A Note on the Domain of Attraction for the Stationary Solution to a Singularly Perturbed Parabolic Equation

We consider  a boundary-value problem  for a singularly  perturbed parabolic  equation with an initial function  independent of a perturbation parameter in the case where a degenerate stationary equation  has smooth possibly intersecting roots.  Before, the existence of a stable stationary solution to  this  problem  was proved  and  the  domain  of attraction of this  solution  was investigated — due  to exchange  of stabilities, the  stationary solution  approaches  the  non-smooth  (but  continuous) composite root  of the  degenerate equation  as the  perturbation parameter gets smaller,  and  its domain  of attraction contains  all initial  functions  situated strictly  on one side of the other  non-smooth  (but  continuous) composite  root of the degenerate equation.   We show that if the  initial  function  is out of the boundary of this family of initial  functions  near some point,  the problem cannot  have a solution inside the domain of the problem,  i.e. this boundary is the true  boundary of the attraction domain.  The proof uses ideas of the nonlinear  capacity  method.

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