Convergence of semilinear parabolic flows with general initial data

We analyze the long-time behavior of solutions to semilinear parabolic equations in Euclidean space that arise as gradient flows of an energy functional. We prove that, for general initial data (including data without compact support) the flow converges to a unique ground state. The argument relies on a sharp stability estimate for almost critical points of the energy, providing a flexible framework for establishing convergence of gradient flows associated with constrained minimization problems in R^n. As an application, we strengthen previous convergence results of Cortazar (1999) and Feireisl (1997).