Penerapan Teorema Densitas Ruang Sobolev pada Konstruksi Skema Galerkin dan Eksistensi Lokal Solusi Aproksimasi untuk Sistem Reaksi-Difusi Schnakenberg

Abstract

This study examines the density of smooth functions  in the Sobolev  and demonstrates its application to the analysis of a reaction–diffusion system. The density theorem is established using mollification and partition-of-unity techniques on a bounded open domain  . As an application, the density property is used to construct the weak formulation of the Schnakenberg reaction–diffusion system with homogeneous Neumann boundary conditions, in particular to extend the validity of test functions from  to . Based on this formulation, a Faedo–Galerkin approximation scheme is constructed. The solution is approximated using a basis of eigenfunctions of the Neumann–Laplace operator, which reduces the partial differential system to a finite-dimensional system of ordinary differential equations for the approximation coefficients. The right-hand side of this system is shown to be polynomial, so that the local Lipschitz property is satisfied. Local existence and uniqueness of the Galerkin system solution are established through the Picard–Lindelöf Theorem. These results show that the density theorem serves as a structural component in building a finite-dimensional approximation framework for a nonlinear reaction–diffusion system, and provide a foundation for the study of global existence and solution boundedness in future work.