Existence of spiky stationary solutions to a mass-conserved reaction-diffusion model

Cells realize various life activities such as migrations and the ingestion of extracellular substances by deforming their cytoplasm and cell membrane. Many reaction-diffusion systems with mass conservation have been used for describing these cell activities. Among them, a mass-conserved three-component reaction-diffusion system was proposed to describe the dynamics of wavelike actin polymerization in macropinocytosis (Yochelis et al. in Phys. Rev. E 101:022213, 2020). This system numerically exhibits dynamical patterns such as annihilation, crossover, and nucleation of pulses in a relatively large interval. In this article, to investigate the dynamics of wavelike actin polymerization in macropinocytosis, we first establish the condition for the diffusion driven instability in the system. We then rigorously prove the existence of spiky stationary solutions to the system in a bounded interval with the Neumann condition. These solutions play a crucial role in the nucleation of pulses although it is only numerically demonstrated. By reducing the stationary problem to a scalar second order nonlinear equation with a nonlocal term, we construct the desired solution by converting the equation into an integral equation.