Nonlocal Kirchhoff-Type Parabolic Equations in L²: Reduction to Finite Dimension

This paper investigates the finite-dimensional reduction of the asymptotic dynamics of one-dimensional nonlocal Kirchhoff-type parabolic equations, where the diffusion coefficient depends on the L² L 2 -norm of the gradient of the solution. Specifically, we establish the existence (without uniqueness) of an ε ε -regular mild solution and a global attractor for initial data in L² L 2 . Importantly, by employing a time transformation (one for each solution) that converts the nonlocal term into a nonlinear term, we construct an inertial manifold for initial data in H₀¹ H 0 1 , which implies that the global attractor is the graph of a Lipschitz function over a finite-dimensional subspace of the phase space, thereby revealing that the long-time behavior of this infinite-dimensional system can be completely described by a finite-dimensional system of ordinary differential equations.