (u) over bar(x), we study the long-time behavio ur of solutions of the heat equation [GRAPHICS] with the Dirichlet or Neumann boundary conditions at I = 0. We find a criti cal parameter lambda(D) > 0 such that the solution subjected to the Dirichl et boundary condition tends to a spatially localized wave travelling to inf inity in the space variable. On the other hand, there exists a lambda(N) > 0 such that the corresponding solution of the Neumann problem converges to a non-trivial strictly positive stationary solution. Consequently, the dyna mics is considerably influenced by the choice of boundary conditions.