Attractor Continuity for Semilinear Parabolic Equations on Thin Domains with Degenerating Outward Peaks

In this work, we analyze the asymptotic behavior of the attractors associated with a semilinear parabolic equation subject to homogeneous Neumann boundary conditions and defined on a thin domain $R^{\epsilon }\subset {\mathbb{R}}^{1+n}$. We assume that the thin domain exhibits a cusp, known as an outward peak, whose geometry is characterized by a nonnegative function that vanishes at a point on the boundary. Our objective is to rigorously establish the continuity of the attractors as $\epsilon \to 0$ and to determine their rate of convergence.