Invariant manifolds of the Hoff equation

The Hoff equation $(\lambda+\Delta)u_t=-\alpha u-\beta u^3$ models the buckling of a T-shaped beam, where $\lambda$, $\alpha$, and $\beta\in\mathbb R_+$ are the parameters of the model. The existence of a finite-dimensional local invariant manifold is established in the neighborhood of zero.