The unstable invariant set in the neighbourhood of a non-hyperbolic fixed point of a nonlinear operator is studied. Lower semicontinuity of the attractor for a gradient system having non-hyperbolic stationary points is proved. The result is applied to the semigroup generated by the viscous Cahn-Hilliard equation.

The unstable invariant set near a non‐hyperbolic stationary point of an abstract parabolic equation is studied. Lower semicontinuity of the attractor for the Chafee‐Infante problem in the case of the non‐hyperbolic zero stationary solution is proved.

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