We study the emergent dynamics of the spatially extended continuum Winfree model on the whole domain and derive the uniform-in-time continuum limit from the infinite lattice model. In previous literature, the asymptotic convergence of the Winfree model has been studied in \(\ell ^1\) -type topology such as the order parameter. Since we are dealing with the whole domain, where each point of the space represents an individual with one oscillator, it is more natural to employ \(L^\infty \) topology to analyze diverse emergent patterns such as an oscillator death, a phase-locking state and a quasi-steady state. Our key analysis lies in the uniform-in-time stability with respect to the initial data. From this, under a general network structure, a sufficiently large coupling strength leads to the exponential convergence of Winfree oscillators to an equilibrium. Moreover, uniform-in-time continuum limit from the infinite lattice model can be proved using the contraction property of extremal phases and stability estimates with respect to initial data and system parameters.
