Global dynamics of a discontinuous Galerkin approximation to a class of reaction-diffusion equation (Q1904026)

The subject of this paper is the nonlinear equation (1) \(\dot U- \gamma\Delta U= - f(U)\) (called the Allen-Cahn equation), \(\gamma\) being a positive constant. The unknown function \(U(x, t)\) is considered for \(t> 0\) and \(x\) belonging to a bounded convex domain \(\Omega\subset \mathbb{R}^d\) \((d= 1,2,3)\) with smooth boundary \(\Gamma\). Assuming the initial condition (2) \(U(.,0)= U_0\) on \(\Omega\) and the boundary condition (3) \(U= 0\) on \(\Gamma\), the authors construct a fully discretized (with respect to both \(x\) and \(t\)) Galerkin scheme to solve approximately problem (1), (2), (3). This scheme is proved to be uniquely solvable. Further investigations are concerning the global dynamic structure of this scheme, namely its absorbing sets, global attractors and the related discrete semigroup operator.