Global wellposedness of nutrient-taxis systems derived by a food metric

This paper deals with the nutrient-taxis system derived by a food metric. The system was proposed in [Sun-Ho Choi and Yong-Jung Kim: Chemotactic traveling waves by metric of food, SIAM J. Appl. Math. 75 (2015), 2268-2289] using geometric ideas without gradient sensing, and has a simple form but contains a singular diffusive coefficient on the equation for the organism side. To overcome the difficulty arising from this singular structure, we use a weighted Lp-estimate involving a weighted Gagliardo-Nirenberg type inequality. In the one dimensional setting, it turns out that the system is shown to be globally well-posed in certain Sobolev spaces and the solutions are uniformly bounded. Moreover, the zero viscosity limit of the equation for the nutrient side is considered. For the same initial data and any given finite time interval, a diffusive solution converges to a non-diffusive solution when the diffusion coefficient vanishes.