Dynamic behaviors of a receptor-based model

In this paper, we consider a diffusion equation coupled to an ordinary differential equation, modeling the interaction between receptors and ligands. We first prove the existence of global-in-time solutions for all bounded and non-negative initial conditions. Then large-time behavior of solutions is studied. Our result shows that each positive solution converges to the constant solutions as time [Formula: see text] tends to infinity. Finally, we construct both traveling back and traveling front solutions by applying the blow up method.