UNBOUNDED STURM GLOBAL ATTRACTORS FOR SEMILINEAR PARABOLIC EQUATIONS ON THE CIRCLE

We consider one-dimensional semilinear parabolic equations under periodic boundary conditions. A positive linear growth for the nonlinearity is assumed, yielding a nondissipative dynamic on the semiflow. We account for the growth of unbounded solutions and obtain the existence of some limiting objects at infinity, regarded as equilibria and frozen waves at infinity. Despite the nondissipativity, the solutions on the associated noncompact global attractor remain bounded backwards in time. This allows for a more accurate description of the heteroclinic connectivity on the attractor. We conclude with an important remark on the analog result for the Neumann problem.